.

Устойчивость движения в нелинейных системах

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19

Предмет:

“Теория автоматического управления”

Тема:

“Устойчивость движения в нелинейных системах”

1. Общие понятия и определения

При анализе устойчивости движений в нелинейных системах исследуют
устойчивость в особых точках, характеризующих равновесные состояния и на
предельных циклах, характеризующих автоколебания. Если в линейных
системах работоспособными оказываются только устойчивые системы, то в
нелинейных системах наличие автоколебаний является нормальным режимом ее
функционирования.

Рассмотрим некоторые понятия об устойчивости движения в нелинейных
системах. Существенный вклад в развитие теории нелинейных систем внес
А.М. Ляпунов.

Устойчивость в малом – устойчивость при малых отклонениях. Для
определения устойчивости в малом используют первый метод Ляпунова –
метод линеаризации.

Устойчивость в целом – устойчивость, которая не зависит от величины
начальных условий. Для определения устойчивости в целом используют
второй (или прямой) метод Ляпунова.

2. Исследование устойчивости движения в окрестности особых точек

Ляпунов сформулировал теоремы об устойчивости линеаризованных систем.
Движение в окрестности особой точки может быть асимптотически устойчивым
(рис. 1а) или устойчивым в смысле Ляпунова (рис. 1б).

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а) б)

Рис. 1

Пусть имеется особая точка в начале координат. Устойчивость определяется
в окрестности этой точки.

Если может быть найдена такая окрестность -, чтобы движение, начавшись в
пределах окрестности, заканчивалось в точке, характеризующей состояние
равновесия, то такое движение называется асимптотически устойчивым.

Если внутри окрестности точки может быть найдена такая область, чтобы
движение, начавшись вблизи окрестности, заканчивалось в пределах области
точки, то такое движение называется устойчивым по Ляпунову.

Р????м ????ую ???у ???о ?ря?а, ???я ?????я ???? ????й:

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01000000ec0008000000320a96077603010000003d0008000000320ad6026e0301000000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 (1)

Условие особых точек:
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 (2)

Каждое из этих уравнений может быть представлено в виде линии на
плоскости x0y. Если система линейная (рис. 2а), то оба уравнения линейны
и линии пересекаются в одной (особой) точке, которая расположена, как
правило, в начале координат. Для нелинейных систем (рис. 2б) каждое
уравнение это уравнение кривой. Они могут пересекаться в нескольких
точках, т.е. особых точек может быть сколь угодно.
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fffffffffffffffffc010101ffffffffffffffffffffffffffffffffffffffffffffffff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Рис. 2

Для определения устойчивости в окрестности какой-либо точки можно
воспользоваться методом линеаризации.
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 (3)

Переходим к уравнениям в изображениях
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 (4)

где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- постоянные
коэффициенты, при этом x = x0, а y = y0.

Система линеаризованных уравнений
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 (5)

Исключив одну переменную, можно получить уравнение второго порядка
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 (6)

Для определения устойчивости анализируем корни характеристического
уравнения
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 (7)

Если корни характеристического уравнения расположены в левой
полуплоскости, то линеаризованная система устойчива, а соответствующая
ей исходная нелинейная система асимптотически устойчива в окрестности,
рассматриваемой особой точки.

Если корни расположены в правой плоскости, то линеаризованная система
неустойчива, а движение в окрестности особой точки является
неустойчивым.

Если корни расположены на мнимой оси, то линеаризованная система не
устойчива, а для определения устойчивости нелинейной системы необходимо
провести дополнительные исследования нелинейной системы, т.е. уравнения
в первом приближении не дают точного представления об устойчивости
нелинейной системы.

3. Второй (прямой) метод Ляпунова

Если хотя бы один из корней характеристического уравнения расположен на
мнимой оси, то первый метод Ляпунова не дает ответа на вопрос об
устойчивости движения в окрестности особой точки, при этом используется
второй (прямой) метод Ляпунова, позволяющий определить устойчивость в
большом.

Метод основан на использовании специальных функций, называемых функциями
Ляпунова. Чтобы выяснить смысл функций Ляпунова рассмотрим фазовый
портрет в окрестности некоторой особой точки (рис. 3). Рассмотрим радиус
– вектор -, который изменяется по модулю в функции времени.

picscalex100010009000003120f000000009009000000009009000026060f001613574d
46430100000000000100a6360000000001000000f412000000000000f412000001000000
6c00000000000000000000009e000000830000000000000000000000d71800009f140000
20454d4600000100f41200000c0000000100000000000000000000000000000020030000
5802000040010000f000000000000000000000000000000000e2040080a9030046000000
2c00000020000000454d462b014001001c000000100000000210c0db0100000060000000
6000000046000000c0060000b4060000454d462b224004000c000000000000001e400900
0c00000000000000244001000c000000000000003040020010000000040000000000803f
214007000c00000000000000084000050c060000000600000210c0db0100000000000000
0000000000000000000000000100000089504e470d0a1a0a0000000d494844520000009f
00000084080200000001686eef000000017352474200aece1ce900000009704859730000
0ec300000ec301c76fa8640000058949444154785eed9cdd8ee4280c85b7f6fddfb92733
486e442830e07f5c17abd22c31b63f1f4352a43f3f3f3fffe52768063e51e97e3e9f1a59
0913fe316ad44d95fe1fa96a6ba2353ff85ebe5c82f6893414dd48954a124b64baa051d0
f4f3e51ee1a676491462d74864eddacdba9467a13ad5b7c67bdb56198a27b52ba5238d79
aea07bd54eaaaea22be896802f647c115d8dd6a83ce715746fbbcd8dbfab7a88c2475941
7ad347bb23ba737dfd563fd13af3855ba7416b884657af0b5a9c39e95aa442e553d2a5ca
a4453b49d722152a9f922e55262dda49ba16a950f99474a93269d14ed2b54885caa7a44b
95498b7692ae452a543e255daa4c5ab493742d52a1f229e95265d2a29da46b910a954f49
972a9316ed245d8b54a87c4aba5499ecdb695e34e59dec653de90a275c743adfe7aadeca
b076f246f738a627ba5396ddf7ed45c5d29b4c11b003baf877bc9a3ce22f1c54403172d2
12926e9bde6f7f23612cc4f13b80ab846aae27844eae3d6c3ce67655e588f943023e8711
16e53d9f8dedeb6a419cbb4a6bc110dd9a2b6d90c55a01bcc198c319199b26d6ddf3b5ad
240bd9033786612e19d48d560f50d62eb75ebb12d9ebd27b6a834541a56768d285f5752f
71025795e2c308b7bbafd6922c64468d2e32654c0891f245eeeca0086a6fd5d1fe5daa54
9c6042bb64763018bae83839dd61dd3dc4e1c6a2e9ea4d690e9c54d32e932849cc22254b
3217ab917be9229bf33bfbf5625ceaa01943de0e6b83e53bfc773c9702dda5fec95ada4b
c6a750e1ae7ac92c7230502c5d1a9943e97517e91632e666d88671e425c86163b7a98c74
f7e7dda945b54b12de1ef8a5abde7b60f266bbe44f7775c05810a58b7148710cb2f72a7a
b8dafc6fa70b0f9fe1d18a3599423135cf559adb24fdceac5ef5b50375fbed6e7dad795b
3b594a700af83aed7a69bf4d21169680f3fda55b8b727b66812dd5dee32701c74aeac526
fa6de662cb8c406ccd1475e39a3e5364cd43f184758a5bd6dd77ef954fab9135fbb7334f
9768231ea71b4d0606b51b67dd3de97edcabc6896f27d51c64cfcc8de724c58ad746a09b
68bf15907bba8976d01b7cd37581169e71cab768af749b87ae8789dbfe25ff705eeecbbd
d2e5ce4b0cfb2ee92af63a5fd4fdd175b1d61a29027f74391217b5629cd18d8a81a3641f
9bcee8326521aad9a4cb4b56b7d9245d851fd5790baab2ee89aeae0ec490104ee4892e61
d8bf6753fefd15070ecb166c7aa24bfebc90bb1970db9f169027bad3609606a8a77ec9db
bdc1cee852c95700adc01453e4cee84ee3c10cb090778c9fe763e4e852c9ae7e3764237e
19b432b34cc397a33b75053f00de8d87c3ad986b697f12c6cca83e46f47e80afa2e1b86e
f316739d5fb13b1fbe3057cb4594eee31c6be4f5916c3196ef8cb3c6b804d865671e4408
4d7b290b8483eda07d828a469790530053499712a229e12a6897eabe8892495c5ba95d32
b6d684aba0dd67ca942f5941cd0ca5766719c2fd7f83c2d5d12e2e5d398a20033ada0dd6
9c6d0a37b54ba00fcb2674b41b696f6556b8cada0dd09f2da355a6eb5ac12e7e4f54ebcc
b05c7954b0979710f5e95ade95747d33de8d6b9f4dd075245f4768f5d75d47fdd9c542db
341b13da2d3e5956b09785d62e5d9b803d4a16181bd2ae41053b952cd0953e3587dc21bf
cf38622e24dcf21407148fde61e29d8e31a7dda9c739009f01a3dafded2d2b1aa2d22e95
1d3c06a691d6e996b0917d728f4a7d0a1ad67ea6740b9bf54117c9f84df74dee9d5fef8b
eba0623cd1ad19234510981c2603fee88e4a35f49f49c0e034fd34632380bc649081bc23
8a5c1e4937e946ce40e4d852bb4937720622c796da4dba91331039b6d46ed28d9c81c8b1
a576936ee40c448e2db59b74236720726ca9dda41b390391634bed26ddc819881c5b6a37
e946ce40e4d8429d9a8b0c6a2bb6eccc5b69737251d275026acbcd3fc3dee5a989888b44
0000000049454e44ae4260820840010824000000180000000210c0db0100000003000000
0000000000000000000000001b40000040000000340000000100000002000000000000bf
000000bf00001f430000044303000000000000b3000000b3ffff1e43000000b3000000b3
ffff03432100000008000000620000000c00000001000000150000000c00000004000000
150000000c0000000400000051000000d00a000000000000000000009e00000083000000
000000000000000000000000000000009f00000084000000500000003000000080000000
500a0000000000002000cc009f00000084000000280000009f0000008400000001000100
00000000000000000000000000000000020000000000000000000000ffffff00ffffffff
fffffffffffffffffffffffffffffffefffffffffffffffffffffffffffffffffffffffe
fffffffffffffffffffffffffffffffffffffffefffffffffffffffff7ffffffffffffff
fffffffefffffffffffffffff7fffffffffffffffffffffefffffffffffffffff7ffffff
fffffffffffffffefffffffffffffffff7fffffffffffffffffffffeffffffffffffffff
f7fffffffffffffffffffffefffffffffffffffff7fffffffffffffffffffffeffffffff
fffffffff7fffffffffffffffffffffefffffffffffffffff7fffffffffffffffffffffe
fffffffffffffffff7fffffffffffffffffffffefffffffffffffffff7ffffffffffffff
fffffffefffffffffffffffff7fffffffffffffffffffffefffffffffffffffff7ffffff
fffffffffffffffefffffffffffffffff7fffffffffffffffffffffeffffffffffffffff
f7fffffffffffffffffffffefffffffffffffffff7fffffffffffffffffffffeffffffff
fffffffff7fffffffffffffffffffffefffffffffffffffff7fffffffffffffffffffffe
fffffffffffffffff7fffffffffffffffffffffefffffffffffffffff7ffffffffffffff
fffffffefffffffffffffffff7fffffffffffffffffffffefffffffffffffffff7ffffff
fffffffffffffffeffffffffffffffe0003ffffffffffffffffffffeffffffffffffff9f
f7c003fffffffffffffffffefffffffffffffe7ff7fffc0ffffffffffffffffeffffffff
fffff9fff7fffff7fffffffffffffffeffffffffffffc7fff7fffff9fffffffffffffffe
ffffffffffffc7fff7fffff9fffffffffffffffeffffffffffffbffff7fffffeffffffff
fffffffeffffffffffff7ffff7ffffff3ffffffffffffffefffffffffffefffff7ffffff
dffffffffffffffefffffffffffdfffff7ffffffe7fffffffffffffefffffffffffbffff
f7fffffffbfffffffffffffefffffffffff7fffff7fffffffdfffffffffffffeffffffff
fff7fffff7fffffffefffffffffffffeffffffffffeffffff7ffffffff3ffffffffffffe
ffffffffffdffffff7ffffffffdffffffffffffeffffffffffbffffff7ffffffffefffff
fffffffeffffffffff7ffffff7fffffffff7fffffffffffeffffffffff7ffffff7ffffff
fff7fffffffffffefffffffffefffffff7fffffffffbfffffffffffefffffffffeffffff
f7fffffffffbfffffffffffefffffffffefffffff7fffffffffdfffffffffffeffffffff
fdfffffff7fffffffffdfffffffffffefffffffffdfffffff7fffffffffefffffffffffe
fffffffffbfffffff7fffffffffefffffffffffefffffffffbfffffff7ffffffffff7fff
fffffffefffffffffbfffffff7ffffffffff7ffffffffffefffffffffbfffffff0ffffff
ffffbffffffffffefffffffff7ffffffc703ffffffffbffffffffffefffffffff7fffffe
37fcffffffffdffffffffffefffffffff7fffff1f7ff7fffffffdffffffffffeffffffff
efffffeff7ffbfffffffeffffffffffeffffffffefffffeff7ffdfffffffeffffffffffe
ffffffffefffffdff7ffeffffffff7fffffffffeffffffffefffffdff7fff7fffffff7ff
fffffffeffffffffefffffbff7fffbfffffffbfffffffffeffffffffefffffbff7fffdff
fffffbfffffffffeffffffffefffffdff7fffdfffffffbfffffffffeffffffffefffffdf
f7fffefffffffbfffffffffeffffffffefffffdff7fffefffffffdfffffffffeffffffff
efffffdff7fffefffffffdfffffffffeffffffffefffffeff7fffefffffffdfffffffffe
ffffffffefffffeff7ffff7ffffffdfffffffffeffffffffeffffff7f7ffff7ffffffdff
fffffffeffffffffeffffff9f7ffff7ffffffdfffffffffeffffffffeffffff9f7ffff7f
fffffdfffffffffeffffffffeffffffef7fffefffffffdfffffffffeffffffffefffffff
77fffefffffffe7ffffffffeffffffffefffffffb7fffefffffffe7ffffffffeffffffff
efffffffd7fffefffffffe7ffffffffeffffffffefffffffe7fffefffffffe7ffffffffe
e0000000000000000000000000000000000003feffffffffeffffffff47ffefffffffe7f
fffffffeffffffffeffffffff787fefffffffe7ffffffffeffffffffeffffffff7f8fdff
ffffffbffffffffefffffffff7fffffff7ff1dffffffffbffff0c3fefffffffff7ffffff
f7ffe1ffffffffbffffbe7fefffffffff7fffffff7fffc7fffffffbffffccffeffffffff
f7fffffff7fffb8fffffffbfffff5ffefffffffff7fffffff7fffbf1ffffffbfffff1ffe
fffffffffbfffffff7fffbffdfffffbfffff3ffefffffffffdfffffff7fff7ffe3fffe7f
fffc5ffefffffffffefffffff7fff7fffc7f7e7ffff8effefffffffffefffffff7ffefff
ff8f3e7ffff043fefffffffffefffffff7ffeff1ff8f3e7ffff043feffffffffff7fffff
f7ffeffdfff01e7ffffffffeffffffffffbffffff7ffdffc3ffe0e7ffffffffeffffffff
ffcffffff7ffdffddffc067ffffffffefffffffffff3fffff7ffbffddffc01fffffffffe
fffffffffffcfffff7fe7ffddffffdfffffffffeffffffffffff3ffff7fdfffddffffdff
fffffffeffffffffffffc3fff7f3fffcdffffdfffffffffefffffffffffffc7ff78fffff
3ffffdfffffffffeffffffffffffff81f07ffffffffffdfffffffffefffffffffffffffe
07fffffffffffbfffffffffefffffffffffffffff7fffffffffffbfffffffffeffffffff
fffffffff7fffffffffffbfffffffffefffffffffffffffff7fffffffffffffffffffffe
fffffffffffffffff7fffffffffffffffffffffefffffffffffffffff7ffffffffffffff
fffffffefffffffffffffffff7fffffffffffffffffffffefffffffffffffffff7ffffff
fffffffffffffffefffffffffffffffff7fffffffffffffffffffffeffffffffffffffff
f7fffffffffffffffffffffefffffffffffffffff7fffffffffffffffffffffeffffffff
fffffffff7fffffffffffffffffffffefffffffffffffffff7fffffffffffffffffffffe
fffffffffffffffff7fffffffffffffffffffffefffffffffffffffff7ffffffffffffff
fffffffefffffffffffffffff7fffffffffffffffffffffefffffffffffffffff7ffffff
fffffffffffffffefffffffffffffffff7fffffffffffffffffffffeffffffffffffffff
f7fffffffffffffffffffffefffffffffffffffff7fffffffffffffffffffffeffffffff
fffffffff3fffffffffffffffffffffefffffffffffffffff707fffffffffffffffffffe
fffffffffffffffff7f7fffffffffffffffffffefffffffffffffffff7fbffffffffffff
fffffffefffffffffffffffff7fbfffffffffffffffffffefffffffffffffffff7f9ffff
fffffffffffffffefffffffffffffffff7f1fffffffffffffffffffeffffffffffffffff
f7f1fffffffffffffffffffefffffffffffffffff7c6fffffffffffffffffffeffffffff
fffffffff7c6fffffffffffffffffffefffffffffffffffff7c6fffffffffffffffffffe
fffffffffffffffff78f7ffffffffffffffffffefffffffffffffffff78f7fffffffffff
fffffffeffffffffffffffffff063ffffffffffffffffffeffffffffffffffffffffffff
fffffffffffffffe4c0000006400000000000000000000009e0000008300000000000000
000000009f000000840000002900aa0000000000000000000000803f0000000000000000
0000803f0000000000000000000000000000000000000000000000000000000000000000
220000000c000000ffffffff460000001c00000010000000454d462b024000000c000000
000000000e00000014000000000000001000000014000000040000000301080005000000
0b0200000000050000000c0284009f00030000001e000400000007010400040000000701
04004d050000410b2000cc0084009f000000000084009f0000000000280000009f000000
840000000100010000000000000000000000000000000000020000000000000000000000
ffffff00fffffffffffffffffffffffffffffffffffffffeffffffffffffffffffffffff
fffffffffffffffefffffffffffffffffffffffffffffffffffffffeffffffffffffffff
f7fffffffffffffffffffffefffffffffffffffff7fffffffffffffffffffffeffffffff
fffffffff7fffffffffffffffffffffefffffffffffffffff7fffffffffffffffffffffe
fffffffffffffffff7fffffffffffffffffffffefffffffffffffffff7ffffffffffffff
fffffffefffffffffffffffff7fffffffffffffffffffffefffffffffffffffff7ffffff
fffffffffffffffefffffffffffffffff7fffffffffffffffffffffeffffffffffffffff
f7fffffffffffffffffffffefffffffffffffffff7fffffffffffffffffffffeffffffff
fffffffff7fffffffffffffffffffffefffffffffffffffff7fffffffffffffffffffffe
fffffffffffffffff7fffffffffffffffffffffefffffffffffffffff7ffffffffffffff
fffffffefffffffffffffffff7fffffffffffffffffffffefffffffffffffffff7ffffff
fffffffffffffffefffffffffffffffff7fffffffffffffffffffffeffffffffffffffff
f7fffffffffffffffffffffefffffffffffffffff7fffffffffffffffffffffeffffffff
fffffffff7fffffffffffffffffffffeffffffffffffffe0003ffffffffffffffffffffe
ffffffffffffff9ff7c003fffffffffffffffffefffffffffffffe7ff7fffc0fffffffff
fffffffefffffffffffff9fff7fffff7fffffffffffffffeffffffffffffc7fff7fffff9
fffffffffffffffeffffffffffffc7fff7fffff9fffffffffffffffeffffffffffffbfff
f7fffffefffffffffffffffeffffffffffff7ffff7ffffff3ffffffffffffffeffffffff
fffefffff7ffffffdffffffffffffffefffffffffffdfffff7ffffffe7fffffffffffffe
fffffffffffbfffff7fffffffbfffffffffffffefffffffffff7fffff7fffffffdffffff
fffffffefffffffffff7fffff7fffffffefffffffffffffeffffffffffeffffff7ffffff
ff3ffffffffffffeffffffffffdffffff7ffffffffdffffffffffffeffffffffffbfffff
f7ffffffffeffffffffffffeffffffffff7ffffff7fffffffff7fffffffffffeffffffff
ff7ffffff7fffffffff7fffffffffffefffffffffefffffff7fffffffffbfffffffffffe
fffffffffefffffff7fffffffffbfffffffffffefffffffffefffffff7fffffffffdffff
fffffffefffffffffdfffffff7fffffffffdfffffffffffefffffffffdfffffff7ffffff
fffefffffffffffefffffffffbfffffff7fffffffffefffffffffffefffffffffbffffff
f7ffffffffff7ffffffffffefffffffffbfffffff7ffffffffff7ffffffffffeffffffff
fbfffffff0ffffffffffbffffffffffefffffffff7ffffffc703ffffffffbffffffffffe
fffffffff7fffffe37fcffffffffdffffffffffefffffffff7fffff1f7ff7fffffffdfff
fffffffeffffffffefffffeff7ffbfffffffeffffffffffeffffffffefffffeff7ffdfff
ffffeffffffffffeffffffffefffffdff7ffeffffffff7fffffffffeffffffffefffffdf
f7fff7fffffff7fffffffffeffffffffefffffbff7fffbfffffffbfffffffffeffffffff
efffffbff7fffdfffffffbfffffffffeffffffffefffffdff7fffdfffffffbfffffffffe
ffffffffefffffdff7fffefffffffbfffffffffeffffffffefffffdff7fffefffffffdff
fffffffeffffffffefffffdff7fffefffffffdfffffffffeffffffffefffffeff7fffeff
fffffdfffffffffeffffffffefffffeff7ffff7ffffffdfffffffffeffffffffeffffff7
f7ffff7ffffffdfffffffffeffffffffeffffff9f7ffff7ffffffdfffffffffeffffffff
effffff9f7ffff7ffffffdfffffffffeffffffffeffffffef7fffefffffffdfffffffffe
ffffffffefffffff77fffefffffffe7ffffffffeffffffffefffffffb7fffefffffffe7f
fffffffeffffffffefffffffd7fffefffffffe7ffffffffeffffffffefffffffe7fffeff
fffffe7ffffffffee0000000000000000000000000000000000003feffffffffefffffff
f47ffefffffffe7ffffffffeffffffffeffffffff787fefffffffe7ffffffffeffffffff
effffffff7f8fdffffffffbffffffffefffffffff7fffffff7ff1dffffffffbffff0c3fe
fffffffff7fffffff7ffe1ffffffffbffffbe7fefffffffff7fffffff7fffc7fffffffbf
fffccffefffffffff7fffffff7fffb8fffffffbfffff5ffefffffffff7fffffff7fffbf1
ffffffbfffff1ffefffffffffbfffffff7fffbffdfffffbfffff3ffefffffffffdffffff
f7fff7ffe3fffe7ffffc5ffefffffffffefffffff7fff7fffc7f7e7ffff8effeffffffff
fefffffff7ffefffff8f3e7ffff043fefffffffffefffffff7ffeff1ff8f3e7ffff043fe
ffffffffff7ffffff7ffeffdfff01e7ffffffffeffffffffffbffffff7ffdffc3ffe0e7f
fffffffeffffffffffcffffff7ffdffddffc067ffffffffefffffffffff3fffff7ffbffd
dffc01fffffffffefffffffffffcfffff7fe7ffddffffdfffffffffeffffffffffff3fff
f7fdfffddffffdfffffffffeffffffffffffc3fff7f3fffcdffffdfffffffffeffffffff
fffffc7ff78fffff3ffffdfffffffffeffffffffffffff81f07ffffffffffdfffffffffe
fffffffffffffffe07fffffffffffbfffffffffefffffffffffffffff7fffffffffffbff
fffffffefffffffffffffffff7fffffffffffbfffffffffefffffffffffffffff7ffffff
fffffffffffffffefffffffffffffffff7fffffffffffffffffffffeffffffffffffffff
f7fffffffffffffffffffffefffffffffffffffff7fffffffffffffffffffffeffffffff
fffffffff7fffffffffffffffffffffefffffffffffffffff7fffffffffffffffffffffe
fffffffffffffffff7fffffffffffffffffffffefffffffffffffffff7ffffffffffffff
fffffffefffffffffffffffff7fffffffffffffffffffffefffffffffffffffff7ffffff
fffffffffffffffefffffffffffffffff7fffffffffffffffffffffeffffffffffffffff
f7fffffffffffffffffffffefffffffffffffffff7fffffffffffffffffffffeffffffff
fffffffff7fffffffffffffffffffffefffffffffffffffff7fffffffffffffffffffffe
fffffffffffffffff7fffffffffffffffffffffefffffffffffffffff7ffffffffffffff
fffffffefffffffffffffffff3fffffffffffffffffffffefffffffffffffffff707ffff
fffffffffffffffefffffffffffffffff7f7fffffffffffffffffffeffffffffffffffff
f7fbfffffffffffffffffffefffffffffffffffff7fbfffffffffffffffffffeffffffff
fffffffff7f9fffffffffffffffffffefffffffffffffffff7f1fffffffffffffffffffe
fffffffffffffffff7f1fffffffffffffffffffefffffffffffffffff7c6ffffffffffff
fffffffefffffffffffffffff7c6fffffffffffffffffffefffffffffffffffff7c6ffff
fffffffffffffffefffffffffffffffff78f7ffffffffffffffffffeffffffffffffffff
f78f7ffffffffffffffffffeffffffffffffffffff063ffffffffffffffffffeffffffff
fffffffffffffffffffffffffffffffe0c00000040092900aa0000000000000084009f00
00000000040000002701ffff030000000000

Рис. 3

Если при
picscalex100010009000003720100000200150000000000050000000902000000000400
000002010100050000000102ffffff00040000002e011800050000003102010000000500
00000b0200000000050000000c02400260191200000026060f001a00ffffffff00001000
0000c0ffffffd7ffffff20190000170200000b00000026060f000c004d61746854797065
0000600015000000fb0240fe0000000000009001000000000402001054696d6573204e65
7720526f6d616e000066040000002d01000008000000320a80013f180100000030000800
0000320a8001290301000000300010000000fb0240fe0000000000009001000000020002
001053796d626f6c0002040000002d01010004000000f001000008000000320a8001e616
010000003c0008000000320a80018f0f01000000de0008000000320a8001450c01000000
ae0008000000320a80014e0801000000a50008000000320a8001390601000000ae000800
0000320a8001d001010000003e0015000000fb0240fe0000000000009001010000000402
001054696d6573204e657720526f6d616e000066040000002d01000004000000f0010100
08000000320a8001101502000000647408000000320a80011014010000002f0008000000
320a8001b21101000000640008000000320a80014c0e01000000300008000000320a8001
dc0902000000202008000000320a8001a409010000002c0008000000320a800143050100
0000740008000000320a8001780402000000202008000000320a80010804010000003b00
10000000fb0240fe0000000000009001010000020002001053796d626f6c000204000000
2d01010004000000f001000008000000320a8001921201000000720008000000320a8001
ca0a01000000720008000000320a80014e000100000072000a00000026060f000a00ffff
ffff01000000000010000000fb021000070000000000bc02000000cc0102022253797374
656d00cc040000002d01000004000000f0010100030000000000, то движение
асимптотически устойчиво.

Если при
picscalex100010009000003150100000200150000000000050000000902000000000400
000002010100050000000102ffffff00040000002e011800050000003102010000000500
00000b0200000000050000000c024002a00c1200000026060f001a00ffffffff00001000
0000c0ffffffd7ffffff600c0000170200000b00000026060f000c004d61746854797065
0000600010000000fb0240fe0000000000009001000000020002001053796d626f6c0002
040000002d01000008000000320a8001770b010000002d0008000000320a8001d4070100
0000a50008000000320a8001bf0501000000ae0010000000fb0240fe0000000000009001
010000020002001053796d626f6c0002040000002d01010004000000f001000008000000
320a8001f50901000000720008000000320a80014e0001000000720015000000fb0240fe
0000000000009001010000000402001054696d6573204e657720526f6d616e0000810400
00002d01000004000000f001010008000000320a8001070902000000202008000000320a
8001c90401000000740008000000320a80016e0401000000200008000000320a8001fe03
010000003b0008000000320a8001110301000000300008000000320a8001a40101000000
3e0008000000320a800157010100000020000a00000026060f000a00ffffffff01000000
000010000000fb021000070000000000bc02000000cc0102022253797374656d00cc0400
00002d01010004000000f0010000030000000000увеличиваетсяpicscalex1000100090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, то движение не устойчиво.

Если при
picscalex1000100090000030e0100000200150000000000050000000902000000000400
000002010100050000000102ffffff00040000002e011800050000003102010000000500
00000b0200000000050000000c024002400e1200000026060f001a00ffffffff00001000
0000c0ffffffd7ffffff000e0000170200000b00000026060f000c004d61746854797065
0000600010000000fb0240fe0000000000009001000000020002001053796d626f6c0002
040000002d01000008000000320a80015a0c01000000ae0008000000320a8001be080100
0000a50008000000320a8001a90601000000ae0008000000320a8001d001010000003e00
10000000fb0240fe0000000000009001010000020002001053796d626f6c000204000000
2d01010004000000f001000008000000320a8001df0a01000000720008000000320a8001
4e0001000000720015000000fb0240fe0000000000009001010000000402001054696d65
73204e657720526f6d616e0000a2040000002d01000004000000f001010008000000320a
8001f10902000000202008000000320a8001b30501000000740009000000320a80017804
030000002020200008000000320a80010804010000003b0008000000320a80011b030100
000030000a00000026060f000a00ffffffff01000000000010000000fb02100007000000
0000bc02000000cc0102022253797374656d00cc040000002d01010004000000f0010000
030000000000constpicscalex100010009000003ed00000002001500000000000500000
00902000000000400000002010100050000000102ffffff00040000002e0118000500000
0310201000000050000000b0200000000050000000c024002000a1200000026060f001a0
0ffffffff000010000000c0ffffffd7ffffffc0090000170200000b00000026060f000c0
04d617468547970650000600015000000fb0240fe0000000000009001010000000402001
054696d6573204e657720526f6d616e000081040000002d01000008000000320a8001cd0
801000000300008000000320a8001ac0502000000647408000000320a8001ac040100000
02f0008000000320a80014e0201000000640010000000fb0240fe0000000000009001000
000020002001053796d626f6c0002040000002d01010004000000f001000008000000320
a8001820701000000a30008000000320a80012b0001000000de0010000000fb0240fe000
0000000009001010000020002001053796d626f6c0002040000002d01000004000000f00
1010008000000320a80012e030100000072000a00000026060f000a00ffffffff0100000
0000010000000fb021000070000000000bc02000000cc0102022253797374656d00cc040
000002d01010004000000f0010000030000000000, то движение устойчиво в
смысле Ляпунова.

Ляпунов доказал, что надо найти такую произвольную функцию H (x, y),
которая бы играла роль радиус-вектора , и была бы положительной для всех
точек за исключением, быть может, начала координат, где она может быть
равной нулю. Такая функция называется функцией Ляпунова.

Знакоопределенной функцией называется функция, которая при всех
значениях аргументов за исключением, может быть, начала координат, где
она равна нулю, имеет определенный знак (рис. 4а).

Знакопостоянной функцией называется функция, которая при всех значениях
аргументов (за исключением нескольких точек, где она равна нулю)
сохраняет постоянный знак (рис. 4б).
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d3010000a20000000000000000000000d4010000a30000002900aa000000000000000000
0000803f00000000000000000000803f0000000000000000000000000000000000000000
000000000000000000000000220000000c000000ffffffff460000001c00000010000000
454d462b024000000c000000000000000e00000014000000000000001000000014000000
0400000003010800050000000b0200000000050000000c02a300d401030000001e000400
00000701040004000000070104003f130000410b2000cc00a300d40100000000a300d401
0000000028000000d4010000a30000000100010000000000000000000000000000000000
020000000000000000000000ffffff00ffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffffffffffffffffffffffffffffffffffffffffffffffffff8fffffffffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffffffffffffffff
ffffffffffffffffffffffffffffffffff97ffffe1ffffffffffffffffffffffffffffff
fffffffffffffffffffffffffc7ff001ffe1ffffffffffffffffffffffffffffffffffff
ffffffffff9fffffccffffffffffffffffffffffffffffffffffffffffffffffffffffff
fcbff001ffccffffffffffffffffffffffffffffffffffffffffffffff9fffffdeffffff
fffffffffffffffffffffffffffffffffffffffffffffffffcfff001ffdeffffffffffff
ffffffffffffffffffffffffffffffffff9fffff9e7fffffffffffffffffffffffffffff
fffffffffffffffffffffffffcfff001ff9e7fffffffffffffffffffffffffffffffffff
ffffffffff9fffff9e7fffffffffffffffffffffffffffffffffffffffffffffffffffff
fcfff001ff9e7fffffffffffffffffffffffffffffffffffffffffffff9fffff9e7fffff
fffffffffffffffffffffffffffffffffffffffffffffffffcfff001ff9e7fffffffffff
ffffffffffffffffffffffffffffffffff9fffff9e7fffffffffffffffffffffffffffff
fffffffffffffffffffffffffcfff001ff9e7fffffffffffffffffffffffffffffffffff
ffffffffff07ffff9e7fffffffffffffffffffffffffffffffffffffffffffffffffffff
fcfff001ff9e7fffffffffffffffffffffffffffffffffffffffffffff9fffff9e7fffff
fffffffffffffffffffffffffffffffffffffffffffffffff83ff001ff9e7fffffffffff
ffffffffffffffffffffffffffffffffffdfffff9e7fffffffffffffffffffffffffffff
fffffffffffffffffffffffffcfff001ff9e7fffffffffffffffffffffffffffffffffff
ffffffffffdfffffdeffffffffffffffffffffffffffffffffffffffffffffffffffffff
fefff001ffdeffffffffffffffffffffffffffffffffffffffffffffffffffffccffffff
fffffffffffffffffffffffffffffffffffffffffffffffffefff001ffccffffffffffff
ffffffffffffffffffffffffffffffffffffffffe1ffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffe1ffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffc0000000000000000000000000000000000000
000000000001ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffdfffffffffffffffffffffffffffffffffffffffffffffffffffff80000000
0000000000000000000000000000000000000000000000000003f001ffdfffffffffffff
ffffffffffffffffffffffffffffffffffffffffdfffffffffffffffffe7ffffffffffff
fffffe7ffffffffffffffffe3ffff001ffdfffffffffffffffffffffffffffffffffffff
ffffffffffffffffdfffffffffffffffffe7fffffffffffffffff99ffffffffffffffffd
fffff001ffdfffffffffffffffffffffffffffffffffffffffffffffffffffffdfffffff
ffffffffff9bfffffffffffffffff9effffffffffffffffdfffff001ffdfffffffffffff
ffffffffffffffffffffffffffffffffffffffffdfffffffffffffffff9cffffffffffff
fffff9f7fffffffffffffffbfffff001ffdfffffffffffffffffffffffffffffffffffff
ffffffffffffffffdfffffffffffffffff7f7ffffffffffffffff7fbffffffffffffffe7
fffff001ffdfffffffffffffffffffffffffffffffffffffffffffffffffffffdfffffff
fffffffffeffbffffffffffffffff7ff3ffffff3ffffffdffffff001ffdfffffffffffff
ffffffffffffffffffffffffffffffffffffffffdffffffffffffffffeffdfffffffffff
ffffefffdffffff3ffffffbffffff001ffdfffffffffffffffffffffffffffffffffffff
ffffffffffffffffdffffffffffffffff9ffdfffffffffffffffefffe7fffff3fffffe7f
fffff001ffdfffffffffffffffffffffffffffffffffffffffffffffffffffffdfffffff
fffffffff9ffe7ffffff9fffffffeffffbffff807ffff9fffffff001ffdfffffffffffff
fffffffffffffffffffffc0fffffffffffffffffdffffffffffffffff7fffbffffff9fff
ffffdffffdfffff3fffff7fffffff001ffdfffffffffffffffffffffffffffffffffc3f0
7fffffffffffffffdffffffffffffffff7fffbffffff9fffffffdffffefffff3ffffcfff
fffff001ffdffffffffffffffffffffffffffffffffe3fff9fffffffffffffffdfffffff
ffffffffeffffdfffffc03ffffff3fffff1ffff3ffff3ffffffff001ffdfffffffffffff
fffffffffffffffffff9ffffefffffffffffffffdfffffffffffffffeffffeffffff9fff
ffff3fffffe1fffffffefffffffff001ffdffffffffffffffffffffffffffffffff7ffff
f7ffffffffffffffdfffffffffffffffdfffff3fffff9ffffffefffffffe1ffffff1ffff
fffff001ffdfffffffffffffffffffffffffffffff8ffffffbfffffffbffffffdfffffff
ffffffffdfffff3fffff9ffffffeffffffffe3ffff0ffffffffff001ffdfffffffffffff
ffffffffffffffffff7ffffffdffffffffffffffdfffffffffffffff3fffffdfffffffff
fffdfffffffffc3f80fffffffffff001ffdffffffffffffffffffffffffffffffcffffff
fffffffff7ffffffdfffff3fffffffff3fffffeffffffffffffdffffffffffc07fffffff
fffff001ffdffffffffffffffffffffffffffffffffffffffe7fffffcfffffffdfffff3f
fffffffeffffffeffffffffffffbfffffffffffffffffffffffff001ffdfffffffffffff
fffffffffffffffffbffffffffbfffffbfffffffdffff807fffffffdfffffffcffffffff
ffdffffffffffffffffffffffffff001ffdffffffffffffffffffffffffffffff7ffffff
ffcffffc7fffffffdfffff3ffffffffbffffffff7fffffffffdfffffffffffffffffffff
fffff001ffdfffffffffffffffffffffffffffffeffffffffff3ffe3ffffffffdfffff3f
ffffffe7ffffffff8fffffffffbffffffffffffffffffffffffff001ffdfffffffffffff
ffffffffffffffffdffffffffffc7f1fffffffffdfffff3fffffffe7fffffffff7ffffff
fe7ffffffffffffffffffffffffff001ffdfffffffffffffffffffffffffffffffffffff
ffff80ffffffffffdfffffffffffffdffffffffffbfffffffdffffffffffffffffffffff
fffff001ffdfffffffffffffffffffffffffffffbfffffffffffffffffffffffdfffffff
ffffffdffffffffffc7ffffff3fffffffffffffffffffffffffff001ffdfffffffffffff
ffffffffffffffff7fffffffffffffffffffffffdfffffffffffffbfffffffffffbfffff
8ffffffffffffffffffffffffffff001ffdffffffffffffffffffffffffffffeffffffff
ffffffffffffffffdffffffffffffe7fffffffffffc7fffe7fffffffffffffffffffffff
fffff001ffdfffffffffffffffffffffffffffffffffffffffffffffffffffffdfffffff
fffffdfffffffffffff80ff1fffffffffffffffffffffffffffff001ffdfffffffffffff
fffffffffffffffdffffffffffffffffffffffffdffffffffffffbfffffffffffffff00f
fffffffffffffffffffffffffffff001ffdfffffffffffffffc07ffffffffffbffffffff
ffffffffffffffffdffffffffffff7ffffffffffffffffffffffffffffffffffffffffff
fffff001ffdfffffffffffffffbf9ffffffffff7ffffffffffffffffffffffffdfffffff
ffff8ffffffffffffffffffffffffffffffffffffffffffffffff001ffdfffffffffffff
fe7fe3ffffffffffffffffffffffffffffffffffdfffffffffff7fffffffffffffffffff
fffffffffffffffffffffffffffff001ffdffffffffffffffdfffdffffffffefffffffff
ffffffffffffffffdffffffffffeffffffffffffffffffffffffffffffffffffffffffff
fffff001ffdffffffffffffff3fffe7fffffffdfffffffffffffffffffffffffdfffffff
fff7fffffffffffffffffffffffffffffffffffffffffffffffff001ffdfffffffffffff
efffffbfffffffbfffffffffffffffffffffffffdfffffffffcfffffffffffffffffffff
fffffffffffffffffffffffffffff001ffdfffffffffffffdfffffcfffffff7fffffffff
ffffffffffffffffdffffffffe3fffffffffffffffffffffffffffffffffffffffffffff
fffff001ffdfffffffffffffbffffff0fffffcffffffffffffffffffffffffffdfffffff
f1fffffffffffffffffffffffffffffffffffffffffffffffffff001ffdffffffffffffe
7fffffff1ffff3ffffffffffffffffffffffffffdfffffffcfffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffdfffffffffffffffffffffe1fc0fffffffffff
ffffffffffffffffdffffffc3fffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffdffffffffffffdfffffffffe03ffffffffffffffffffffffffffffdffffff3
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffdffffffffffff3
ffffffffffffffffffffffffffffffffffffffffdfffff8fffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffdfffffffffffffffffffffffffffffffffffff
ffffffffffffffffdfff807fffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffdfffffffffffefffffffffffffffffffffffffffffffffffffffffd8007fff
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffdfffffffffff9f
ffffffffffffffffffffffffffffffffffffffffc7ffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffdfffffffffffffffffffffffffffffffffffff
ffffffffffffffffdfffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffdfffffffffff7fffffffffffffffffffffffffffffffffffffffffdfffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffdffffffffffcff
ffffffffffffffffffffffffffffffffffffffffdfffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffdfffffffffffffffffffffffffffffffffffff
ffffffffffffffffdfffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffdfffffffffe7ffffffffffffffffffffffffffffffffffffffffffdfffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffdfffffffffdfff
ffffffffffffffffffffffffffffffffffffffffdfffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffdfffffffffbfffffffffffffffffffffffffff
ffffffffffffffffdfffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffdffffffffe7fffffffffffffffffffffffffffffffffffffffffffdfffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffdfffffffffffff
ffffffffffffffffffffffffffffffffffffffffdfffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffd9fffffff9ffffffffffffffffffffffffffff
ffffffffffffffffdfffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffde7fffffe7ffffffffffffffffffffffffffffffffffffffffffffdfffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffdfbfffffdfffff
ffffffffffffffffffffffffffffffffffffffffdfffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffdfcfffff3fffffffffffffffffffffffffffff
ffffffffffffffffdfffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffdff7fffcffffffffffffffffffffffffffffffffffffffffffffffdfffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffdff9fffbffffff
ffffffffffffffffffffffffffffffffffffffffdfffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffdffefc07ffffffffffffffffffffffffffffff
ffffffffffffffffdfffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffdfff03ffffffffffffffffffffffffffffffffffffffffffffffffdfffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffdfffffffffffff
ffffffffffffffffffffffffffffffffffffffffdfffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffdfffffffffffffffffffffffffffffffffffff
ffffffffffffffffdfffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffdfffffffffffffffffffffffffffffffffffffffffffffffffffffdfffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffdfffffffffffff
ffffffffffffffffffffffffffffffffffffffffdfffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffdfffffffffffffffffffffffffffffffffffff
ffffffffffffffffdfffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffdfffffffffffffffffffffffffffffffffffffffffffffffffffffdfffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffdfffffffffffff
ffffffffffffffffffffffffffffffffffffffffdfffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffdfffffffffffffffffffffffffffffffffffff
ffffffffffffffffdfffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffdfffffffffffffffffffffffffffffffffffffffffffffffffffffdfffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffdfffffffffffff
ffffffffffffffffffffffffffffffffffffffffdfffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffdfffffffffffffffffffffffffffffffffffff
ffffffffffffffffdfffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffdfffffffffffffffffffffffffffffffffffffffffffffffffffffdfffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffdfffffffffffff
ffffffffffffffffffffffffffffffffffffffffdfffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffdfffffffffffffffffffffffffffffffffffff
ffffffffffffffffdfffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffdfffffffffffffffffffffffffffffffffffffffffffffffffffffdfffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffdfffffffffffff
ffffffffffffffffffffffffffffffffffffffffdfffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffdfffffffffffffffffffffffffffffffffffff
ffffffffffffffffdfffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffdfffffffffffffffffffffffffffffffffffffffffffffffffffffdfffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffdfffffffffffff
ffffffffffffffffffffffffffffffffffffffffdfffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffdfffffffffffffffffffffffffffffffffffff
ffffffffffffffffdffffdfdffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffdffffdfdffffffffffffffffffffffffffffffffffffffffffffffdffffbfe
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffdffffbfeffffff
ffffffffffffffffffffffffffffffffffffffffdffff3ff7fffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffdffff3ff7fffffffffffffffffffffffffffff
ffffffffffffffffdfffe7ff3fffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffdfffe7ff3fffffffffffffffffffffffffffffffffffffffffffffdc0fe78f
3ffffffffffffffffffffffffffffffffffffffffffffffffffff001ffdc0fe78f3fffff
ffffffffffffffffffffffffffffffffffffffffdf3fef97bfffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffdf3fef97bfffffffffffffffffffffffffffff
ffffffffffffffffdf3fcf9f9fffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffdf3fcf9f9fffffffffffffffffffffffffffffffffffffffffffffdf3fcf9f
9ffffffffffffffffffffffffffffffffffffffffffffffffffff001ffdf3fcf9f9fffff
ffffffffffffffffffffffffffffffffffffffffdf3dcf9f9fffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffdf3dcf9f9fffffffffffffffffffffffffffff
ffffffffffffffffdf3dcf9f9fffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffdf3dcf9f9fffffffffffffffffffffffffffffffffffffffffffffdf01cf9f
9ffffffffffffffffffffffffffffffffffffffffffffffffffff001ffdf01cf9f9fffff
ffffffffffffffffffffffffffffffffffffffffdf3dcf9f9fffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffdf3dcf9f9fffffffffffffffffffffffffffff
ffffffffffffffffdf3def07bfffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffdf3def07bfffffffffffffffffffffffffffffffffffffffffffffdf3fe79f
3ffffffffffffffffffffffffffffffffffffffffffffffffffff001ffdf3fe79f3fffff
ffffffffffffffffffffffffffffffffffffffffdf3f67df3fffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffdf3f67df3fffffffffffffffffffffffffffff
ffffffffffffffffdf3f77df7fffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffdf3f77df7ffffffffffffffffffffffffffffffffffffffffffffffc007bfe
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffdc007bfeffffff
fffffffffffffffffffffffffffffffffffffffffffffdfdffffffffffffffffffffffff
fffffffffffffffffffffffffffff001fffffffdfdffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff001ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffff001ffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffff001ffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffff0010c00000040092900aa00000000000000a300d40100000000040000002701ffff
030000000000

а) б)

Рис. 4

4. Теоремы Ляпунова об устойчивости нелинейных систем

1. Если можно найти такую знакоопределенную функцию H (x, y), что
picscalex1000100090000039d0000000200150000000000050000000902000000000400
000002010100050000000102ffffff00040000002e011800050000003102010000000500
00000b0200000000050000000c02e001a0051200000026060f001a00ffffffff00001000
0000c0ffffffd7ffffff60050000b70100000b00000026060f000c004d61746854797065
0000300015000000fb0240fe0000000000009001010000000402001054696d6573204e65
7720526f6d616e000066040000002d01000008000000320a8001fa030200000064740800
0000320a8001fa02010000002f0008000000320a800139000200000064480a0000002606
0f000a00ffffffff01000000000010000000fb021000070000000000bc02000000cc0102
022253797374656d00cc040000002d01010004000000f0010000030000000000 тоже
знакоопределенная функция противоположного знака, то движение в
окрестности рассматриваемой особой точки асимптотически устойчиво.

2. Если можно найти такую знакоопределенную функцию H (x, y), что
picscalex1000100090000039d0000000200150000000000050000000902000000000400
000002010100050000000102ffffff00040000002e011800050000003102010000000500
00000b0200000000050000000c02e001a0051200000026060f001a00ffffffff00001000
0000c0ffffffd7ffffff60050000b70100000b00000026060f000c004d61746854797065
0000300015000000fb0240fe0000000000009001010000000402001054696d6573204e65
7720526f6d616e000066040000002d01000008000000320a8001fa030200000064740800
0000320a8001fa02010000002f0008000000320a800139000200000064480a0000002606
0f000a00ffffffff01000000000010000000fb021000070000000000bc02000000cc0102
022253797374656d00cc040000002d01010004000000f0010000030000000000 будет
знакопостоянной функцией противоположного знака, то движение в
окрестности рассматриваемой особой точки будет устойчивым в смысле
Ляпунова.

3. Если существует такая функция H (x, y) > 0, что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 > 0, то
такое движение неустойчиво.

Э? ???ы ?????уют ?????е ???я ?????? ??ения в ????? ????. Ф??? Ля???
???ю?я в ?? ?ад???? ??ц? ?и ?????ая ???я ?юс ???? ??дя ? ??, ?о
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picscalex100010009000003900200000300150000000000050000000902000000000400
000002010100050000000102ffffff00040000002e011800050000003102010000000500
00000b0200000000050000000c02e004c0291200000026060f001a00ffffffff00001000
0000c0ffffffc0ffffff80290000a00400000b00000026060f000c004d61746854797065
0000e00009000000fa02000010000000000000002200040000002d010000050000001402
b0025810050000001302b0028b12050000001402b0027214050000001302b00289160500
00001402b002a41f050000001302b002d72115000000fb0240fe00000000000090010000
00000402001054696d6573204e657720526f6d616e000081040000002d01010008000000
320a20032929010000002e0008000000320a2003eb2601000000290008000000320a2003
632501000000280008000000320a8004542001000000320008000000320a20033f1d0100
0000290008000000320a2003881b010000002c0008000000320a2003071a010000002800
09000000320a20032f17030000002020200008000000320a2003cd16010000003b000800
0000320a8004141501000000320008000000320a8004081101000000320008000000320a
2003f30d01000000290008000000320a20033c0c010000002c0008000000320a2003bb0a
01000000280008000000320a2003530802000000202008000000320a2003060801000000
3a0015000000fb0240fe0000000000009001000000cc0402001054696d6573204e657720
526f6d616e00cc81040000002d01020004000000f00101000b000000320a200339000800
0000cde0eff0e8ece5f015000000fb02c0fe000000000000900100000000040200105469
6d6573204e657720526f6d616e000081040000002d01010004000000f001020008000000
320a9004b92301000000300008000000320a3501e62001000000320008000000320a3b01
981501000000320008000000320a35019a1101000000320010000000fb02c0fe00000000
00009001000000020002001053796d626f6c0002040000002d01020004000000f0010100
08000000320a3203dc2301000000f20010000000fb0240fe000000000000900100000002
0002001053796d626f6c0002040000002d01010004000000f001020008000000320a2003
5022010000002b0008000000320a20033d1e010000003d0008000000320a200304130100
00002b0008000000320a2003f10e010000003d0015000000fb02c0fe0000000000009001
010000000402001054696d6573204e657720526f6d616e000081040000002d0102000400
0000f001010008000000320ac001c82301000000780015000000fb0240fe000000000000
9001010000000402001054696d6573204e657720526f6d616e000081040000002d010100
04000000f001020008000000320a20038c2702000000647808000000320a200319260100
0000780008000000320a20039f24010000007a0008000000320afe01e71f010000007900
08000000320a2003661c01000000790008000000320a2003bd1a01000000780008000000
320a2003861801000000480008000000320a0402a01401000000780008000000320afe01
9b1001000000790008000000320a20031a0d01000000790008000000320a2003710b0100
0000780008000000320a20033a090100000048000a00000026060f000a00ffffffff0100
0000000010000000fb021000070000000000bc02000000cc0102022253797374656d00cc
040000002d01020004000000f0010100030000000000 (8)

Это самое слабое место метода, так как нет способа выбора функций
Ляпунова, ее выбор представляет трудности, надо полагаться на интуицию и
некоторые рекомендации.

Р????м ???ы ???а ?????? ???ия в ??с??? ??? то?к с ??????? ???в Ля???.

Пример 1. Пусть задана схема нелинейная система (рис. 5). Определить
особые точки и проанализировать их устойчивость, если нелинейные звенья
описываются функциями: f(x) = x3, f(y) = y3.

’’

’’

Рис. 5

Решение

Запишем дифференциальные уравнения системы
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Определим особые точки системы (рис. 6).

Рис. 6

Используем для анализа устойчивости 1-ый метод Ляпунова (метод
линеаризации или метод оценки устойчивости по первому приближению).

Выполним линеаризацию:
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Перейдем к изображениям:
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.

Корни характеристического уравнения линеаризованной системы расположены
на мнимой оси, т.е. для определения достаточного условия устойчивости
применим 2-ой метод Ляпунова.

Необходимо найти функцию Ляпунова (т.е. знакоопределенную функцию Н (x,
y) > 0). Пусть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.

Найдем производную:
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Это знакоопределенная функция отрицательного знака, следовательно,
состояние равновесия асимптотически устойчиво.

Пример 2. Дана система (рис. 7). Проанализировать устойчивость особых
точек.

Рис. 7

Решение

Запишем исходные уравнения нелинейной системы:

picscalex810100090000037d02000003001500000000000500000009020000000004000
00002010100050000000102ffffff00040000002e0118000500000031020100000005000
0000b0200000000050000000c02e00de0141200000026060f001a00ffffffff000010000
000c0ffffffa5ffffffa0140000850d00000b00000026060f000c004d617468547970650
000400309000000fa02000010000000000000002200040000002d0100000500000014025
e028a0a0500000013025e02fa11050000001402440b4000050000001302440b700305000
0001402440b1c05050000001302440bf10615000000fb0240fe000000000000900101000
0000402001054696d6573204e657720526f6d616e000066040000002d010100080000003
20ab40b380d010000002e0008000000320ab40b580c01000000300008000000320ab40ba
b0801000000780008000000320a150d500502000000647408000000320a980a350502000
000647808000000320a860d890002000000647408000000320a980a94020100000078000
8000000320a980a590001000000640008000000320a60071414010000003b00080000003
20a6007261301000000300008000000320a6007eb1001000000290008000000320a6007b
70f01000000700008000000320a60079f0e01000000280008000000320a6007f10d01000
0007a0008000000320a6007c70b01000000290008000000320a6007930a0100000070000
8000000320a60077b0901000000280008000000320a6007df07020000007078080000003
20a6007920501000000290008000000320a60075e0401000000700008000000320a60074
60301000000280008000000320a60078a0201000000780008000000320a6007710001000
000700008000000320ace024f12010000003b0008000000320a2f046a110100000029000
8000000320a2f045a1001000000310008000000320a2f04cc0c020000007054080000003
20a2f04b40b01000000280008000000320a2f04db0a01000000700008000000320ab201d
a0d01000000310008000000320ace02b70901000000290008000000320ace02830801000
000700008000000320ace026b0701000000280008000000320ace02bd06010000007a000
8000000320ace02560301000000290008000000320ace022202010000007000080000003
20ace020a0101000000280008000000320ace024e0001000000780015000000fb0280fe0
000000000009001010000000402001054696d6573204e657720526f6d616e00006604000
0002d01020004000000f001010008000000320aeb0a9d0901000000330008000000320ab
d0c2a0201000000320008000000320acf09900101000000320008000000320a970686010
1000000320010000000fb0240fe0000000000009001000000020002001053796d626f6c0
002040000002d01010004000000f001020008000000320ab40b0a0b010000003d0008000
000320ab40b4d07010000002b0008000000320ab40bcc03010000002b0008000000320a6
007d811010000003d0008000000320a6007930c010000002b0008000000320a60075e060
10000002b0008000000320a2f04260f010000002b0008000000320ace02a605010000002
d0008000000320ace024304010000003d000a00000026060f000a00ffffffff010000000
00010000000fb021000070000000000bc02000000cc0102022253797374656d00cc04000
0002d01020004000000f0010100030000000000

Определим особые точки и проанализируем их устойчивость:
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Используем 1-й метод Ляпунова (линеаризации) или метод устойчивости по
первому приближению. Можно линеаризовать либо уравнение, либо систему
уравнений.

1). Линеаризуем уравнение

picscalex810100090000038104000003001500000000000500000009020000000004000
00002010100050000000102ffffff00040000002e0118000500000031020100000005000
0000b0200000000050000000c02a00ee02c1200000026060f001a00ffffffff000010000
000c0ffffffa6ffffffa02c0000460e00000b00000026060f000c004d617468547970650
000b00209000000fa02000010000000000000002200040000002d0100000500000014020
f0340000500000013020f03d7090500000014020f03be0b0500000013020f03271405000
0001402d0084000050000001302d0086a04050000001402d0085106050000001302d0083
809050000001402d008a719050000001302d008d11d050000001402d008b81f050000001
302d0089f2215000000fb0240fe0000000000009001010000000402001054696d6573204
e657720526f6d616e000066040000002d01010008000000320ae50d622c010000002e000
8000000320ae50d822b01000000310008000000320ae50dc626010000007000090000003
20ae50d4525030000002020200008000000320ae50dd524010000003b0008000000320ae
50de72301000000300008000000320ae50d5f2001000000700009000000320ae50dde1e0
30000002020200008000000320ae50d6e1e010000003b0008000000320ae50d801d01000
000300008000000320ae50d4c1b01000000290008000000320ae50d3b1a0100000031000
8000000320ae50d971701000000700008000000320ae50d7f16010000002800080000003
20ae50da61501000000700009000000320ae50db513040000002020202008000000320ae
50d4513010000003b0008000000320ae50d571201000000300008000000320ae50dba0f0
1000000700008000000320ae50d9b0b01000000700009000000320ae50d1a0a030000002
020200008000000320ae50daa09010000003b0008000000320ae50dbc080100000030000
8000000320ae50d5b0502000000707808000000320ae50d8602010000007800080000003
20ae50d710001000000700008000000320a40095125010000003b0008000000320a40096
32401000000300008000000320aa00a752002000000647408000000320a2408c22101000
000780008000000320a2408d11f01000000640008000000320a0f0b6f1a0200000064740
8000000320a2408f41c01000000780008000000320a2408c0190100000064000a0000003
20a400977170500000020202020200008000000320a40090717010000003b00080000003
20a4009191601000000300008000000320a4009a61102000000787808000000320a4009a
e0f01000000330008000000320a40092d0b01000000780008000000320aa00a0e0702000
000647408000000320a24085b0801000000780008000000320a24086a060100000064000
8000000320a0f0b080102000000647408000000320a24088d03010000007800080000003
20a2408590001000000640008000000320a7f03f320010000003b0008000000320a7f030
52001000000300008000000320a7f03af1c01000000290008000000320a7f03971b01000
000780008000000320a7f03e01601000000780008000000320a7f03eb150100000028000
8000000320adf043c0f02000000647408000000320a49029713010000002900080000003
20a49027f1201000000780008000000320a4902c80d01000000780008000000320a4902d
30c01000000280008000000320a4902d70b01000000640008000000320a4e05be0302000
000647408000000320a4902470901000000290008000000320a49022f080100000078000
8000000320a4902780301000000780008000000320a49028302010000002800080000003
20a4902590001000000640015000000fb0280fe000000000000900101000000040200105
4696d6573204e657720526f6d616e000066040000002d01020004000000f001010008000
000320a550ec02701000000320008000000320a550e472101000000310008000000320a1
c0dae0c01000000320008000000320a1c0d840101000000320008000000320a460a0e1c0
1000000320008000000320a5b07f61a01000000320008000000320a77086113010000003
20008000000320af809361301000000300008000000320a7508e70c01000000330008000
000320ab009f90b01000000300008000000320a460aa70201000000320008000000320a5
b078f0101000000320008000000320ab602531d01000000330008000000320aef03ac170
1000000300008000000320ab902940e01000000300008000000320a85045d05010000003
20008000000320ab902440401000000300008000000320a80018f0101000000320010000
000fb0240fe0000000000009001000000020002001053796d626f6c0002040000002d010
10004000000f001020008000000320ae50d8d2a010000002d0008000000320ae50d2d290
10000003d0008000000320ae50d9c22010000003d0008000000320ae50d351c010000003
d0008000000320ae50de918010000002b0008000000320ae50d0c11010000003d0008000
000320ae50d1b0e010000002b0008000000320ae50d7107010000003d0008000000320ae
50dbc03010000002b0008000000320a40091823010000003d0008000000320a2408b1200
1000000440008000000320a40094a1e010000002b0008000000320a2408e31b010000004
40008000000320a4009ce14010000003d0008000000320a4009951001000000440008000
000320a40094e0e010000002b0008000000320a4009b109010000002b0008000000320a2
4084a0701000000440008000000320a4009e304010000002b0008000000320a24087c020
1000000440008000000320a7f03ba1e010000003d0008000000320a7f03861a010000004
40008000000320a7f031f19010000002b0008000000320a7f03a014010000002b0008000
000320a49026e1101000000440008000000320a49020710010000002b0008000000320a7
f03500a010000002b0008000000320a49021e0701000000440008000000320a4902b7050
10000002b000a00000026060f000a00ffffffff01000000000010000000fb02100007000
0000000bc02000000cc0102022253797374656d00cc040000002d01020004000000f0010
100030000000000

2). Линеаризуем систему
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Поскольку один из корней находится на границе устойчивости, то для того,
чтобы определить достаточное условие устойчивости необходимо
использовать 2-й метод Ляпунова. Необходимо найти знакоопределенную
функцию H (x, y) > 0. Выбираем функцию Ляпунова в виде “квадратичная
форма плюс интеграл” и находим ее производную.
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Состояние равновесия устойчиво по Ляпунову, так как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 < 0 при всех значениях y и не зависит от х, т.е. х = 0, так как производная функции Ляпунова не включает координату х, то она знакопостоянная.П??р 3. Для ???? ???ы (?с. 8) ????? ??? ??и и п???и???? ? ??????.Рис. 8РешениеЗапишем исходные уравнения:picscalex1000100090000036c0200000300150000000000050000000902000000000400 000002010100050000000102ffffff00040000002e011800050000003102010000000500 00000b0200000000050000000c02e00860261200000026060f001a00ffffffff00001000 0000c0ffffffb0ffffff20260000900800000b00000026060f000c004d61746854797065 0000000209000000fa02000010000000000000002200040000002d010000050000001402 7004540b0500000013027004cc0d05000000140270046e0f05000000130270049c120500 0000140273022d1e050000001302730203200500000014029906ac1d0500000013029906 891f10000000fb0240fe0000000000009001000000020002001053796d626f6c00020400 00002d01010008000000320add06c51c01000000ef0008000000320a6d08c51c01000000 ee0008000000320a9d03c51c01000000ef0008000000320a2d05c51c01000000ed000800 0000320aed01c51c01000000ec0008000000320a09076221010000002d0008000000320a 09070220010000003d0008000000320ae3027c20010000003d0008000000320ae0047514 010000002d0008000000320ae0041513010000003d0008000000320ae004870a01000000 d70008000000320ae004a005010000002d0008000000320ae0044004010000003d001500 0000fb0240fe0000000000009001010000000402001054696d6573204e657720526f6d61 6e000066040000002d01020004000000f001010008000000320a0907362502000000292e 08000000320a09071e2401000000780008000000320a0907292301000000280008000000 320a09077a22010000007a0008000000320a6908e41d02000000647408000000320ae705 c51d0200000064790a000000320ae3024f230500000020202020200008000000320ae302 df22010000003b0008000000320ae302062201000000790008000000320a4304621e0200 0000647408000000320ac701461e0200000064780b000000320ae0044c19080000002020 20202020202008000000320ae004491802000000293b08000000320ae004311701000000 780008000000320ae0043c1601000000280008000000320ae0048d15010000007a000800 0000320aaf06b80f02000000647408000000320ac403bf1101000000780008000000320a c403870f01000000640008000000320ae0048e0e02000000202008000000320ae0041e0e 010000003b0008000000320aaf06a50b01000000700008000000320ac403270c01000000 310008000000320ae004b30901000000290008000000320ae0047f080100000070000800 0000320ae004670701000000280008000000320ae004b806010000007a0008000000320a e004570301000000290008000000320ae004230201000000700008000000320ae0040b01 01000000280008000000320ae0044e0001000000780015000000fb0280fe000000000000 9001010000000402001054696d6573204e657720526f6d616e000066040000002d010100 04000000f001020008000000320ae605571101000000320008000000320afb02bd100100 0000320008000000320ae605b80c0100000032000a00000026060f000a00ffffffff0100 0000000010000000fb021000070000000000bc02000000cc0102022253797374656d00cc 040000002d01020004000000f0010100030000000000Определим устойчивость по второму методу Ляпунова. Выбираем функцию Ляпунова в видеpicscalex100010009000003920100000300150000000000050000000902000000000400 000002010100050000000102ffffff00040000002e011800050000003102010000000500 00000b0200000000050000000c022006c0131200000026060f001a00ffffffff00001000 0000c0ffffffb6ffffff80130000d60500000b00000026060f000c004d61746854797065 0000500109000000fa02000010000000000000002200040000002d010000050000001402 10030d090500000013021003650b15000000fb0240fe0000000000009001010000000402 001054696d6573204e657720526f6d616e000066040000002d01010008000000320a8003 2913010000002e0008000000320a8003771102000000647808000000320a8003eb100100 0000290008000000320a8003d30f01000000780008000000320a8003de0e010000002800 08000000320a80032f0e010000007a0008000000320ae104cd0901000000320008000000 320a5c02500901000000790008000000320a8003b50601000000290008000000320a8003 960501000000790008000000320a8003d804010000002c0008000000320a8003f8030100 0000780008000000320a8003030301000000280008000000320a80039701010000004800 09000000320a80034000030000002020200015000000fb0280fe00000000000090010100 00000402001054696d6573204e657720526f6d616e000066040000002d01020004000000 f001010008000000320a70011e0d01000000780008000000320ad005000d010000003000 08000000320a9301500a01000000320010000000fb0280fd000000000000900100000002 0002001053796d626f6c0002040000002d01010004000000f001020008000000320a1404 140d01000000f20010000000fb0240fe0000000000009001000000020002001053796d62 6f6c0002040000002d01020004000000f001010008000000320a8003c10b010000002b00 08000000320a8003a207010000003d000a00000026060f000a00ffffffff010000000000 10000000fb021000070000000000bc02000000cc0102022253797374656d00cc04000000 2d01010004000000f0010200030000000000Найдем производную:picscalex1000100090000038c0200000300150000000000050000000902000000000400 000002010100050000000102ffffff00040000002e011800050000003102010000000500 00000b0200000000050000000c026005c0251200000026060f001a00ffffffff00001000 0000c0ffffffa2ffffff80250000020500000b00000026060f000c004d61746854797065 0000200109000000fa02000010000000000000002200040000002d010000050000001402 b0024000050000001302b002c502050000001402b0027705050000001302b002a4080500 00001402b002a70a050000001302b0027d0c050000001402b002360f050000001302b002 6312050000001402b0026614050000001302b002431615000000fb0240fe000000000000 9001010000000402001054696d6573204e657720526f6d616e000066040000002d010100 08000000320a20032f25010000002e0008000000320a20034f2401000000300008000000 320a2003c72101000000790008000000320a2003112101000000290008000000320a2003 f91f01000000780008000000320a2003041f01000000280008000000320a2003551e0100 00007a0008000000320a2003a31b01000000790008000000320a2003ed1a010000002900 08000000320a2003d51901000000780008000000320a2003e01801000000280008000000 320a20033118010000007a0008000000320a80049e1402000000647408000000320afe01 7f1402000000647908000000320a8004391101000000790008000000320a8004a6100100 0000200008000000320a0402d71001000000480008000000320a04026010010000002000 08000000320a8004dc0a02000000647408000000320a0402c00a02000000647808000000 320a8004730701000000780008000000320a8004f50601000000200008000000320a0402 180701000000480008000000320a0402a10601000000200008000000320a8004cc000200 0000647408000000320a0402590002000000644810000000fb0240fe0000000000009001 000000020002001053796d626f6c0002040000002d01020004000000f001010008000000 320a20030423010000003d0008000000320a2003e01c010000002d0008000000320a2003 bc16010000003d0008000000320a2003991301000000d70008000000320ac0039b120100 0000f70008000000320a1a039b1201000000f70008000000320adf049b1201000000f800 08000000320afb019b1201000000f60008000000320ac0034f0e01000000e70008000000 320a1a034f0e01000000e70008000000320adf044f0e01000000e80008000000320afb01 4f0e01000000e60008000000320a2003f60c010000002b0008000000320a2003da090100 0000d70008000000320a4903dc0801000000f70008000000320ab004dc0801000000f800 08000000320a2a02dc0801000000f60008000000320a4903900401000000e70008000000 320ab004900401000000e80008000000320a2a02900401000000e60008000000320a2003 3e03010000003d0010000000fb0240fe0000000000009001010000020002001053796d62 6f6c0002040000002d01010004000000f001020008000000320a8004800f01000000b600 08000000320a04023a0f01000000b60008000000320a8004cf0501000000b60008000000 320a04027b0501000000b6000a00000026060f000a00ffffffff01000000000010000000 fb021000070000000000bc02000000cc0102022253797374656d00cc040000002d010200 04000000f0010100030000000000Так как picscalex100010009000003ac0000000300150000000000050000000902000000000400 000002010100050000000102ffffff00040000002e011800050000003102010000000500 00000b0200000000050000000c02800420031200000026060f001a00ffffffff00001000 0000c0ffffffb2ffffffe0020000320400000b00000026060f000c004d61746854797065 0000e00009000000fa02000010000000000000002200040000002d010000050000001402 500240000500000013025002c40215000000fb0240fe0000000000009001010000000402 001054696d6573204e657720526f6d616e000066040000002d01010008000000320a2004 cc0002000000647408000000320aa50159000200000064480a00000026060f000a00ffff ffff01000000000010000000fb021000070000000000bc02000000cc0102022253797374 656d00cc040000002d01020004000000f0010100030000000000 0 во всем фазовом пространстве, то состояние равновесия устойчиво по Ляпунову. Фазовый портрет представляет семейство эллипсов. Особая точка типа "центр".Пример. Для заданной системы (рис. 9) определить особые точки и проанализировать их устойчивостьxОпределим устойчивость по 2-му методу Ляпунова. Выбираем функцию Ляпунова в виде "квадратичная функция плюс интеграл":picscalex83010009000003cd02000003001500000000000500000009020000000004000 00002010100050000000102ffffff00040000002e0118000500000031020100000005000 0000b0200000000050000000c02e00a80201200000026060f001a00ffffffff000010000 000c0ffffffb2ffffff40200000920a00000b00000026060f000c004d617468547970650 000800209000000fa02000010000000000000002200040000002d0100000500000014021 403a6070500000013021403f7090500000014021403d50b0500000013021403100d05000 0001402bc084000050000001302bc08c402050000001402bc089c04050000001302bc085 606050000001402bc085e0e050000001302bc08990f050000001402bc088113050000001 302bc08bc14050000001402bc08341c050000001302bc086f1d15000000fb0240fe00000 00000009001010000000402001054696d6573204e657720526f6d616e000066040000002 d01010008000000320a2c09f21f010000002e0008000000320a2c09ef1d0100000079000 8000000320a8c0a311c01000000540008000000320a1108691c010000003100080000003 20a2c09af1801000000290008000000320a2c09971701000000780008000000320a2c09a 21601000000280008000000320a2c09f315010000007a0008000000320a8c0a7e1301000 000540008000000320a1108b61301000000310008000000320a2c0994110100000079000 8000000320a8c0a5b0e01000000540008000000320a1108930e010000003100080000003 20a2c09710c01000000790008000000320a2c09610a01000000790008000000320a2c09a 20701000000790008000000320a8c0ad90401000000540008000000320a1108ca0402000 0007a7808000000320a8c0acc0002000000647408000000320a110859000200000064480 8000000320a84037e13010000003b0008000000320a8403cc11020000006478080000003 20a8403401101000000290008000000320a8403281001000000780008000000320a84033 30f01000000280008000000320a8403840e010000007a0008000000320ae404d20b01000 000540008000000320a69020a0c01000000310008000000320ae40462080100000032000 8000000320a6502e90701000000790008000000320a84035e05010000002900080000003 20a84033f0401000000790008000000320a84038703010000002c0008000000320a8403a 70201000000780008000000320a8403b20101000000280008000000320a8403470001000 000480015000000fb0280fe0000000000009001010000000402001054696d6573204e657 720526f6d616e000066040000002d01020004000000f001010008000000320a6308eb1e0 1000000320008000000320a7401790d01000000780008000000320ad4055c0d010000003 00008000000320a9c01e50801000000320010000000fb0240fe000000000000900100000 0020002001053796d626f6c0002040000002d01010004000000f001020008000000320a2 c09f01a010000002d0008000000320a2c099419010000003d0008000000320a2c091c150 1000000d70008000000320a2c09b81201000000d70008000000320a2c090e10010000002 d0008000000320a2c09950d01000000d70008000000320a2c09850b01000000d70008000 000320a2c09db08010000002d0008000000320a2c09b60601000000d70008000000320a2 c093903010000003d0008000000320a84036c0a010000002b0008000000320a840343060 10000003d0010000000fb0280fd0000000000009001000000020002001053796d626f6c0 002040000002d01020004000000f001010008000000320a18046f0d01000000f2000a000 00026060f000a00ffffffff01000000000010000000fb021000070000000000bc0200000 0cc0102022253797374656d00cc040000002d01010004000000f0010200030000000000Эта система называется абсолютно устойчивой, т.е. она устойчива для любого типа нелинейного элемента, лишь бы его характеристика находилась в секторе z = 0; z = kx.П??р 5. Для ???ой ???ы (?с. 10) ??и ??? ??и и оп???? ? ??????, ?? f(x) = x+3x2.РешениеЗапишем выражение для выходной величиныpicscalex880100090000038901000003001500000000000500000009020000000004000 00002010100050000000102ffffff00040000002e0118000500000031020100000005000 0000b0200000000050000000c022005e0151200000026060f001a00ffffffff000010000 000c0ffffffb5ffffffa0150000d50400000b00000026060f000c004d617468547970650 000300109000000fa02000010000000000000002200040000002d0100000500000014025 002060b0500000013025002211515000000fb0240fe00000000000090010100000004020 01054696d6573204e657720526f6d616e00007d040000002d01010008000000320ac0025 b15010000002e0008000000320a7504911401000000290008000000320a7504801301000 000310008000000320a7504051101000000700008000000320a7504470d0100000070000 8000000320a7504300c01000000280008000000320a7504570b010000007000080000003 20aa2010e1101000000310008000000320aa201930e01000000700008000000320ac0024 e0a01000000290008000000320ac0021a0901000000700008000000320ac002030801000 000280008000000320ac002980601000000550008000000320ac00256030100000029000 8000000320ac002220201000000700008000000320ac0020b01010000002800080000003 20ac0024e0001000000780015000000fb0280fe000000000000900101000000040200105 4696d6573204e657720526f6d616e00007d040000002d01020004000000f001010008000 000320aac033e0e01000000320010000000fb0240fe00000000000090010000000200020 01053796d626f6c0002040000002d01010004000000f001020008000000320a750443120 10000002b0008000000320a75047b0f010000002b0008000000320aa201d10f010000002 b0008000000320ac002bf05010000002d0008000000320ac0024f04010000003d000a000 00026060f000a00ffffffff01000000000010000000fb021000070000000000bc0200000 0cc0102022253797374656d00cc040000002d01020004000000f0010100030000000000Запишем дифференциальное уравнение системыpicscalex85010009000003c701000003001500000000000500000009020000000004000 00002010100050000000102ffffff00040000002e0118000500000031020100000005000 0000b0200000000050000000c028005201b1200000026060f001a00ffffffff000010000 000c0ffffffabffffffe01a00002b0500000b00000026060f000c004d617468547970650 000100109000000fa02000010000000000000002200040000002d010000050000001402f 0024000050000001302f0026203050000001402f0024905050000001302f002770805000 0001402f0025e0a050000001302f002340c050000001402f002540f050000001302f002f 21015000000fb0240fe0000000000009001010000000402001054696d6573204e6577205 26f6d616e000066040000002d01010008000000320a6003fa1902000000292e080000003 20a6003e21801000000780008000000320a6003ed1701000000280008000000320a60031 b1701000000660008000000320a60037e1401000000290008000000320a6003661301000 000780008000000320a6003711201000000280008000000320a60039f110100000066000 8000000320ac0046d0f02000000647408000000320a4402a20f010000006400080000003 20ac004930a02000000647408000000320a4402770a02000000647808000000320a2f059 30502000000647408000000320a44029a0701000000780008000000320a4402620501000 000640008000000320a2f058a0002000000647408000000320a440285020100000078000 8000000320a4402590001000000640015000000fb0280fe0000000000009001010000000 402001054696d6573204e657720526f6d616e000066040000002d01020004000000f0010 10008000000320a6604320701000000320008000000320a7b01980601000000320008000 000320a6604230201000000330008000000320a7b01890101000000330010000000fb024 0fe0000000000009001000000020002001053796d626f6c0002040000002d01010004000 000f001020008000000320a60036715010000002d0008000000320a60030d0e010000002 d0008000000320a6003ad0c010000003d0008000000320a6003f008010000002b0008000 000320a6003db03010000002b000a00000026060f000a00ffffffff01000000000010000 000fb021000070000000000bc02000000cc0102022253797374656d00cc040000002d010 20004000000f0010100030000000000picscalex100010009000003330200000300150000000000050000000902000000000400 000002010100050000000102ffffff00040000002e011800050000003102010000000500 00000b0200000000050000000c028004a0281200000026060f001a00ffffffff00001000 0000c0ffffffb3ffffff60280000330400000b00000026060f000c004d61746854797065 0000e00009000000fa02000010000000000000002200040000002d010000050000001402 500240000500000013025002de010500000014025002c40705000000130250029a090500 000014025002910a0500000013025002670c05000000140250024d120500000013025002 231405000000140250021a150500000013025002f01605000000140250021e2005000000 13025002f42115000000fb0240fe0000000000009001010000000402001054696d657320 4e657720526f6d616e000087040000002d01010008000000320ac0027d2702000000292e 08000000320ac002652601000000780008000000320ac002462501000000360008000000 320ac002ff2201000000310008000000320ac002342201000000280008000000320a2104 532002000000647408000000320aa401372002000000647808000000320ac002c61d0100 0000290008000000320ac0027d1b01000000780008000000320ac002811a010000003300 08000000320ac002251801000000780008000000320ac002301701000000280008000000 320a2104331502000000647808000000320aa401841501000000640008000000320a2104 821202000000647408000000320aa401661202000000647808000000320ac002f50f0100 0000290008000000320ac002dd0e01000000780008000000320ac002e80d010000002800 08000000320ac002170d01000000660008000000320a2104aa0a02000000647808000000 320aa401fb0a01000000640008000000320a2104f90702000000647408000000320aa401 dd0702000000647808000000320ac0026c0501000000290008000000320ac00254040100 0000780008000000320ac0025f0301000000280008000000320ac0028e02010000006600 08000000320a2104590002000000647408000000320aa4018e0001000000640015000000 fb0280fe0000000000009001010000000402001054696d6573204e657720526f6d616e00 0087040000002d01020004000000f001010008000000320af701761c0100000032001000 0000fb0240fe0000000000009001000000020002001053796d626f6c0002040000002d01 010004000000f001020008000000320ac0021124010000002b0008000000320ac002b31e 010000003d0008000000320ac0023e19010000002b0008000000320ac0026a1401000000 d70008000000320ac002e210010000003d0008000000320ac002e10901000000d7000800 0000320ac0025906010000003d000a00000026060f000a00ffffffff0100000000001000 0000fb021000070000000000bc02000000cc0102022253797374656d00cc040000002d01 020004000000f0010100030000000000Подставим в дифференциальное уравнениеpicscalex100010009000003dd0100000300150000000000050000000902000000000400 000002010100050000000102ffffff00040000002e011800050000003102010000000500 00000b0200000000050000000c028005c01e1200000026060f001a00ffffffff00001000 0000c0ffffffabffffff801e00002b0500000b00000026060f000c004d61746854797065 0000100109000000fa02000010000000000000002200040000002d010000050000001402 f0024000050000001302f0025f03050000001402f0024505050000001302f00271080500 00001402f002b510050000001302f0028a1215000000fb0240fe00000000000090010100 00000402001054696d6573204e657720526f6d616e000066040000002d01010008000000 320a60032a1e010000002e0008000000320a60034a1d01000000300008000000320a6003 161b01000000290008000000320a6003ff1901000000780008000000320a600321160100 0000780008000000320a6003261501000000330008000000320a60034d14010000002800 08000000320ac004ea1002000000647408000000320a4402ce1002000000647808000000 320a6003e50f01000000290008000000320a6003ce0e01000000780008000000320a6003 b00d01000000360008000000320a6003140b01000000320008000000320a6003340a0100 0000280008000000320a2f058f0502000000647408000000320a44029507010000007800 08000000320a44025e0501000000640008000000320a2f05890002000000647408000000 320a4402830201000000780008000000320a4402590001000000640015000000fb0280fe 0000000000009001010000000402001054696d6573204e657720526f6d616e0000660400 00002d01020004000000f001010008000000320a9702171701000000320008000000320a 66042d0701000000320008000000320a7b01930601000000320008000000320a66042102 01000000330008000000320a7b01880101000000330010000000fb0240fe000000000000 9001000000020002001053796d626f6c0002040000002d01010004000000f00102000800 0000320a6003ff1b010000003d0008000000320a60038418010000002b0008000000320a 60030313010000002b0008000000320a60035f0c010000002b0008000000320a6003ea08 010000002b0008000000320a6003d803010000002b000a00000026060f000a00ffffffff 01000000000010000000fb021000070000000000bc02000000cc0102022253797374656d 00cc040000002d01020004000000f0010100030000000000Линеаризуем полученное нелинейное уравнение. Найдем уравнение установившегося режимаpicscalex100010009000003090200000300150000000000050000000902000000000400 000002010100050000000102ffffff00040000002e011800050000003102010000000500 00000b0200000000050000000c026005401e1200000026060f001a00ffffffff00001000 0000c0ffffffa7ffffff001e0000070500000b00000026060f000c004d61746854797065 0000200109000000fa02000010000000000000002200040000002d010000050000001402 5900df140500000013020705df140500000014025900a3140500000013020705a3141500 0000fb0240fe0000000000009001010000000402001054696d6573204e657720526f6d61 6e000086040000002d01010008000000320a71047f1d010000003b0008000000320a7104 9f1c01000000330008000000320a7104a51b010000002f0008000000320a71047b1a0100 0000310008000000320a7104181501000000780008000000320ad101ce1a010000003b00 08000000320ad101e11901000000300008000000320ad101991501000000780008000000 320a2003cb13010000003b0008000000320a2003de1201000000300008000000320a2003 b10f01000000290008000000320a20039a0e01000000780008000000320a2003bb0a0100 0000780008000000320a2003bf0901000000330015000000fb0280fe0000000000009001 010000000402001054696d6573204e657720526f6d616e000086040000002d0102000400 0000f001010008000000320ae104f51502000000323008000000320a4102641602000000 313008000000320a9003211001000000300008000000320a5702b10b0100000032001000 0000fb0240fe0000000000009001000000020002001053796d626f6c0002040000002d01 010004000000f001020008000000320a71048719010000002d0008000000320a71042718 010000003d0008000000320ad1019618010000003d0008000000320a2003931101000000 3d0008000000320a20031e0d010000002b0008000000320a2003200701000000a5000800 0000320a20030a0501000000ae0015000000fb0240fe0000000000009001000000000402 001054696d6573204e657720526f6d616e000086040000002d01020004000000f0010100 0a000000320ad1013e1b0500000020202020200008000000320a20032509010000002800 08000000320a2003530802000000202009000000320a20033b0303000000202074001500 0000fb0240fe0000000000009001000000cc0402001054696d6573204e657720526f6d61 6e00cc86040000002d01010004000000f001020009000000320a2003390003000000cff0 e8000a00000026060f000a00ffffffff01000000000010000000fb021000070000000000 bc02000000cc0102022253797374656d00cc040000002d01020004000000f00101000300 00000000picscalex100010009000003e30100000200150000000000050000000902000000000400 000002010100050000000102ffffff00040000002e011800050000003102010000000500 00000b0200000000050000000c026002801e1200000026060f001a00ffffffff00001000 0000c0ffffffb7ffffff401e0000170200000b00000026060f000c004d61746854797065 0000600015000000fb0240fe0000000000009001010000000402001054696d6573204e65 7720526f6d616e0000a8040000002d01000008000000320aa001ee1d010000002e000800 0000320aa0010e1d01000000300008000000320aa0018d1a01000000780008000000320a a001f61701000000290008000000320aa001de1601000000780008000000320aa001bf15 01000000360008000000320aa0013e1301000000310008000000320aa001731201000000 280008000000320aa001f20f01000000780008000000320aa0015b0d0100000029000800 0000320aa001430c01000000780008000000320aa001240b01000000360008000000320a a001870801000000320008000000320aa001a70701000000280008000000320aa0012605 01000000780008000000320aa0014a0101000000780015000000fb0280fe000000000000 9001010000000402001054696d6573204e657720526f6d616e0000a8040000002d010100 04000000f001000008000000320a1002661801000000300008000000320a1002cb0d0100 0000300010000000fb0240fe0000000000009001000000020002001053796d626f6c0002 040000002d01000004000000f001010008000000320aa001c31b010000003d0008000000 320aa0017c1901000000440008000000320aa0016d14010000002b0008000000320aa001 2811010000002b0008000000320aa001e10e01000000440008000000320aa001d2090100 00002b0008000000320aa0015c06010000002b0008000000320aa0011504010000004400 08000000320aa001ae02010000002b0008000000320aa001390001000000440011000000 fb0240fe000000000000900100000002040200104d542045787472610002040000002d01 010004000000f001000008000000320a9301191001000000260008000000320a93017e05 01000000260008000000320a93011c0501000000260008000000320a9301d30101000000 260008000000320a9301710101000000260008000000320a93010f010100000026000a00 000026060f000a00ffffffff01000000000010000000fb021000070000000000bc020000 00cc0102022253797374656d00cc040000002d01000004000000f0010100030000000000Получили два состояния равновесия, проанализируем их устойчивость.При x0 = 0; picscalex100010009000003850100000300150000000000050000000902000000000400 000002010100050000000102ffffff00040000002e011800050000003102010000000500 00000b0200000000050000000c02800540131200000026060f001a00ffffffff00001000 0000c0ffffffabffffff001300002b0500000b00000026060f000c004d61746854797065 0000100109000000fa02000010000000000000002200040000002d010000050000001402 f0024000050000001302f0026203050000001402f0024905050000001302f00277080500 00001402f002890b050000001302f0025f0d15000000fb0240fe00000000000090010100 00000402001054696d6573204e657720526f6d616e000094040000002d01010008000000 320a6003b512010000002e0008000000320a6003d51101000000300008000000320a6003 540f01000000780008000000320ac004be0b02000000647408000000320a4402a20b0200 0000647808000000320a6003570a01000000320008000000320a2f059305020000006474 08000000320a44029a0701000000780008000000320a4402620501000000640008000000 320a2f058a0002000000647408000000320a4402850201000000780008000000320a4402 590001000000640015000000fb0280fe0000000000009001010000000402001054696d65 73204e657720526f6d616e000094040000002d01020004000000f001010008000000320a 6604320701000000320008000000320a7b01980601000000320008000000320a66042302 01000000330008000000320a7b01890101000000330010000000fb0240fe000000000000 9001000000020002001053796d626f6c0002040000002d01010004000000f00102000800 0000320a60038a10010000003d0008000000320a6003d80d010000002b0008000000320a 6003f008010000002b0008000000320a6003db03010000002b000a00000026060f000a00 ffffffff01000000000010000000fb021000070000000000bc02000000cc010202225379 7374656d00cc040000002d01020004000000f0010100030000000000Характеристическое уравнение s3 +s2 +2s+1 = 0 по Гурвицу не содержит корней в правой полуплоскости (21>1), следовательно, состояние
равновесия устойчиво.

При x0 = -1/3;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Характеристическое уравнение системы s3 +s2 –1 = 0 по Гурвицу содержит
корни в правой полуплоскости, следовательно, состояние равновесия не
устойчиво.

Устойчивость движения в предельных циклах

Предельные циклы характеризуют режимы автоколебаний. Для анализа
устойчивости автоколебаний вводится понятие орбитальной устойчивости.

Пусть имеем предельный цикл, вблизи которого выбираем окрестность -,
представляющую криволинейный цилиндр (рис. 12).

Если может быть выбрана такая окрестность -, чтобы движение, начавшись в
пределах окрестности, сколь угодно близко приближалось к предельному
циклу, то такое движение называется орбитально-асимптотически
устойчивым.

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643010000000000010048e60000000002000000002000007c0900007c290000010000006
c0000000000000000000000e9000000c600000000000000000000008f240000171f00002
0454d46000001007c2900000c00000001000000000000000000000000000000200300005
802000040010000f000000000000000000000000000000000e2040080a90300460000002
c00000020000000454d462b014001001c000000100000000210c0db01000000600000006
000000046000000b80e0000ac0e0000454d462b224004000c000000000000001e4009000
c00000000000000244001000c000000000000003040020010000000040000000000803f2
14007000c0000000000000008400005040e0000f80d00000210c0db01000000000000000
000000000000000000000000100000089504e470d0a1a0a0000000d49484452000000ea0
00000c7080200000098c458f1000000017352474200aece1ce9000000097048597300000
ec400000ec401952b0e1b00000d8049444154785eed9ce19625290a84b7f6fddfb9b7669
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4dfdfefc597c3437cb83e9c01f1fd70f1a753dfdb7af9ea365dc0af3a18cec46b84b9b6f
bfeea9bff24291f3faed197bbb82ab079f4f6cdad7ae1d8feaae19aaf9ff366afdd3e397
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fe470972ec47797f21bf6cd4dd770efd48977114c7c0d4b79ae2bdba67b4e1b26bee7326
7159947d36d12bcbe0d135f2b480ef56338e922192e2698f82245a1cda10a98bd846ecf6
f719bd99e2f12805493e61b98e29a42ba2f924bd32624bed2f75c4501d482a68529c2f5f
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04d6773a1146270f749cd0fc5b72c3f587b915e0aff88d0cd1a4b0b8f6c34b401d57823f
54d73d06d7e0481b51ba6f3667016be992a45da22651f0fd9f4a36253f061bd9e60448d3
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0edcd257866d6dd3c98a8d6c9cadbffe375a7deeef19b9aefe1927e766045cf379b995b1
ed9adc0573a20ea0ab000dffcee85140084d53ceca1c37903d1adc2703b5db9d32af7e12
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4c022e58ba4ce2081cf180bf89bdcc21b4a5c761b7caf7f512b0bd62c5e7e934b98d6f69
3c4e015fd94a501bee79cc5ed95eb7fad0c2718b7d48de3bfabd45be0e5c62dd58533c05
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2e4fd8cdfc8131123327ee402ae4d66c83991bedbac01630adf3521fa41867816b5de72e
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fb986c6f304cf7b4885cf6db8bed71b1e8024cbdea62bc617c9ea6b368f67310e7a162a2
dd13134a976bed62de7f8becf925add95f064ccf572a8bb0cfb8a79580a871e4136dfc31
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9baa8ffac0c06dc6b8cafd35907e3db6ef6467039fc899efe8ff955d1d2ac26e0eddace0
730c6777e8fe81e32c1e598580e7fe469578907f87ebcf596ef5809e2b7571611c122e32
619f31e760167bb2fbbaf40cf3e346aa4d40b05a15f6adac397ad575a747c9025b2526d9
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7191fe70f979af7be935a71f9710a7078982d896dfbb4f5369bdbf1eb89af6f8914384ad
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420b002c43770f1183af12503811520be818bc7d0892f1908ac00f10d5c3c864e7cc9406
005886fe0e23174e24b06022b407c03178fa1135f32105801e21bb8780c9df89281c00a1
0dfc0c563e8c4970c045680f8062e1e4327be6420b002c43770f118fa3dffca0e6bf9410
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fc00101fffffffffffffffffffffffffffffffffffffffffffffe7effffffffffc00101f
ffffffffffffffffffffffffffffffffffffffffffffe7effffffffffc00101fffffffff
ffffffffffffffffffffffffffffffffffff800ffffffffffc00101fffffffffffffffff
fffffffffffffffffffffffffffffffffffffffffc00101fffffffffffffffffffffffff
fffffffffffffffffffffffffffffffffc00101fffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffc001010c00000040092900aa00000000000000c700ea000
0000000040000002701ffff030000000000

Рис. 12

Если внутри окрестности предельного цикла может быть найдена такая
область, чтобы движение, начавшись вблизи окрестности, заканчивалось в
пределах области, то такое движение называется орбитально-устойчивым по
Ляпунову.

Для определения автоколебаний их устойчивости и параметров существует
ряд методов: метод точечных преобразований, метод гармонической
линеаризации и др.

5. Метод точечных преобразований

Метод точечных преобразований позволяет определить наличие предельных
циклов и их устойчивость.

Идея метода. Пусть задана фазовая траектория, выбираем линию, которая
пересекает фазовую траекторию (рис. 13). Обычно это отрезок оси – 0х.
Пусть начальное положение, изображающие точки на оси 0х –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.
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f5f34a2d7bbba00a85c520b09e005abf9e312bfc4da06cd35f951da5264d20b094005abf
142f93ff43a014faf2392d802000011d0268bd0ee7a357c9db6f12855cd4cff76d8ec68a
f310f842806dce5f6831762581b1bd986347adf483b92160910075bdc5a838b549a74ed7
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6cf56de5096d9c85a165ea5804d0fa58f1b4ea4dd98baf6a7c3e466b3568d8158a005a1f
2a9c069da1c762302898742001b4fec0a0ebb93cd363f91d9bbef4cc652508c42580d6c7
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3b9417659fcf30430e3c8a80efba9e4d1a16927557479e068e85e8638317025eb53e3704
ca76015bf476a59db2ec52cbef0a34ebfa25e055ebaf1f9d4f3f51161dbf8117b75ced3d
c308bd78ec98f004025eb5fe84d8b8f351e742abb38a3bf8180c813601c75a7fddba870a
90ee108000046e0938d67a220a01084000029d04dc687de3ddb9e9912c457d67c8190601
081c48c08dd61f181b5c860004202045c0bdd653ce4ba5c2a279f2eb2af3e7607981e522
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36d9c5b26711c81df9b3dc2ebc45eb8f0d3d8e43e00802273c77ed09245adf4389311080
804b02a90178c2a3d7d7f0a0f5af8818000108f82340d3a68a195aef2f89435a9c1e9a95
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bf4962f3ca91841eb99f49308eb54c60be4cfceaddb1679380d6cf446ba09caf421be00d
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ff7ffc43000000b3000000b3ffff50432100000008000000620000000c00000001000000
150000000c00000004000000150000000c0000000400000051000000c034000000000000
00000000f8010000d000000000000000000000000000000000000000f9010000d1000000
50000000300000008000000040340000000000002000cc00f9010000d100000028000000
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00000000ffffff00ffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff80
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffff80ffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffff80ffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffff80ffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffff80ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff80ffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffff80ffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffff80ffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffdfffffffefff80ffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffbffffffff7ff80
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffff3ffffffffbff80ffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffe7ffffffff9ff80ffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffe784388f879ff80ffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffefce731
7cfdff80ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
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ff1ffffffffeffffffffffffffffffffffffffffffeffffffff9fdffffffffffffffffff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Рис. 13 Рис. 14

За один оборот начальная точка траектории М
(wmetafile8010009000003b300000002001700000000000500000009020000000004000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) переходит в
конечную точку М
(wmetafile8010009000003b300000002001700000000000500000009020000000004000
00002010100050000000102ffffff00040000002e0118000500000031020100000005000
0000b0200000000050000000c02a00280020e00000026060f001200ffffffff000008000
000c0ffb0ff400250020b00000026060f000c004d617468547970650000600015000000f
b0200fe0000000000009001010000000002001054696d6573204e657720526f6d616e000
200040000002d01000009000000320ae0014000010000007800e60017000000fb02c0fe0
000000000009001000000cc0002001054696d6573204e657720526f6d616e2043797200d
701040000002d01010004000000f00100000a000000320a600206010200000031309700a
0000a00000026060f000a00ffffffff01000000000010000000fb021000080000000000b
c02000000000102022253797374656d0006040000002d01000004000000f001010003000
00000000000000000000000000000000000000000000000000000000000), которая
является начальной для следующего оборота
M(wmetafile8010009000003b30000000200170000000000050000000902000000000400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). Таким
образом, можно построить зависимость конечных значений изображающей
точки от начальных, т.е. так называемую кривую точечного преобразования
(рис. 14). Эта кривая позволяет определить наличие предельных циклов и
их устойчивость. Точка пересечения кривой точечного преобразования с
линией под углом 45 характеризует наличие предельного цикла. При этом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wmetafile8010009000003b3000000020017000000000005000000090200000000040000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 (рис. 15а).

Возьмем точку справа от цикла: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, т.е. колебания будут затухать (рис. 15б). Возьмем
точку слева от цикла: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, т.е. колебания будут увеличиваться по амплитуде (рис. 15в).

picscalex100010009000003d62c00000000c511000000001610000026060f002220574d
4643010000000000010005ac000000000200000000200000601500006035000001000000
6c0000000000000000000000e50100008c0000000000000000000000ef4b000007160000
20454d4600000100603500000c0000000100000000000000000000000000000020030000
5802000040010000f000000000000000000000000000000000e2040080a9030046000000
2c00000020000000454d462b014001001c000000100000000210c0db0100000060000000
60000000460000003c10000030100000454d462b224004000c000000000000001e400900
0c00000000000000244001000c000000000000003040020010000000040000000000803f
214007000c0000000000000008400005880f00007c0f00000210c0db0100000000000000
0000000000000000000000000100000089504e470d0a1a0a0000000d49484452000001e6
0000008d0802000000edb54eb5000000017352474200aece1ce900000009704859730000
0ec300000ec301c76fa86400000f0449444154785eed9ddb921d2bae454f9dffffe7ddab
9b0e9ac80b29408024463d38ec2a92cb98d24490cbf6df3ffffcf37f7c41000210808007
02ffef6192cc110210800004fe4d00cb260e20000108b8218065bb918a89420002108860
d97f7f7f0809010840e004027fee5e3ffe0c3acfb934ebf4cdfc1d77eb3a21da58230420
3048c077959d7df9f21bfc7a302c781c0210b049c0b765db64caac200001084c22e0deb2
2ff721e5b5c92464740b010840601701f796bd0b1ce342000210584f208265e7429b127b
7d003122040609f089af2680112cbb69c1348600049611a8dbf1efa7a9416e96be838957
040a62d97c446459122e1b88bc5d867ad7407ce2ab837c10cb4e2bc7b83b22c0e023145f
0645614a460884b26c234c99462b814b414df1d50ad06f7b3ef1d5aa5d10cbe6c563abf0
b48700043c12f06dd9bcacf018739239537c4928c568c327be9a74f46dd979a9dc6237a9
4e630840c02901df96fd73eaf4e5943ed3ae10a0f83a273c4861b9d6be2d5bbe4e5a4200
02c60960dc1281b06c0925daec21400eefe1cea8860960d986c5616aff218071870f043e
f1259718cb96b3a2250420a049804f7c75d0c4b23ba0f1c82202145f8b40ef1e8683945c
012c5bce8a968b08507c2d02bd7b183ef1d5a10096dd018d471611a0f85a049a61fc10c0
b2fd6875cc4c29be8e919a853613c0b29b91f100042000815d04b0ec5de4191702108040
33012cbb19190f40000210d84500cbde459e71210001083413f873f152fef3ff9472b18a
66718e7980cf5f47951a65d595b56bd9a54d978efc18046f8dd579d1e10c0224f60caa16
fa44597515cc597636dfb7c2f933083e7b5087488783043e351dec9fc77711405975f286
2c3b59ede715873c08e42dd5b1d2e12781fa65d767187cf64f83f5042a9a22a8961c262c
5b68d669cd4d464cc5ad15282afdc82fbbb8e952013eaf13616609b3f5edff6b9e377fbf
3defb76ca1a819716bfb0ea3f72ba7d999d777e5baa6427730bbf6781393e7a0bc6549a9
efa9789c1f57b4cdb2bb55e97e10ef5e1cd3f2c3935c53799f8b177bc2701d7ba75cd90b
40847e8ba83d96dd2d64ebc5c8f336f5b767d5276475df61a8351ec8e7f5b1d4aa914a85
84d077a157ff559a9f067dda2bc6e8ef4d489a86629f74559af56c89d3bf1b85820ba2ae
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a) б) в)

Рис. 15

Таким образом, рассматриваемый цикл будет устойчивым. Устойчивость можно
определить по направлению стрелок точечного преобразования.

П??р 6. П?? ??? ??ая ???? ??????? (?с. 16). П? э?м ?зо?й ??р? ?????н ?
?с. 17.

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fffffffdffffffffffefffffffdfffffffffff80ff7fffffffffffffffffcff7ffcfffff
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ffdfffffffffff80ff7fffffffffffffffffcff7fff7fffffffffefffff3ffff7fffffff
ffffffffffffdfffffffdffffffffffdffffffffffefffffffdfffffffffff80ff7fffff
ffffffffffffe7f7fffbfffffffffeffffe5ffff7fffffffffffffffffffdfffffffdfff
fffffffdffffffffffefffffffdfffffffffff80ff7fffffffffffffffffe7f7fffdffff
fffffeffffe5ffff7fffffffffffffffffffdfffffffeffffffffffdffffffffffdfffff
ffdfffffffffff80ff7ffffffffffffffffff3fbffff7ffffffffeffffceffff7fffffff
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fffffffffffff1fbffffbffffffffeffffceffff7fffffffffffffffffffdfffffffefff
fffffffdffffffffffdfffffffdfffffffffff80ff7ffffffffffffffffff1fbffffdfff
ffffffffff9effff7fffffffffffffffffffefffffffeffffffffffdffffffffffdfffff
ffbfffffffffff80ff7ffffffffffffffffff8fbffffefffffffffffff1f7fff7fffffff
ffffffffffffeffffffff7fffffffffdffffffffffbfffffffbfffffffffff80ff7fffff
fffffffffffffc7bffe12bffffffffffff3f7fff7fffffffffffffffffffeffffffff7ff
fffffffdffffffffffbfffffffbfffffffffff80ff7ffffffffffffffffffc7bffe0ecff
fffffffffc0e1fff7fffffffffffffffffffeffffffffbfffffffffdffffffffff7fffff
ffbfffffffffff80ff7ffffffffffffffffffe3bffe1f77fffffffffffffffff7fffffff
fffffffffffff7fffffffbfffffffffdffffffffff7fffffff7fffffffffff80ff7fffff
fffffffffffffe3bff83f7bfffffffffffffffff7ffffffffffffffffffff7fffffffbff
fffffffdffffffffff7fffffff7fffffffffff80ff7fffffffffffffffffff1bff67f7df
ffffffffffffffff7ffffffffffffffffffff7fffffffdfffffffffdfffffffffeffffff
ff7fffffffffff80ff7fffffffffffffffffff8be3fff7effffffeffffffffff7fffffff
fffffffffffffbfffffffdfffffffffdfffffffffefffffffeffffffffffff80ff7fffff
ffffffffffffff8b9ffffbf7fffffeffffffffff7ffffffffffffffffffffbfffffffeff
fffffffdfffffffffdfffffffeffffffffffff80ff7fffffffffffffffffffc27ffffbfb
fffffefffffbffff7ffffffffffffffffffffbfffffffefffffffffdfffffffffdffffff
feffffffffffff80ff7fffffffffffffffffffc1fffffbfdfffffefffff7ffff7fffffff
fffffffffffffdffffffff7ffffffffdfffffffffbfffffffdffffffffffff80ff7fffff
fffffffffffffff0fffffbff7ffffeffffefffff7ffffffffffffffffffffdffffffff7f
fffffffdfffffffffbfffffffdffffffffffff80ff7ffffffffffffffffffff87ffffdff
9ffffeffffdfffff7ffffffffffffffffffffdffffffffbffffffffdfffffffff7ffffff
fdffffffffffff80ff7ffffffffffffffffffffc3ffffdffeffffeffffbfffff7fffffff
fffffffffffffeffffffffdffffffffdffffffffeffffffffbffffffffffff80ff7fffff
fffffffffffffffe1ffffdfff7ffffffff7fffff7ffffffffffffffffffffeffffffffdf
fffffffdffffffffeffffffffbffffffffffff80ff7fffffffffffffffffffff1ffffdff
fbfffffffeffffff7fffffffffffffffffffff7fffffffeffffffffdffffffffdfffffff
f7ffffffffffff80ff7fffffffffffffffffffff87fffefffefffffffbffffff7fffffff
ffffffffffffff7ffffffff7fffffffdffffffffbffffffff7ffffffffffff80ff7fffff
ffffffffffffffffc3fffeffff7ffffff7ffffff7fffffffffffffffffffff7ffffffffb
fffffffdffffffff7ffffffff7ffffffffffff80ff7fffffffffffffffffffffe1fffeff
ffbffffff7ffffff7fffffffffffffffffffffbffffffffdfffffffdfffffffeffffffff
efffffffffffff80ff7ffffffffffffffffffffff0fffeffffdfffffefffffff7fffffff
ffffffffffffffbffffffffefffffffdfffffffdffffffffefffffffffffff80ff7fffff
fffffffffffffffffc7fff7fffc7feffbfffffff7fffffffffffffffffffffdffffffffe
fffffffdfffffffdffffffffdfffffffffffff80ff7ffffffffffffffffffffffe1fff7f
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dfffffffffffff80ff7fffffffffffffffffffffff0fff7ff039fefd7fffffff7fffffff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fffffffffffffffffffffffffffffffdffffffffffffffffffffffffffffff80ff7fffff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Рис. 16 Рис. 17

Литература

1. Грумондз В.Т. Динамика нелинейных систем: Некоторые задачи
устойчивости и колебаний – 2-е изд. Вуз. книга, 2009. – 182c.

2. Мирошник И.В. Теория автоматического управления: Нелинейные и
оптимальные системы. Издательство: ПИТЕР, 2006. – 272c.

3. Пащенко Ф.Ф. Введение в состоятельные методы моделирования систем. В
2-х ч. Ч. 2. Идентификация нелинейных систем Изд-во: ФИНАНСЫ И
СТАТИСТИКА, 2007. – 288c.

4. Сборник задач по теории автоматического регулирования и управления/
Под редакцией В.А. Бесекерского. – M.: Наука, 1978.

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