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Zbl 0931.34029
Schneider, K.; Kostadinov, S.I.; Stamov, G.T.
Integral manifolds of impulsive differential equations defined on torus.
(English)
[J] Proc. Japan Acad., Ser. A 75, No.4, 53-57 (1999). ISSN 0386-2194

The authors study a system of differential equations with impulses of the form $$\dot\varphi =a(\varphi),\quad \dot x=A(\varphi)x+f(\varphi),\quad \varphi\notin \Gamma ,\quad \Delta x=I( \varphi),\quad \varphi\in \Gamma, \tag 1$$ where $\varphi =(\varphi _1,\dots ,\varphi _m)$ is in the $m$-dimensional torus $T^m$, $x\in \bbfR^n$, $\Gamma$ is an $(m-1)$-dimensional closed submanifold of $T^m$, the functions $a(\varphi)$, $A(\varphi)$, $f(\varphi)$ and $I( \varphi)$ are continuous and $2\pi$-periodic in $\varphi _k$, $k=1,\dots ,m$, $\Delta x(t)=x(t+0)-x(t-0)$. The dynamics of the system (1) is as follows. Between two successive meetings of the point $(t,\varphi (t),x(t))$ with $\Gamma$ the motion proceeds along the trajectory of the system $\dot\varphi =a(\varphi)$, $\dot x=A(\varphi)x+f(\varphi)$. When the point $(t,\varphi (t),x(t))$ meets $\Gamma$ it is momentarily transfered to the point $(t,\varphi (t),x(t)+I(\varphi (t)))$. For system (1) sufficient conditions are obtained for the existence of an integral manifold of the form $J=\{(\varphi ,x): x=u(\varphi)$, $\varphi\in T^m$, $x\in \bbfR^n \}$. Some continuous properties of $u(\varphi)$ are investigated.
[E.Ershov (St.Peterburg)]
MSC 2000:
*34C45 Method of integral manifolds
34A37 Differential equations with impulses

Keywords: differential equations with impulses; integral manifolds; Green function; invariant tori

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