THE TOPOLOGICAL PRINCIPLE AND COMPARISON OF SOLUTION OF SYSTEMS ODE
Abstract
The present paper deals with an ordering relation for systems ODE at infinity.Some criteria for ordering (preceding or succeeding) of two systems are obtained.The proofs of the main results are based on topological principle in the theory of ODE.
A relationship between (asymptotical) stability of zero solution of system ODE and introduced relation are established.We prove, that if a system generates a monotone flow then its zero solution is stable if and only if the system is preceded of zero-system ($\dot x = 0$).
At the end, we prove some classical results for stability of zero solution of a given system, by the using of considered relations. The proofs are based on comparison of investigated system and a "model" system (usually zero-system).
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