
ВИДЕОТЕКА 
Конференция «Hyperbolic Dynamics and Structural Stability», посвященная 85летию Д. В. Аносова



Structural stability for dynamical systems on time scales S. G. Kryzhevich^{} ^{} Gdansk University of Technology 

Аннотация: We consider a system on a time scale \begin{equation} x^\Delta=f(t,x), \quad x \in {\mathbb R}^n, \quad t\in {\mathbb T} \end{equation} where the time scale Definition. We say that the system (1) is structurally stable if for any $$\varphi_f(t,x_0)\varphi_{f+g}(t,h(x_0))<\varepsilon$$ for any \begin{equation} x^\Delta=f(t,x)+g(t,x) \end{equation} with initial conditions For systems of ordinary differential equations, conditions for global structural stability were obtained in [1], see also [2]. It was proved that a system is structurally stable if its linearizations are uniformly hyperbolic on families of segments. We formulate and prove an analog of this statement for time scale systems. Although the result is very similar to that for ordinary differential equations, the proof for the time scale case is significantly different. We need to use specific approaches of time scale systems theory [3]. Remarkably, the classical results for structural stability of autonomous systems of ODEs, obtained by C.Robinson [4], are, in general, nonapplicable for systems on time scales (even for the autonomous ones). [1] S. Kryzhevich, V. Pliss, Structural stability of nonautonomous systems, Diff. equations, 39:10, (2003), 1395 – 1403. [2] V. A. Pliss, Relation between various conditions of structural stability, Differents. Uravn., 17:5 (1981), 828 – 835. [3] M. Bohner, A.A. Martynyuk, Elements of stability theory of A. M. Liapunov for dynamic equations on time scales, Nonlinear Dyn. Syst. Theory, 7:3, (2007), 225 – 251. [4] C. Robinson, Structural stability of vector fields, Ann. of Math., 99:3, (1974), 447 – 493. Язык доклада: английский 