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Dynamical phenomena in systems with structurally unstable PoincarĂ© homoclinic orbits. (English) Zbl 1055.37578
Summary: Recent results describing non-trivial dynamical phenomena in systems with homoclinic tangencies are represented. Such systems cover a large variety of dynamical models known from natural applications and it is established that so-called quasiattractors of these systems may exhibit rather non-trivial features which are in a sharp distinction with that one could expect in analogy with hyperbolic or Lorenz-like attractors. For instance, the impossibility of giving a finite-parameter complete description of dynamics and bifurcations of the quasiattractors is shown. Besides, it is shown that the quasiattractors may simultaneously contain saddle periodic orbits with different numbers of positive Lyapunov exponents. If the dimension of a phase space is not too low (greater than four for flows and greater than three for maps), it is shown that such a quasiattractor may contain infinitely many coexisting strange attractors.

MSC:
37C29 Homoclinic and heteroclinic orbits for dynamical systems
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
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