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Nonautonomous systems, cocycle attractors and variable time-step discretization. (English) Zbl 0886.65077
Nonautonomous systems have been investigated using G. R. Sell’s skew-product flows [Topological dynamics and ordinary differential equations (1971; Zbl 0212.29202)], which satisfy a semigroup property in a Cartesian product of the original state space and a certain function space. An alternative formulation, which retains the original state space and is thus particularly suitable for discretization studies, involves a generalization of the semigroup property to a cocycle property. It has been central to the development of the theory of random dynamical systems, which are intrinsically nonautonomous, and many results concerning their asymptotic behaviour including the cocycle attractor concept can be usefully transferred to and reinterpreted in the context of deterministic nonautonomous systems. The purpose of the present article is to illustrate how this can been done.
MSC:
65L05Initial value problems for ODE (numerical methods)
37-99Dynamic systems and ergodic theory (MSC2000)
37D45Strange attractors, chaotic dynamics