Динамический хаос и гомоклинические траектории Пуанкаре
In this work, we give an overview of some fundamental results in the theory of dynamical systems, which led to the discovery of dynamic chaos and its three forms, two classical ones - this is "conservative chaos" and "dissipative chaos", and also a third, completely new one - this is so called "mixed dynamics", in which the sets of attractors and repellers have a non-empty intersection. Most of the work is devoted to homoclinic Poincaré trajectories, those. doubly asymptotic trajectories to saddle periodic ones, as the main elements of dynamic chaos. Using simple examples, we show how such trajectories appear under periodic perturbations of two-dimensional conservative systems. As is known, homoclinic trajectories were discovered by Poincaré. In this paper, we discuss the very problem (the plane restricted circular three-body problem) that was used to solve this discovery, and also, in the appendix, we present some interesting facts about its history.