No abstract

The article is devoted to interrelations between an existence of trivial and nontrivial basic sets of A-diffeomorphisms of surfaces. We prove that if all trivial basic sets of a structurally stable diffeomorphism of surface M2 are source periodic points α1,…αk, then the non-wandering set of this diffeomorphism consists of points α1,…,αk and exactly one one-dimensional attractor Λ. We give some sufficient conditions for attractor Λ to be widely situated. Also, we prove that if a non-wandering set of a structurally stable diffeomorphism contains a nontrivial zero-dimensional basic set, then it also contains source and sink periodic points.

We consider the system of two coupled one-dimensional parabola maps. It is well known that the parabola map is the simplest map that can exhibit chaotic dynamics, chaos in this map appears through an infinite cascade of period-doubling bifurcations. For two coupled parabola maps we focus on studying attractors of two types: those which resemble the well-known discrete Lorenz-like attractors and those which are similar to the discrete Shilnikov attractors. We describe and illustrate the scenarios of occurrence of chaotic attractors of both types.