с матрицей
![](data:image/png;base64,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)
. Так как
![](data:image/png;base64,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)
, то
![](data:image/png;base64,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)
. Мультипликаторы являются собственными числами матрицы
![](data:image/png;base64,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)
,
где
![](data:image/png;base64,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)
— решение уравнения (6), удовлетворяющее начальным условиям
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAD8AAAAXCAMAAACLfme2AAAAAXNSR0ICQMB9xQAAAHtQTFRFAAAAAQEBExMuLhMTPyY/IyNPIiJOTiIiTyMjTDVdXTVMQCdAQ1h4RFl5Xn6XeFhDeVlEaURpakVql35ekHeEmay9vayZ2c7Cws7Z09zc3+bf3OPm2+Ll5uPc5eLb/v7+8fHu7/Ly8vLv7vHx+fn4+Pn5+Pj39/j4////DFnREAAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAEsSURBVDjL5ZRrU4UgEIblXPRUduxkXMoMIe3s//+FwQKCAzPV9K34wIAvD/vuAlbwu1b9DV5uv+JUylyIK2TK892QitNFATBCBOjLexFHMeH9zDV+NDivlW4EjIehgHsx8MaMlKtTbaPPnZF5u8B4ozJ8FR3/0R1fGtJevdpT02Fc7Ng546No+blrF06hx/1M+NvBOTRD6zEYkK7BRrS8XcAojHsBq5gs8Yn2xLVzxttEDD931JU28rilbmihfF5EvlaOF+jO8THFwCf+t/lbI4Z/Ozw/1Cr4xxIzW5KC/yhi/dnuldHJhJmfVKifyeo6IVk6wFV0589JVdkbhLw7v3jFCueX3T/rHwLvDaR3+cv3k/Iw3sUN9EnAT3nQjyFmHH3n/ffkfvln/69PzzbJAX9aL8gAAAAASUVORK5CYII=)
, а
![](data:image/png;base64,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)
— решение уравнения (6), удовлетворяющее начальным условиям
![](data:image/png;base64,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)
. Пусть
![](data:image/png;base64,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)
— характеристическое уравнение для определения мультипликаторов. Так как
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADgAAAAXCAMAAABponzPAAAAAXNSR0ICQMB9xQAAAGlQTFRFAAAAAQEBFBQvIiJOTyMjXTVMXjZNQEAnQ1h4RFl5Xn6XeFhDaURpl35ekHeEmay9pI6CvayZv7i4wc3YzsHB2c7Cws7Z2M3B3OPm/v7++Pj3+Pn58vLv8fHu+fn47/Ly7vHx9/j4////o7Z1BAAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAC2SURBVDjL3ZTdEoIgEIUh+zGIyjJjkaR8/4dsGUY3G4eMuvJccWA/5szOAmsTxeYMAqSBRuZ1CmiX7A2EQZwIWhCoBN600mF/YH4FIehrUPEg8f+oPg2BAFOi3rC2kZnrDTaeHz5Fze5GsnPpmr0jUzq7cyODo/JHBx43fKEV3u/B3vgDPTIAHM8D6MuDcEUGdaqjQ049ROq1oZcq/jrsuqs1CPamvQqY+qwU31K4AvtYzeLreAIruJrpiEx5jgAAAABJRU5ErkJggg==)
, то оно принимает вид
![](data:image/png;base64,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)
, где
![](data:image/png;base64,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)
.
2. Устойчивость решений систем дифференциальных уравнений.
2.1. Устойчивость по Ляпунову.
Вводя определение устойчивости по Лагранжу и Пуассону в пункте 1.3, описывались свойства одной отдельно взятой траектории. Понятие устойчивости по Ляпунову характеризует траекторию с точки зрения поведения соседних траекторий, располагающихся в ее окрестности. Предположим, что система при старте из начальной точки
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAYCAMAAAArvOYAAAAAAXNSR0ICQMB9xQAAAE5QTFRFAQEBExMuLhMTRFl5Xn6XeFhDeVlEeXlqaWmGl35ego6kmay9pI6CvayZr7fGucK5xrev2c7Cws7Z3OPm5uPc/v7++Pn5+fn49/j4////S7fCbQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAABpSURBVCjPxZBLDoAgDESr4rdWQYra+1/UTxAFtybOYhYvnWlTkFTwP3HuMk8MZHa3Yn6kSJGKezjHpHkpp7PvJro+QgToiTPIzarHTnp7EgIlSwVIkxiMLtRWeIjIe4Y753uC2rDri/9sZtAksbeaUZsAAAAASUVORK5CYII=)
порождает траекторию
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB0AAAAVCAMAAACNHhVQAAAAAXNSR0ICQMB9xQAAAIdQTFRFAAAAAQEBExMuFBQvLhMTPyY/IyNPIiJOTiIiTyMjTDVdXTVMQCdARFl5Q1h4Xn6XeFhDeVlEaURpakVqeXlqaWmGl35ekHeEmay9vayZr7fGxrev2c7Cws7Z3OPm2+Ll5uPc5eLb/v7+8fHu7/Ly8vLv7vHx6uzn+fn4+Pn59/j45uvp////uXOv1gAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAC+SURBVCjPrZHhEoIgEISlskwrzcoggYigrHj/50uOQlOnmWbixzLwcTvLXWC+reAnKlvao+dc1arzyxClUwW7CHmfaqg04mBErDyV0kmB7ekUHo0h6ZtSNOK1LK+J9SuCALFXsXMmqX1LZ2AHDjpinoqJvSFANTjoCHtaZcxTsrp1aLm1IcC5yiBa4ywpFrt7ycFTxI+9Mk0qglJTrRF2eSgac9P+UdONhH90tNNJsXBYz9nQFPRGef15vv+jT8d9XQc1Kl/iAAAAAElFTkSuQmCC)
. Рассмотрим другую траекторию той же системы
![](data:image/png;base64,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)
, стартовая точка которой близка к
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAYCAMAAAArvOYAAAAAAXNSR0ICQMB9xQAAAE5QTFRFAQEBExMuLhMTRFl5Xn6XeFhDeVlEeXlqaWmGl35ego6kmay9pI6CvayZr7fGucK5xrev2c7Cws7Z3OPm5uPc/v7++Pn5+fn49/j4////S7fCbQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAABpSURBVCjPxZBLDoAgDESr4rdWQYra+1/UTxAFtybOYhYvnWlTkFTwP3HuMk8MZHa3Yn6kSJGKezjHpHkpp7PvJro+QgToiTPIzarHTnp7EgIlSwVIkxiMLtRWeIjIe4Y753uC2rDri/9sZtAksbeaUZsAAAAASUVORK5CYII=)
. Если обе траектории остаются близкими в любой последующий момент времени, то траектория
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB0AAAAVCAMAAACNHhVQAAAAAXNSR0ICQMB9xQAAAIdQTFRFAAAAAQEBExMuFBQvLhMTPyY/IyNPIiJOTiIiTyMjTDVdXTVMQCdARFl5Q1h4Xn6XeFhDeVlEaURpakVqeXlqaWmGl35ekHeEmay9vayZr7fGxrev2c7Cws7Z3OPm2+Ll5uPc5eLb/v7+8fHu7/Ly8vLv7vHx6uzn+fn4+Pn59/j45uvp////uXOv1gAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAC+SURBVCjPrZHhEoIgEISlskwrzcoggYigrHj/50uOQlOnmWbixzLwcTvLXWC+reAnKlvao+dc1arzyxClUwW7CHmfaqg04mBErDyV0kmB7ekUHo0h6ZtSNOK1LK+J9SuCALFXsXMmqX1LZ2AHDjpinoqJvSFANTjoCHtaZcxTsrp1aLm1IcC5yiBa4ywpFrt7ycFTxI+9Mk0qglJTrRF2eSgac9P+UdONhH90tNNJsXBYz9nQFPRGef15vv+jT8d9XQc1Kl/iAAAAAElFTkSuQmCC)
называется устойчивой по Ляпунову.
Наглядная иллюстрация устойчивости по Лагранжу, Пуассону и Ляпунову приводится на рис. 2. Когда говорят просто об устойчивой траектории, то всегда имеется в виду устойчивость по Ляпунову.
![](data:image/jpeg;base64,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)
Рис. 2. Качественная иллюстрация устойчивости по Лагранжу (траектория остается в замкнутой области), по Пуассону (траектория многократно возвращается в -окрестность стартовой точки) и по Ляпунову (две близкие на старте траектории остаются близкими всегда)
Рассмотрим уравнение
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEcAAAAVCAMAAAAjKAPRAAAAAXNSR0ICQMB9xQAAAJBQTFRFAAAAAQEBExMuFBQvLhMTLxQUPyY/IyNPIiJOTyMjTiIiXTVMTDVdTTZeQCdARFl5Q1h4Xn6XeVlEeFhDaURpakVqeXlqaWmGl35ekHeEmay9vayZr7fGxrev2c7Cws7Z2M3B3OPm2+Ll4NfT5uPc5eLb/v7+8fHu7/Ly8vLv7vHx+Pn5+fn49/j45uvp////ERk0rwAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAE2SURBVDjL1ZTbUsMgEIbBQ21qPdBWrWCJUYrQRnn/t5PzwZB443TGvQiZwP+x/8IGqL8JcEIO526ozv3g8HHMRwOxGTZiOCc3h4JDz7oxjLzu6N1R0StRm2WzruDAdoyzxW5ji2EvA9BSRI4tAK/7f50fEm0/eyuKZh4EBQ6F2hWFOnkzYyOs7lcAGEPanIEBEPJOGpeQ80WQo6otdIGSLftK5yJlpQqNXLSRwy7wWJXtKkUsx2UVCuM1coEjp1+30XbhK5TRccj9MXGCJufsHr2RgS8vtb76dZ510CRfnGL29Lmr3aAgtY7Y8utZhJOKmlhnonfvVxBPXTNbYQrPje5S55Zp0rlPBEHpVseb855n7tsFTEL2N7EZ2K1Xywbl59m0v/Y7hXlPyYdan4aPp/z//EfON/SP+RCFAHCrAAAAAElFTkSuQmCC)
(1)
где
![](data:image/png;base64,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)
и функция f удовлетворяет в G условию Липшица локально:
![](data:image/png;base64,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)
и
![](data:image/png;base64,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)
, где
![](data:image/png;base64,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)
— константа, не зависящая от выбора точек
![](data:image/png;base64,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)
и
![](data:image/png;base64,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)
.
Предположим, что уравнение (1) имеет решение
![](data:image/png;base64,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)
, определенное при
![](data:image/png;base64,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)
, и что
![](data:image/png;base64,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)
. Чтобы перейти к исследованию нулевого решения, выполним в (1) замену
![](data:image/png;base64,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)
. В результате получим уравнение
![](data:image/png;base64,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)
, (2)
где
![](data:image/png;base64,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)
определена в области, содержащей множество
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMUAAAAbCAMAAAD70rudAAAAAXNSR0ICQMB9xQAAANJQTFRFAAAAAQEBExMuFBQvLxQULhMTPz8mPyY/IyNPIiJOTiIiTyMjTDVdXTVMTTZeQEAnXUw1Q1h4RFl5RWpqXn6XX3+YeFhDeVlEaURpakVqeWp5aWmGmH9fl35ehmlph2pqkHeEkXiFhHeQkIR3hYWFg5ubgo6kmay9pI6CrZmZpZulvq2avayZr7fGxrev2c7Cws7ZwcHOzsHB0NDQ09fg3OPm2+Ll4NfT5uPc5eLb+Pn5/v7+7/Ly+fn48vLv7vHx+Pj39/j45uvp8fHu8PLw////EDTv2gAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAOcSURBVFjD7Vdtc9MwDLZZx96BjawF1ozxGtgyXBgpoU4yGPj//yVsSY7tNGmyK7vbh3l3u6sjP9FjSY8UpgYucbhQ93BdRFL/ZwOt8/3iPpJQapoMZ1GOZ+u8Sso7Y1G9KFpYlGfXbbYHmQnI9zacH1nfm2LGo1XPfQSpfMbhrzZH4X6bLPKTd49brj3fysyz1tIoT/toxElPpF87BBa4xDqTxTpajpMWFqNHT9tYbM5Ued5R350PhrLwEQazIEerXctCSkxc8788ayvjfDNRotOXdLYmCw+hn4VEd8s3hb1fMIo540cLSN+ks7oT9akzcfKP67JwCL0syjHnzMm+eJRRLATDq/iSrdAoEIMVOrEWC4fQy0JoXy94HTtoAWCUIrUV+a3zjDpGftmlYDXuZRD5mkXzpHloLRxCL4vq5bUMlGGOLKDMOxy0eZtQ8/65NQ/cACXEbkLY1iIfcc63i5qFfxIg+UaWb3K8VdeP0G8L7bOQKbpZ7e74PITu3swWyFJCSeku0/ADFjFjdTBjrndgF3wQWFTWQhwtgoyaeiepEqpXzw4XqNMNFlML7VjI1Bbtr8nzvcincQxGYqdYTqiY44rCWPgpbs7BoaCzo0V5/ruRUY3iMFcG9QgZ0IyFhbYsZFwLT7qRqWovCoQBNCrqSyjT2YBFUAGmFVZfVciCLKpduAOXUWHtKPWtoPA2ENBvC00sYhZWs714RT2XmReiyedCdWWUKR04nvqTrTlonAm6N1lUbxdheHBfBrGIjS/I4jSsbgtNnEZ8Jr0EoFki0KjqYN4i+mFGaRZGDK0O1Cwwfp7SWovydB6w0PvSVHTiEkEK9j6SEtpQU2ktdF0XU55I6g8GWGwXzX4R65qX+ZNVmm96t35dvn+jW7sL0Ae8cbwAiVdzg81fGFBXF7Cv7YzPuCu4pjQljdJPyJaRbCK0p1F0A+VEA//Zcxlme7c24Pxo5SxkbPUEIrQ66vhSUpZjuhGYH1CjwEKRl15dwH4KIPX5YAKxu8Risl0s9YuUaPD6Hcqfo/qXycNawyRN0vKKYjJ0GtS1YPJELs3yBoF2qV9cdXa9paliOAtQGBIxlLn85K/Vld4PDMsCY+HLpKoR7K72W0NPZsNZzG75refPrvmoTp1I9bKQtqJbR0rdt/yZw0EPYNH+rdeVUut8d8dW6qa8dy4c+n0RxHkoi2CiuMvF9V/Xr5Z1caxuw+JerwcWDyz+7/oHxC4asP/pnF8AAAAASUVORK5CYII=)
. Это уравнение называется уравнением в отклонениях. Пусть
![](data:image/png;base64,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)
— решение (2) с начальными данными
![](data:image/png;base64,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)
.
Определение. Решение
![](data:image/png;base64,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)
уравнения (2) называется устойчивым по Ляпунову, если для
![](data:image/png;base64,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)
, такое, что при
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJkAAAAbCAMAAABY+t1bAAAAAXNSR0ICQMB9xQAAAI1QTFRFAAAAAQEBExMuLxQULhMTPyY/IiJOTiIiTyMjTDVdXTVMXjZNQ1h4RGlpXn6XeFhDaURpaWmGl35ehHeQkHeEg5ubgo6kmay9pI6CpZulvayZr7fGxrevzca52c7Cws7ZwcHOzsHB0NDQ3OPm4NfT5uPc+fn4+Pn58vLv7/Ly6uzn/v7+5+zq8fPx////tWFykwAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAHvSURBVFjD7ZYNT4MwEIbLmOKmDsUPJmqrq2VFtP//50l7LS2lbGyZybKsS5YFyr1P7707hsSxLnQmOzEyFOBDuzKjfU65SeVM9q9kZYoQulmPJON31TCRuXkgMkyEKBAZR4Yvq03JYhf0gGShLGR5mMzsZa9DaLPqEGQF6POkl4gyzYNkmlinJrSKxYAKjhCSOi1ZJksIt2XkPFOg3GDcJgv1MHJW7pLpKN9zRZQ1t1v3PQGVNFclsodhMXFzJiujfOmRZUZaFBPapG3he6yv6JxBlHedXGt2T4AnxFWJzE5+TWvPTQnLPzw3M3tmqAw8ZKgmgygF7OJz6lW9FeBJHlKR8aIJ7ZDJQ3xWXTI2tc8AU69yvJxBFE1WdGZMV8AhY3FEaneXl7PmkuwkXVV9N7E6ieIbrjMdBU5h28NoKgEwrudm7Y0bS1amTw9rwZ7FkgY7oEybZ9oTtakn3d6EKGAjm/08Vu5eV2CwA2JSKx9cMklbEIHzcD/Ld8CEbplnEAWuY7mdxSYJXYHBqdF4250aQqzk1xuF6bj3e3Ml/Nr/Mj9dARjFIyYtu/8FX4I5G0vWRnE6hV+pDvEEdFWMIZtqnwJ1tgPZ1LrtvctY3ArUS7rXGz3zenNXN7cuO1SP+f9Z1Hw2kR3nOpOdEtkfdxieLWfmGd0AAAAASUVORK5CYII=)
.
Решение
![](data:image/png;base64,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)
называется асимптотически устойчивым, если оно устойчиво по Ляпунову и существует
![](data:image/png;base64,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)
такое, что
![](data:image/png;base64,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)
при
![](data:image/png;base64,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)
.
Неустойчивость решения
![](data:image/png;base64,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)
означает следующее: существуют положительное
![](data:image/png;base64,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)
, последовательность начальных точек
![](data:image/png;base64,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)
при
![](data:image/png;base64,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)
, и последовательность моментов времени
![](data:image/png;base64,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)
такие, что
![](data:image/png;base64,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)
.
При исследовании вопроса об устойчивости решений часто прибегают к заменам переменных, позволяющим упростить вид рассматриваемого уравнения. Сделаем в (2) замену
![](data:image/png;base64,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)
, где функция
![](data:image/png;base64,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)
определена при всех
![](data:image/png;base64,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)
и непрерывна по z при
![](data:image/png;base64,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)
равномерно относительно
![](data:image/png;base64,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)
, причем
![](data:image/png;base64,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)
. Пусть уравнение
![](data:image/png;base64,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)
однозначно разрешимо относительно z:
![](data:image/png;base64,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)
, где
![](data:image/png;base64,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)
определена на множестве
![](data:image/png;base64,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)
и непрерывна по y при
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACUAAAAVCAMAAAD2OtecAAAAAXNSR0ICQMB9xQAAAHVQTFRFAQEBFBQvLhMTLxQUPyY/IyNPIiJOTyMjTiIiQ1h4RFl5X3+YeFhDeVlEaWmGmH9fhXiRg5ubpZulvayZr7fGwc3Y2M3Bws7Z2c7CwcHOzsHB0NDQ3OPm5uPc/v7+7/Ly8vLv+fn4+Pn59/j48fHu8PLw////dLszSAAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACnSURBVDjLvZLbEsIgDETxUrVarbUqqCVctPn/TxTS4jioQx/UPDBk5ixLFhgOKfYNCmAAVTN2SFLHTOk5T1CmcITIbU8B2UNMyWnTL0SVzAlI5AVdub2YKUS94MHR92ZnSdDV6g3lj9Wn2JEoOX5QXnBWGDlG9/KU3CNGjjRjvQwzur7a2tcoRH69PeVlikx9yJ6H7AHhknxtuWnXPE1NRs0//tcPqDvIaWiUrks3EwAAAABJRU5ErkJggg==)
равномерно относительно
![](data:image/png;base64,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)
. Пусть уравнение (2) заменой
![](data:image/png;base64,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)
можно преобразовать в уравнение
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEsAAAAVCAMAAAA5FONfAAAAAXNSR0ICQMB9xQAAAJNQTFRFAAAAAQEBExMuFBQvLxQUPyY/IyNPIiJOTiIiTyMjQCdATDVdXTVMXjZNQ1h4Xn6XX3+YeVlEeFhDakVqaURpmH9fl35ekHeEmay9mKu8vayZr7fG2c7Cws7Z2M3Bwc3Y3OPm2+Ll5t/f4NfT5uPc5eLb/v7+8fHu7/Ly8vLv7vHx+fn47Orn5+rs+Pn55uvp/////EEyowAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAE8SURBVDjL3ZTZVoMwEIYTtZbaKl0EMYKlSoeEuuT9n87sC+SmHk8vnIs5pcz/nZmfSRD/u0AXY4EIm9VzsmjEgjSrwAhXJovod3Raw3ZDxKqv2jSMqBc68/qWpmq6u0PEwk2aVeY+948KdXyZwJaDZ4F3ZNR/1rjMSzknHGeHyE6pI7ljnTYYo+vWvfXkej6I369z2RBbyYoCYTeD03VLalkfnxzKxpitY21GVA8o11zqutPhdKZx7ZdFjeK0rXwlCbpzQ2odyyrvl0PFM3az1n9kzSIPXxNUyAq6imfUQiNXM4pOYYIKZyz3AG+J1emzZ1BZC1bK5e8nyk3bTue9J4HZ0Z6qv2XWdheVXETx3bobtbReR9bnnm12bzcL3qNjYpb4rHvCHha2iKboF80v7hyWPtv04vfX/2D9AOWYCPIBsG5qAAAAAElFTkSuQmCC)
.
Лемма. При сделанных предположениях нулевое решение уравнения (2) устойчиво по Ляпунову, асимптотически устойчиво или неустойчиво тогда и только тогда, когда соответственно устойчиво по Ляпунову, асимптотически устойчиво или неустойчиво нулевое решение уравнения
![](data:image/png;base64,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)
.
Пусть уравнение (2) автономно, а его нулевое решение асимптотически устойчиво. Множество
![](data:image/png;base64,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)
называется областью притяжения решения
![](data:image/png;base64,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)
.
2.2. Устойчивость линейных однородных систем.
Пусть
![](data:image/png;base64,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)
(3)
— вещественная система,
![](data:image/png;base64,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)
— ее произвольное решение. Замена
![](data:image/png;base64,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)
приводит (3) к виду
![](data:image/png;base64,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)
, т. е. произвольное решение уравнения (3) переводится в тривиальное решение того же уравнения. Следовательно, все решения уравнения (3) устойчивы по Ляпунову, асимптотически устойчивы или неустойчивы одновременно. Поэтому можно говорить об устойчивости уравнения (3), понимая под этим устойчивость всех его решений, в частности тривиального.
Лемма 1. Пусть
![](data:image/png;base64,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)
и
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAG0AAAAVCAMAAAB7V5DIAAAAAXNSR0ICQMB9xQAAAKJQTFRFAAAAAQEBExMuFBQvLhMTLxQUPyY/IyNPIiJOTiIiTyMjQCdATDVdXTVMXjZNRFl5Q1h4X3+YXn6XeVlEeFhDakVqaURpamqHmH9fl35ekXiFmay9mq2+mKu8vayZvKuYvq2ar7fGwc3Y2M3B2c7Cws7Z19fb2+Ll3OPm5eLb5uPc/v7++Pn58fHu7/Ly8vLv7vHx+fn49/j4+Pj35uvp////AtElLAAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAH4SURBVEjH7ZXdUsIwEIUTRP5EBQSlVqu2VKChLYp5/1dzd7OhKQ3qjDNyY24KzZdzNrvJVui/HOIUbgqHfadqj/pAimFeoBqwo3UgxW5zKeT1xvzOZwU+ytm2afZ2M3oneKx303FFu7AKpJSxV0pwNOt2zAEl3cJMZRdLj9001HrRWmkdjTYOXcGJfABCxD4pdFPrcwnhyrMl+OW3RKxfgBluPW4TmOiAW4BuFW3hRKT4wNmmFLoFMiWBhQSlICT/DgYTTTxuIIMw2bq0gcs+5VcHvcIjBW5ZO+RwIUPl5QrLCDtNMaJif4Rs6edQOITzPqxyaQNHmGTjxpPSkRJcBXLL2umiRwYUFgSaal6Ag6IuBxMMdzfFmrg0wbR1+jPRHina2yO7Ja1VRIgJC5DQdyypyGTt0gTbFVg9j5Sg2GMMVyUgYZDIRhg2MklHrGPSVaMJpqpouB1wnTxSdAMyPpMrzg0cc3UskzU3lyYY8ozhPR2Rsr0kEbF5ixvPhh/3GBefkkY/SVpLs0+XNvAz6Cg85F6pfS+R3EvmeHtpm5CFsbfbzat9OjTD/kmebXTl8mrfQbjvfDUq2sK7V2U7Y1Oq+Q3Y96B8kH7f1S1t4Xwge1tohHcbn5Tni1PaVlroH4ySu7KBF90Cj5zt8IdSJ/m+/bv9enwC3neZ9TA5zSIAAAAASUVORK5CYII=)
или
![](data:image/png;base64,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)
, где
![](data:image/png;base64,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)
— неособая при всех
![](data:image/png;base64,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)
матрица, ограниченная по норме вместе с обратной
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACwAAAAYCAMAAAC2se+IAAAAAXNSR0ICQMB9xQAAAJxQTFRFAAAAAQEBExMuFBQvLhMTLxQUPyY/IiJOIyNPTiIiTyMjQCdATDVdXTVMXjZNRFl5Q1h4X3+YXn6XeVlEeFhDakVqaURpamqHl35emH9fkHeEkXiFmq2+may9vayZvKuYr7fGwc3Y2M3B2c7Cws7Z19fb2+Ll3OPm5uPc5eLb/v7++fn47/Ly8fHu8vLv7vHx+Pn5+Pj35uvp////XchONwAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAEpSURBVDjLvZPrUoMwEIVZby0VrbRUoUq0gG1YIIB5/3czbBKato5TR0f+hNn5cs4mZ+PJH3ze72HE8+FqOanPV67W38KIrrWFTQkPYYwBYGM2oYWbFS1i1bpwBi9S5oqulmrTvNNwNjX6/Ha7hzMohiV+6Ixfv1ZazaNmyzfJg9bCjR9SNR7vIFHqMk5JtrxRsmxhYXaxO4Kp0zuqJh4oXx7UGq4ibS/8hXtPudlKBsIvNCz8VJ/SK1yYabghA80omF8R3Efz7guYke0Ii9lgj6+XO3naRhWl6LYhGWxQpVIcZqsPyIOP51ruDzjcFEB4MgnJ4JwDGbJwDKV/R3cErPT91v6a3CmUGUxaxPKpO6RtygoobNz5tObXNBLHjQg7SPWfPat/hz8BJVC82fJn48UAAAAASUVORK5CYII=)
. Тогда
![](data:image/png;base64,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)
ограничена, не ограничена или бесконечно мала по норме при
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACwAAAAQCAMAAABa4m3lAAAAAXNSR0ICQMB9xQAAAJZQTFRFAAAAAQEBFBQvExMuLhMTPyY/IiJOXTVMTDVdTTZeXUw1RFl5Q1h4Xn6XeFhDeWp5mH9fl35eh2pqkXiFhoaGgo6kmay9mq2+pI6CpY+DvayZvq2ar7fGxrevws7Zwc3Y2c7CzsHBwcHO2M3B2+Ll3OPm5eLb5uPc/v7+8vLv+Pn5+fn4+Pj37/Ly7vHx8fHu5uvp////5uPxVQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAC4SURBVCjPvZHrDoIwDIU7vN8QvCAiKipj5abu/V/OAYUFTUyM0fNjSU++tSctyA8E38P4Do78tuuGzTdEegjGqBe2YdEYHjPCqMvYoYYtoEJr16losZbxfDxJ0gWvY9hOM5OUmXAsnL1iAlCdhE9wPOAlGzAtgHLYKVf2NFHMmWCvqNrKzCpY0dnuX6W8EZyazsuqPINTZryw1QzRpcxieF/mT9sY1YaKtpGW3kbAqI3Wlv/q3H+HH0MZeJdcBSF3AAAAAElFTkSuQmCC)
тогда и только тогда, когда
![](data:image/png;base64,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)
обладает таким свойством.