Пример. Решение
![](data:image/gif;base64,R0lGODlhLQATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABAApAAsAhAAAAAAAAAAAHR0AAAAcSB0dSAAzWh1GbDMAADQ0HTNbgEgcAFozAEhuf2xGHW5GM39/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwVHICCOZGmeKLCkbJsiAeHOtEPfrixOgTAGpYBwCMTZTAZAEkfTjSKDA1PEIA6XgGNpZapaA9hm6aGczrRms7OoCkjTJ1gPEAIAOw==)
уравнения
![](data:image/gif;base64,R0lGODlhPwAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwA6AA4AhAAAAAAAAAAAHR0AAAAcSB0dSAAzWh1GbDMAADIAMjQ0HTNbgEgcAFozHVozAFtISF1/f0huf2xGHW5GM25GRn9/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwVvICCOZGmeaJoiwaG+cIxScm3XjSncJMPbmMAPwCIMT4vjSaIsGZonIzRAFQV3IiGJyq2amMAAFqBFPZ9D6e2ssgxcQ/DI0eWiR264ymei1wN3JGo2fCoTAIE3cjaHiVB9g49kIwwtPCxiNWWVeiMhADs=)
неустойчиво справа, т.к. все решения
![](data:image/gif;base64,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)
,
![](data:image/gif;base64,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)
,
![](data:image/gif;base64,R0lGODlhSQAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABQBFAA8AhAAAAAAAAAAAHR0AAAAAMwAcSB0dSAAzWh0zWh1GbDMAADQ0HTNGbjNbgEgcAFozAFozHUZfRkhuf11/f2xGHW5GM39/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwECAwWRICCO5OiUaKquLHq2oxIUcG2zMm1TdXb/hxLvp1M9Ar/aMYgqwoYpR6CxCggkgMyshlRBbU6RQlDjUZg4cqsYaLu7o+8IDhuUA2hvMjznu35bK356LAMELAsldCkXhilyMIMkCnkjCQAGABUACAAQNZRNNjJWSRR4cwBQl0mkakmwsbKzO5a0t7Kdn7i8SYuyIQA7)
, приближаются к
![](data:image/gif;base64,R0lGODlhFQARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwARAAsAgwAAAAAAAAAAHQAcSAAzWjMAAEgcAEceM1ozAF1/f0huf2xGHX9/XWaIiIiIZgECAwQkEMhJp6mYljCyX9nhAd1YgZjolUDgvsEZjqwpoZWa1eYWCIAIADs=)
при
![](data:image/gif;base64,R0lGODlhNQAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAwAAkAhQAAAAAAAB0AHQAAHR0AAAAAMx0AMgAcSB0zWgAzWh1GbDIdADIAHTQdNDQ0NDNGbjNbgEgdHUgcAEceM1ozHVozAEZGbkhuf11/f1l/bmxGHW5GM25GRn9uSH9/XWaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZmQICwIywajwAKcslsAgROJNQ5OWoCAyEhEChWuAFI1JgdD4sNwKF4EVbMR0mXWWXOkW0jCMzvc49+XRxFIEcWQmtwRXJNdUgVBUgSimSKREJdCWwABgCOZp1USH8AX3dYeWZKlFFBADs=)
. Каждое решение
![](data:image/gif;base64,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)
так же, как и решение
![](data:image/gif;base64,R0lGODlhIwATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABAAfAAsAhAAAAAAAAAAAHR0AAAAcSB0dSAAzWh1GbDMAADQ0HTNbgEgcAFozAEhuf2xGHW5GM39/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwVBICCOZGmay6muJRIQbKw6cm3C4hQIY1AGwKDPRDMZAEfbrRQZHJTEU8rEEAaTJdzogYSWit4YQzsELAJPr2sHCAEAOw==)
, является асимптотически устойчивым справа (рис. 2).
![](data:image/gif;base64,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)
Проведем в системе замену переменных
![](data:image/gif;base64,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)
. Новая система будет иметь вид
![](data:image/gif;base64,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)
вводя обозначение
![](data:image/gif;base64,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)
получим систему
![](data:image/gif;base64,R0lGODlhTQAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwBIAA4AhQAAAAAAAAAAHR0AHR0AAAAAMx0AMgAcSB0dSAAzWh1GbDMAADIAHTIAMjIyADQ0HTNGbjNbgEgcAEgdHUceM1ozAEhuf11/f1l/bmxGHW5GM39ZSH9/XX9uSG6AbmaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwbOQIBwSCwaj5mAcqgUFBvHKBEqHTqrWABIYBkuiqBCNtoZZANjrGRIMVLTRIjQAK8bD4DJUWEXmoUZUQFXAAF4QhVKikoJVYNHDEWPQoZGiUtzUYGBfUZfRmFGm1h8XhFRBJ1uDo1GYkapVZFEn1GnRouKUlAZr0S9tlKctKVGD6pDb39FvkTHUhLFQrVFfAgAGp3NANJS1thHdG3CAa0SaHYLSqXqk0MVQ0nm6ATN7kKt9cjIs1hdUW+E/NvXZ9mRBa0O0kpIsE9AOA+HBAEAOw==)
где
![](data:image/gif;base64,R0lGODlhSQAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwBEAA4AhAAAAAAAAAAAHR0AHR0AAAAAMx0AMgAcSAAzWh1GbDMAADIAHTIAMjIyADNGbjNbgEgcAEgdHUceM1ozAEhuf11/f1l/bmxGHX9ZSH9/XX9uSG6AbmaIiIBbM4iIZgECAwXJICACV2COpjACDBqsMNzGdC2edCdQowIXIyBA8bCtBsZkR+ijQUYS2eg5OiRHhmuNKrLSrBHYZKQZ8ETIWvkMQGhjacCaFlDBFiuhqBmbmF4iWSt+f39uI0srfDCLQYh6QzZjim8/ikUyDYcjF5CQBDZ4igmVjiOgUp0xenNoNl49YoWGMHGtIjMAcaYiXBewMFyxpb0rwJAApCuiVXuYWoLEHcNDJsoKKTDKLisEkHUrF5vEJYDEC2w1GMTs7bpGjYzP7vT19jEhADs=)
при
![](data:image/gif;base64,R0lGODlhIwAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABgAfAA4AhAAAAAAAAB0AHR0AAAAAHQAAMx0AMgAcSAAzWh0zWh1GbDMAADIAHTNGbjNbgEgcAEceM1ozAFozHUhuf11/f1l/bmxGHW5GM39uSH9/XWaIiIBbM4iIZgECAwECAwECAwVZICCOZGmeJnYGaNmgQhscbWxGQe60Q2HmrFtrtAiSIorWhOc7MVq0VfRkmZYe1OEoKbIEuAYARHsKjwEJUoCwJHmBQMRosx7tNuS86DJ66cl3f3lpgnk4SSEAOw==)
. Решение
![](data:image/gif;base64,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)
перешло при рассматриваемой замене переменных в положение равновесия
![](data:image/gif;base64,R0lGODlhJQATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABAAhAAsAhAAAAAAAAB0AHQAAHR0AAAAAMwAcSB0dSAAzWh1GbDMAADQ0HTNbgEgcAFozAF1/f0huf2xGHW5GM39/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwVPICCOZGmaURCcbGsWouLOLDMadC4KJA9YgcFoRVIZVSULbCQjIQBPHWkZI1EICSlJSSKUGifH0Rgd+XYkCVRbsolwbBe8CSACGoFsPGUPAQA7)
новой системы. Задача устойчивости решения
![](data:image/gif;base64,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)
переходит, таким образом, в задачу устойчивости нулевого (тривиального) решения
![](data:image/gif;base64,R0lGODlhJQATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABAAhAAsAhAAAAAAAAB0AHQAAHR0AAAAAMwAcSB0dSAAzWh1GbDMAADQ0HTNbgEgcAFozAF1/f0huf2xGHW5GM39/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwVPICCOZGmaURCcbGsWouLOLDMadC4KJA9YgcFoRVIZVSULbCQjIQBPHWkZI1EICSlJSSKUGifH0Rgd+XYkCVRbsolwbBe8CSACGoFsPGUPAQA7)
системы .
Приведем определение устойчивости нулевого решения системы .
Решение
![](data:image/gif;base64,R0lGODlhJQATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABAAhAAsAhAAAAAAAAB0AHQAAHR0AAAAAMwAcSB0dSAAzWh1GbDMAADQ0HTNbgEgcAFozAF1/f0huf2xGHW5GM39/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwVPICCOZGmaURCcbGsWouLOLDMadC4KJA9YgcFoRVIZVSULbCQjIQBPHWkZI1EICSlJSSKUGifH0Rgd+XYkCVRbsolwbBe8CSACGoFsPGUPAQA7)
системы называется устойчивым в смысле Ляпунова, если для любого
![](data:image/gif;base64,R0lGODlhJQATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMABAAgAAsAhAAAAAAAAB0AHR0AAAAAHQAAMx0AMgAcSAAzWh1GbDMAADNbgEgcAFozAF1/f0huf2xGHX9/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwVWICCOZGmaQbCcbEsWouKOzSwyI1UyaVA2AYRLQCLGEjEhqgUbKVYAg000aL5IslFK2YIEkNfRYMXgunwno8h4mClULZwIIqrNCT+rqw3I9lN4UywpUCEAOw==)
можно указать
![](data:image/gif;base64,R0lGODlhJwATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABAAiAAsAhAAAAAAAAB0AAB0AHQAAMwAdMh0dSAAcSAAzWh1GbDMAADIdADIAMjNbgEgcAFozAF1/f0huf2xGHX9uSH9/XWaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwViICCKQmA2Y6quwLkao8DOKSEqKL3mtDNaLEWCFkDQBimkqgA40CwB58o2wokegZxQBxBQR1+AQiRR+bgt1XciQCmrOuxQ9QYow82ZItAblQFQKVIqD3gsUlYAWCZzaERZIiEAOw==)
такое, что из неравенства
![](data:image/gif;base64,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)
следует неравенство
![](data:image/gif;base64,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)
при
![](data:image/gif;base64,R0lGODlhIwAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABgAfAA4AhAAAAAAAAB0AHR0AAAAAHQAAMx0AMgAcSAAzWh0zWh1GbDMAADIAHTNGbjNbgEgcAEceM1ozAFozHUhuf11/f1l/bmxGHW5GM39uSH9/XWaIiIBbM4iIZgECAwECAwECAwVZICCOZGmeJnYGaNmgQhscbWxGQe60Q2HmrFtrtAiSIorWhOc7MVq0VfRkmZYe1OEoKbIEuAYARHsKjwEJUoCwJHmBQMRosx7tNuS86DJ66cl3f3lpgnk4SSEAOw==)
. Если же, кроме того, всякое решение
![](data:image/gif;base64,R0lGODlhHQAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAYAA4AhAAAAAAAAB0AHQAAHR0AAAAAMx0AMgAcSB0dSAAzWh1GbDIAHTMAADIAMjQ0HTNGbjNbgEgcAEceM1ozAF1/f0huf1l/bmxGHW5GM39/XX9uSG5/WWaIiIBbM4iIZgECAwVkICCO5NiUZ6mOXaFqwrqm5CMacqmUsXiRgcEocAAsSJOAcoQb/X6ilmo3YkBIhJKLdCQxVFff1qdiUEUOLSlyFn1HOwQAM6owARLVJZAQRQIjE0BCJH05Il0zhyQ9JW+LJigkIQA7)
, начальные данные которого определяются условием
![](data:image/gif;base64,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)
, обладает свойством
![](data:image/gif;base64,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)
, то нулевое решение называется асимптотически устойчивым в смысле Ляпунова.
Проиллюстрируем идею метода на простейшем примере:
![](data:image/gif;base64,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)
Рассмотрим функцию
![](data:image/gif;base64,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)
. Эта функция положительна всюду, кроме точки
![](data:image/gif;base64,R0lGODlhJAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAeAA4AhAAAAAAAAB0AHR0AAAAAMx0AMgAcSAAzWh1GbDMAADIAHTIAMjNbgEgcAFozAF1/f0huf2xGHX9ZSH9/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwV7ICACyxgEY2qiYqkCxBgDCfPCYi0KUOoCjZThFRwNRYWRY0ThjQSqZm83iowUqVnOlk3pkCKtRUtTjb1eRoRMHrzaaJxI+ow6qaMEQi4qio4pfgCAZQBYQls3R18AFEp7KScpA2SSKgcpUzciEpsjP3ibCZieCZ6nnqAhADs=)
, где она обращается в нуль. В пространстве переменных
![](data:image/gif;base64,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)
уравнение
![](data:image/gif;base64,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)
определяет параболоид с вершиной в начале координат. Линии уровня этой поверхности на плоскости
![](data:image/gif;base64,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)
представляют собой эллипсы. Зададим произвольно малое
![](data:image/gif;base64,R0lGODlhDQAPAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAwAIAAgAhAAAAAAAAAAAHR0AMgAcSAAzWh1GbDMAAEgcAFozAF1/f0huf2xGHX9/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwUcICAiQSkChyEehTicMFC2Iw0TMZCcjHAeJZ8oBAA7)
. Построим на плоскости
![](data:image/gif;base64,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)
круг
![](data:image/gif;base64,R0lGODlhFAAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIACAAPAAwAhAAAAAAAAB0AHR0AAAAAMx0AMgAcSAAzWh1GbDMAADQdHTIAMjNGbjNbgEgcAFozAEhZf11/f0huf1l/bmxGHX9/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwU3ICACCSEGx6gCwdqugDCmsOiMt5jAsriMBZhhBJnBTADFaAhzBExOWm16QlFFlgHiKmJyAQNrCAA7)
радиуса
![](data:image/gif;base64,R0lGODlhDQAPAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAwAIAAgAhAAAAAAAAAAAHR0AMgAcSAAzWh1GbDMAAEgcAFozAF1/f0huf2xGHX9/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwUcICAiQSkChyEehTicMFC2Iw0TMZCcjHAeJZ8oBAA7)
. Возьмем одну из линий уровня --- эллипс, целиком лежащий внутри круга
![](data:image/gif;base64,R0lGODlhFAAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIACAAPAAwAhAAAAAAAAB0AHR0AAAAAMx0AMgAcSAAzWh1GbDMAADQdHTIAMjNGbjNbgEgcAFozAEhZf11/f0huf1l/bmxGHX9/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwU3ICACCSEGx6gCwdqugDCmsOiMt5jAsriMBZhhBJnBTADFaAhzBExOWm16QlFFlgHiKmJyAQNrCAA7)
. Построим другой круг
![](data:image/gif;base64,R0lGODlhFQAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIACAAQAAwAhAAAAAAAAB0AHQAAHQAAMx0AMgAcSAAzWh0zWh1GbDMAADQdHTIAMjNGbjNbgEgcAFozAEg0AFozHVtISEhZf11/f0huf1l/bn9/XX9uSGaIiIBbM4iIZgECAwECAwECAwU6ICACCiEGx6iea7COwpi+41OPCh2LzFjQhhFFRjMBFqOgaBMYiB4BE3QmpFkj1pUtk1UhuitFwCEKAQA7)
целиком лежащий внутри эллипса (рис. 3).
![](data:image/gif;base64,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)
Пусть начальная точка
![](data:image/gif;base64,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)
лежит внутри
![](data:image/gif;base64,R0lGODlhFQAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIACAAQAAwAhAAAAAAAAB0AHQAAHQAAMx0AMgAcSAAzWh0zWh1GbDMAADQdHTIAMjNGbjNbgEgcAFozAEg0AFozHVtISEhZf11/f0huf1l/bn9/XX9uSGaIiIBbM4iIZgECAwECAwECAwU6ICACCiEGx6iea7COwpi+41OPCh2LzFjQhhFFRjMBFqOgaBMYiB4BE3QmpFkj1pUtk1UhuitFwCEKAQA7)
.
Рассмотрим функцию двух переменных
![](data:image/gif;base64,R0lGODlhlAAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABQCPAA8AhQAAAAAAAB0AAB0AHQAAHQAAMx0AMgAcSB0dSAAyMh00NAAzWh0zWhxHTx1JSR1GbDMAADIAHTIAMjMeRzIyADQ0HTQ0NDMzWz09RjNdXTNGbjNbgEgcAEceM0g0AFozAFozHUgzM0YzRltISEhIW1tuSFl/WV1/f0huf1l/bkRubmxGHW5GM25bNW5GRm5bSH9ZSH9uSH9/XW5uRG6AbmaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwb/QIAgEAB4iIEHgHgAOJ2Sp3RKrVqj1mwVq+06Y4LNk4MUr4jeJyTQqEKkBakSnq5bB3ZrPJ9FS1d7Thx2EU5NeFIDKE4QTU4aU1x8fAaTT5KWU2FTC082dh9TMVOIABSOIlOhmZOdk6usVAKqrnyBTyuuNnuQnSxUEYtONgEET0VexYvEjqKVU0jRwMLHSHGzEHtI0GVTc3ZkAbdOzwBxqYN6VZ21aQsrXW+T4/IATUSBpcaGx1P1ebD8OYH34sm3P+MAgDmYZlY8hl4AkaLmxCE0AotKCRHzBRFELRA4CgTwIYGaLAkBpMuzMss/ktGQtDM3ZaUjiwDOODEm8QlOupoTKFmBoOTDHjAzhyX8VSsllQpS1lAJyWcXFXw+Pe2xgRHAPoWbBgIIwQfeUEFSygGLNQVBpClqCb3VJOXrvmxZn3xAxvIjgGdBh3X5xpeMkg5SBAQ6U4uvE2pmLR1cg8QBZY6UFQxhFFOvFsVVNHKaVEgLYksjpGDKU9oJibRsLUWmsjpNDEVZTjNKWuWTGpGTbi+aDcBv7N24FLqMVZuVbyjHsYRLcjyPuCvVqzhOc4YI8Ozgwz8JAgA7)
. Легко видеть, что если вместо
![](data:image/gif;base64,R0lGODlhJAAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIACQAgAAsAhAAAAAAAAAAAHR0AAAAAMwAcSB0dSAAzWh1GbDMAADIAMjQ0HT09RjNbgEgcAEceM1ozHVozAFtISEhuf11/f0RubmxGHW5GM39ZSH9uSH9/XWaIiIBbM4iIZgECAwECAwVSIMAFAmAGZqqq5CQGxZoeAC3fgHVYuJkNiB5uIDQ5irej8FJLEZCAhSqBgiINK8Up5QgEH6rB02QJ2ADVlAsHtq4kwrYpcRZy3O57MW0tBwINIQA7)
подставить решение системы , то полученная таким образом, функция от
![](data:image/gif;base64,R0lGODlhCQAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAGAAkAhAAAAAAAAB0AHQAAHR0AMgAcSB1GbDIAHTNGbjNbgEgcAEceM1ozAEhuf1l/bl1/f2xGHX9uSH9/XWaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwUYICBGYimUTKAmIlOWzVu8ygsQwFIGQwyEADs=)
будет представлять собой полную производную функции
![](data:image/gif;base64,R0lGODlhEAATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABAALAAsAhAAAAAAAAB0AHQAAMwAcSAAzWh0zWh1GbDMAADMeRzNbgEgcAEceM1ozAEgzM1l/WV1/f0huf2xGHW5GM39uSH9/XWaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwUmILAEpAJIJKACwyq6QLFecAPfx70i+ipEgNytldBJAI4eIKC09UIAOw==)
вдоль траектории решения системы . Если эта производная вдоль любой траектории, начинающейся в
![](data:image/gif;base64,R0lGODlhFQAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIACAAQAAwAhAAAAAAAAB0AHQAAHQAAMx0AMgAcSAAzWh0zWh1GbDMAADQdHTIAMjNGbjNbgEgcAFozAEg0AFozHVtISEhZf11/f0huf1l/bn9/XX9uSGaIiIBbM4iIZgECAwECAwECAwU6ICACCiEGx6iea7COwpi+41OPCh2LzFjQhhFFRjMBFqOgaBMYiB4BE3QmpFkj1pUtk1UhuitFwCEKAQA7)
, неположительна, то это будет означать, что траектория не сможет покинуть
![](data:image/gif;base64,R0lGODlhFAAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIACAAPAAwAhAAAAAAAAB0AHR0AAAAAMx0AMgAcSAAzWh1GbDMAADQdHTIAMjNGbjNbgEgcAFozAEhZf11/f0huf1l/bmxGHX9/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwU3ICACCSEGx6gCwdqugDCmsOiMt5jAsriMBZhhBJnBTADFaAhzBExOWm16QlFFlgHiKmJyAQNrCAA7)
, так как иначе между
![](data:image/gif;base64,R0lGODlhIwATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABAAeAAsAhAAAAAAAAB0AHR0AAAAAHQAAMx0AMgAcSAAzWh1GbDMAADIAHTNGbjNbgEgcAEceM1ozAF1/f0huf1l/bmxGHX9/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwVOICCOZGmSQdCcbCsWorKOlts6I1YKNsuPP0hKNRIOh4gSbCQTQXqsZYwkgZqklsHsYDIeA0nST/TDWU1mAGVkADzOJS6gCQgQqnAUERACADs=)
и значением
![](data:image/gif;base64,R0lGODlhIwAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABwAfAA0AhAAAAAAAAB0AHQAAHQAAMx0AMgAcSAAzWh1GbDIAHTNGbjNbgEgcAEceM1ozAEhuf11/f1l/bmxGHX9uSH9/XWaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwVFICBOYmmeaFoKamuypxPMSynPeHDEeG06riBQ9QgaUQbULadDphhHFVTEIJQKgEb0hC1pRYFBcVsKk89ZtHpt+rKVgWQIADs=)
, при котором она попадет на границу
![](data:image/gif;base64,R0lGODlhFAAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIACAAPAAwAhAAAAAAAAB0AHR0AAAAAMx0AMgAcSAAzWh1GbDMAADQdHTIAMjNGbjNbgEgcAFozAEhZf11/f0huf1l/bmxGHX9/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwU3ICACCSEGx6gCwdqugDCmsOiMt5jAsriMBZhhBJnBTADFaAhzBExOWm16QlFFlgHiKmJyAQNrCAA7)
, найдется значение
![](data:image/gif;base64,R0lGODlhJAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAgAA4AhAAAAAAAAB0AAAAAHR0AHR0dAB0AMh0dSAAcSAAzWh1GbDIAHTNGbjNbgEgcAEceM0czHlozAF1/f0huf1l/bmxGHX9uSH9/XWaIiIBbM4iIZgECAwECAwECAwECAwECAwVNICCOZGmeaKqSFgAVzSqXwlDM+PjkvNn2I0aKABQRT5GAMiZKKp+BROmJihQBEYVqUlyoECYnNEqqgFGOK0CLMgB2PTfcFBhweZk6KgQAOw==)
, для которого
![](data:image/gif;base64,R0lGODlhNAApAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMABAAuACIAhAAAAAAAAB0AAB0AHQAAHQAAMx0AMgAcSAAzWh0zWh1GbDMAADIAHTMeRzNAMzNGbjNbgEgcAEceM1ozAEgzM1l/WV1/f0huf1l/bmxGHW5GM39uSH9/XWaIiIBbM4iIZgXUICCOgACJUaCOqkK+cCx7K1m8p6zvr0BmCJKLR9z5SEFRosgkTVQ11EjSrAIOLNLipLE2B6SjCOxtZm4jsadQKTcJ4dzV3VygS6OnnM5bQKNDfCMZUYJVayILhk5EESRYiyJPSTFkABuWkSIBkC93AIqaJBsCnwCfW6I/AZ5hAH+wsbE7rDCWmKp+Oo4jnYumMB4joao8hLXFyco6pHvLLzTIzzImIxvTPSSZy38nTyrOqp3IE9iZ1SIXz2euIr7KcCLNALzTdmkiBgBU9rCvBOpUhQAAOw==)
, поскольку
![](data:image/gif;base64,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)
. То, что ни одна траектория, начинающаяся в
![](data:image/gif;base64,R0lGODlhFQAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIACAAQAAwAhAAAAAAAAB0AHQAAHQAAMx0AMgAcSAAzWh0zWh1GbDMAADQdHTIAMjNGbjNbgEgcAFozAEg0AFozHVtISEhZf11/f0huf1l/bn9/XX9uSGaIiIBbM4iIZgECAwECAwECAwU6ICACCiEGx6iea7COwpi+41OPCh2LzFjQhhFFRjMBFqOgaBMYiB4BE3QmpFkj1pUtk1UhuitFwCEKAQA7)
, не покидает ни при одном
![](data:image/gif;base64,R0lGODlhIwATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABAAeAAsAhAAAAAAAAB0AHR0AAAAAHQAAMx0AMgAcSAAzWh1GbDMAADIAHTNGbjNbgEgcAEceM1ozAF1/f0huf1l/bmxGHX9/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwVUICCOZGmSQdCcbCsWorKOVgm5ozNipWBCAYTLNyICU7NSoAUbyUQ33KBJogIUJAmOEkiUqJbB7OACsogioo6lULXWAMrIAHiYrC3ydRYgaHEtSCIhADs=)
круг
![](data:image/gif;base64,R0lGODlhFAAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIACAAPAAwAhAAAAAAAAB0AHR0AAAAAMx0AMgAcSAAzWh1GbDMAADQdHTIAMjNGbjNbgEgcAFozAEhZf11/f0huf1l/bmxGHX9/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwU3ICACCSEGx6gCwdqugDCmsOiMt5jAsriMBZhhBJnBTADFaAhzBExOWm16QlFFlgHiKmJyAQNrCAA7)
, означает устойчивость тривиального решения.
Итак, мы должны проверить знак
![](data:image/gif;base64,R0lGODlhHAApAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMABAAVACIAhAAAAAAAAB0AAB0AHQAAHQAAMx0AMgAcSAAzWh0zWh1GbDMAADIAHTMeRzNAMzNGbjNbgEgcAEceM1ozAEgzM1l/WV1/f0huf1l/bmxGHW5GM39uSH9/XWaIiIBbM4iIZgWPICCOgACJUaCOqkIC3koW7/mKApkhpHvjL54ocZuoZKiR5HdgkRYnzW9Ayo2oP0CGNrLCCpUsgFC1AZriBbc0Mpp/iyPSJ67b7/i8fs/Xy/+AgH2DhIWGeRsmfDEBhIoiG3peAFh1cidGKm8vaACNIhN2lSVmF2JbVSOdWWSQihF3al0iBgBLdXFyngSmJCEAOw==)
вдоль траектории. Для этого надо знать саму траекторию. Хотя в данном примере это можно сделать, но метод должен быть рассчитан на систему общего вида, для которого
![](data:image/gif;base64,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)
нельзя выписать явно и тем самым нельзя проверить нужное неравенство. Поэтому мы будем требовать, чтобы функция
![](data:image/gif;base64,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)
была неположительной как функция двух независимых переменных
![](data:image/gif;base64,R0lGODlhJAAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIACQAgAAsAhAAAAAAAAAAAHR0AAAAAMwAcSB0dSAAzWh1GbDMAADIAMjQ0HT09RjNbgEgcAEceM1ozHVozAFtISEhuf11/f0RubmxGHW5GM39ZSH9uSH9/XWaIiIBbM4iIZgECAwECAwVSIMAFAmAGZqqq5CQGxZoeAC3fgHVYuJkNiB5uIDQ5irej8FJLEZCAhSqBgiINK8Up5QgEH6rB02QJ2ADVlAsHtq4kwrYpcRZy3O57MW0tBwINIQA7)
по крайней мере в некоторой окрестности
![](data:image/gif;base64,R0lGODlhJAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAeAA4AhAAAAAAAAB0AHR0AAAAAMx0AMgAcSAAzWh1GbDMAADIAHTIAMjNbgEgcAFozAF1/f0huf2xGHX9ZSH9/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwV7ICACyxgEY2qiYqkCxBgDCfPCYi0KUOoCjZThFRwNRYWRY0ThjQSqZm83iowUqVnOlk3pkCKtRUtTjb1eRoRMHrzaaJxI+ow6qaMEQi4qio4pfgCAZQBYQls3R18AFEp7KScpA2SSKgcpUzciEpsjP3ibCZieCZ6nnqAhADs=)
. Это условие можно проверить непосредственно по правым частям системы не зная решения. В нашем примере именно так и будет, поскольку
![](data:image/gif;base64,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)
всюду на плоскости
![](data:image/gif;base64,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)
, а тем самым вдоль любой траектории, и устойчивость тривиального решения гарантирована. Функция
![](data:image/gif;base64,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)
и есть функция Ляпунова для рассмотренного примера. Она имеет вид квадратичной формы, хотя в принципе можно было взять любую другую функцию, лишь бы она была положительной всюду, кроме точки
![](data:image/gif;base64,R0lGODlhJAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAeAA4AhAAAAAAAAB0AHR0AAAAAMx0AMgAcSAAzWh1GbDMAADIAHTIAMjNbgEgcAFozAF1/f0huf2xGHX9ZSH9/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwV7ICACyxgEY2qiYqkCxBgDCfPCYi0KUOoCjZThFRwNRYWRY0ThjQSqZm83iowUqVnOlk3pkCKtRUtTjb1eRoRMHrzaaJxI+ow6qaMEQi4qio4pfgCAZQBYQls3R18AFEp7KScpA2SSKgcpUzciEpsjP3ibCZieCZ6nnqAhADs=)
, где она обращается в нуль, а выражение
![](data:image/gif;base64,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)
было неположительное. Обратимся теперь к формулировке некоторых общих теорем, в основу которых положена эта идея. Будем исследовать тривиальное решение системы .
Все дальнейшие построения будем вести в некоторой
![](data:image/gif;base64,R0lGODlhEQARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwANAAsAhAAAAAAAAB0AAAAAHQAAMwAcSAAzWh1GbDMAADIAHTNGbjNbgEgcAFozAEhZf11/f0huf2xGHX9uSH9/XWaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwU2ICBKQmAGi6g2gaFWQfEGhyrCLsAQtsoMIkTPFgjKhgCUDikCNpGMHs/WQCZ8KgWgFHCKToAQADs=)
-окрестности начала координат в фазовом пространстве. Пусть для определенности
![](data:image/gif;base64,R0lGODlhEQARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwANAAsAhAAAAAAAAB0AAAAAHQAAMwAcSAAzWh1GbDMAADIAHTNGbjNbgEgcAFozAEhZf11/f0huf2xGHX9uSH9/XWaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwU2ICBKQmAGi6g2gaFWQfEGhyrCLsAQtsoMIkTPFgjKhgCUDikCNpGMHs/WQCZ8KgWgFHCKToAQADs=)
задается неравенством
![](data:image/gif;base64,R0lGODlhOAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAwA1AA8AhAAAAAAAAAAAHR0AHR0AAB0AMgAAMwAcSB0dSAAzWgA0SB1GbDMAADQ0HTMzWzNbgEgcAFozAEZGM11/f0huf2xGHW5GM39/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwWNIOAAZGmeKDmmLBC85Xuca4vWbcIGZmHithaQxGNVShFh8JaKBBY2CJSRGpY0AUGsul0OAIaglaTT/U6wZVYZJEBZwGJLARa3IFzjM0UFQGZMNhYAZnkphSQaYSQMgIYmGiQIgUsmEIgEjiotchB7TBUvoi+FWDIApmk0lWesriKvj7E2Y2yzdrcltbMhADs=)
,
![](data:image/gif;base64,R0lGODlhRQATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABAA/AAsAhAAAAAAAAB0AHQAAHR0AAB0AMgAAMwAcSB0dSAAzWgA0SB1GbDMAADVbbjNGbjNbgEgcAFozAEZGM0hbSEhZf1l/WV1/f0huf2xGHW5GRn9uSH9/XW5/WWaIiIBbM4iIZgWnICBiQbkAZCCubOu+cCwDnroWc67v77H6vKAQ8iICGMJkENnSCCQGWaRELSVe1hWkxGI0ADZUyVBhjFuBgdJlBKgBbYjP/H6vMiLCoxWBR9c/LR5/eXl7AAIXAAQtjC1McjFTVVkrGEAshEcnjosrBAFXni2ETICaNAGHRwAaeq2vIjV7nSIQooWADLgOfoGjwCJ9i6tUK1UnSmYBmFthdItp0WFbUSEAOw==)
. Функция
![](data:image/gif;base64,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)
(или короче
![](data:image/gif;base64,R0lGODlhIwAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAeAA4AhAAAAAAAAB0AHQAAHR0AAAAAMx0AMgAcSB0dSAAzWh0zWh1GbDIAHTMAADIAMjMeRzQ0HTNbgEgcAEceM1ozAEgzM1l/WV1/f0huf2xGHW5GM39/XX9uSGaIiIBbM4iIZgVyICAFJGCWpumkrNqaRRulXvyyAsZm7cKut5ahlUgpesEWLyVJTVqMV2CQCqSGpsZM06oFi8WU1iRI2l4EH6thqllumXOr2WqoD5WgPMUNi00UATNJXTccKRRqNxQtViYjin6EAFGTAECWZJNsmT9JmAAhADs=)
) называется положительно определенной в
![](data:image/gif;base64,R0lGODlhEQARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwANAAsAhAAAAAAAAB0AAAAAHQAAMwAcSAAzWh1GbDMAADIAHTNGbjNbgEgcAFozAEhZf11/f0huf2xGHX9uSH9/XWaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwU2ICBKQmAGi6g2gaFWQfEGhyrCLsAQtsoMIkTPFgjKhgCUDikCNpGMHs/WQCZ8KgWgFHCKToAQADs=)
, если
![](data:image/gif;base64,R0lGODlhOwAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwA2AA4AhAAAAAAAAB0AHQAAHR0AAAAAMx0AMgAcSB0dSAAzWh0zWh1GbDMAADIAHTIAMjMeRzQ0HTNbgEgcAEceM1ozAEgzM1l/WV1/f0huf2xGHW5GM39/XX9uSGaIiIBbM4iIZgW3ICAFJGCWpumkrNq+MJuhbNFGqWfHpoDBC55JZ2LIWsHUSmgyxAIJnoR1YEVNCiTTlOF5AtWXIMXxmaamSavxCgxSgZRTSNil7ABjEadpEWFRV3tCDHE1LAw4AGMxeCwESSl6MHUweARDBRYwGY4paIiRQ2A8jABlPwAHFY0wfYJFL7AwoKomFAGKWwAeMBy7L72xt6ItFC2GIgGRszMkz1AtzsAmbFtL1NkwpjGT2t8s2C/i4C0hADs=)
в
![](data:image/gif;base64,R0lGODlhEQARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwANAAsAhAAAAAAAAB0AAAAAHQAAMwAcSAAzWh1GbDMAADIAHTNGbjNbgEgcAFozAEhZf11/f0huf2xGHX9uSH9/XWaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwU2ICBKQmAGi6g2gaFWQfEGhyrCLsAQtsoMIkTPFgjKhgCUDikCNpGMHs/WQCZ8KgWgFHCKToAQADs=)
, причем
![](data:image/gif;base64,R0lGODlhPAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwA3AA4AhAAAAAAAAB0AHQAAHR0AAAAAMx0AMgAcSB0dSAAzWh0zWh1GbDMAADIAHTIAMjMeRzQ0HTNbgEgcAEceM1ozAEgzM1l/WV1/f0huf2xGHW5GM39/XX9uSGaIiIBbM4iIZgWoICAFJGCWpumkrNq+8EtGbdHSpmfHpoDxQNOOgQNkWgvWKggwMGMS1oGVSCmQz+OzxfGlBKmoadJqyAapQMrJopDepCprZ2KkiABNSxeryutFW3x3Xzx0LQRJLHZbcywEOQUWMBmHLGKLimlwKCldPz0pBxUxliZ6f3WNlykZUwBugUEeMByrLa94sJovFC1qJiOaqbedT2ZMS7fLLGBAjMzRLjHK0i0hADs=)
тогда и только тогда, когда
![](data:image/gif;base64,R0lGODlhJQATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABAAhAAsAhAAAAAAAAB0AHQAAHR0AAAAAMwAcSB0dSAAzWh1GbDMAADQ0HTNbgEgcAFozAF1/f0huf2xGHW5GM39/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwVPICCOZGmaURCcbGsWouLOLDMadC4KJA9YgcFoRVIZVSULbCQjIQBPHWkZI1EICSlJSSKUGifH0Rgd+XYkCVRbsolwbBe8CSACGoFsPGUPAQA7)
.