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Zbl 0624.34038
Voskresenskij, Evgenij Viktorovich
Componentwise asymptotics and homeomorphism of differential equations on manifolds. (Russian)
[J] Czech. Math. J. 35(110), 455-466 (1985). ISSN 0011-4642; ISSN 1572-9141/e

The systems of differential equations (1) $\dot x=A(t)x+f(t,x)$, $\dot y=A(t)y$, $x,y\in R\sp n$ are called componentwise asymptotic equivalent on the manifold Q relative to the function $\{\mu\sb i(t)\}$ if there is a map $p: Q\to Q$ that $$ x\sb i(t,t\sb 0,x\sb 0)=y\sb i(t,t\sb 0,y\sb 0)+O(\mu\sb i(t)), $$ where $t\to \infty$, $y\sb 0=Px\sb 0$, $x(t,t\sb 0,x\sb 0)$ and $y(t,t\sb 0,x\sb 0)$ are solutions of the systems (1) and (2). A manifold $Q=\{x\sb j=0\}$ is considered. The article gives sufficient conditions for: 1) componentwise asymptotic equivalence, 2) the map P is one-to-one, 3) the map P is a homeomorphism, 4) systems (1) and (2) are topologically equivalent.
[G.Osipenko]

MSC 2000:: *34C40 ODE on manifolds
34D30 Structural stability of ODE
34C05 Qualitative theory of some special solutions of ODE
34A34 Nonlinear ODE and systems, general

Keywords: topological equivalence; first order differential equation; asymptotic equivalence

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