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Nonexistence of stable trajectories in nonautonomous perturbations of Lorenz-type systems. (English. Russian original) Zbl 1101.37022
Sb. Math. 196, No. 4, 561-594 (2005); translation from Mat. Sb. 196, No. 4, 99-134 (2005).
This interesting paper is well written and organized. The paper deals with the systems of Lorenz-type. By the Lorenz-type system the author means a dynamical system generated by the vector field \(X(x)\) in an invariant subdomain \(U\subset\mathbb{R}^3\) satisfying several additional conditions on the types of equilibrium points and their invariant manifolds and as well as on the properties of the corresponding Poincaré map. If \(\dot x=X(x)\) is a Lorenz-type system, the author investigates the nonautonomous perturbed system \(\dot x=X(x)+h(x,t)\equiv Y(x,t)\) with small function \(h(x,t)\). For such system, the following result is proved: Let \(\dot x=Y(x,t)\) be a system defining a singularly hyperbolic system. Then, there exist no stable trajectories. To prove this result, the author has introduced the notion of singularly hyperbolic flow and as well as the notion of the singular hyperbolicity for nonautonomous systems.

MSC:
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
34D20 Stability of solutions to ordinary differential equations
37C10 Dynamics induced by flows and semiflows
34D10 Perturbations of ordinary differential equations
37C60 Nonautonomous smooth dynamical systems
37D30 Partially hyperbolic systems and dominated splittings
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