The Third Type of Dynamics and Poincaré Homoclinic Trajectories

We present a review of some fundamental results in the theory of dynamical systems, which have led to the discovery of dynamical chaos and its three forms, namely, two classical forms, such as conservative chaos and dissipative chaos, as well as the completely new third form, the so-called mixed dynamics in which the sets of attractors and repellers have non-empty intersection. The major part of the work is devoted to homoclinic Poincaré trajectories, i.e., doubly asymptotic trajectories to saddle periodic ones, as the main elements of dynamical chaos. Using simple examples, we show the appearance of such trajectories during periodic perturbations of two-dimensional conservative systems. As is known, the homoclinic trajectories were discovered by Poincaré. In this work, we discuss the problem (the planar circular restricted three-body problem) solving which this discovery was made. Some interesting facts concerning its history are given in the appendix.