Dynamical phenomena in systems with structurally unstable Poincare homoclinic orbits.

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. (c) 1996 American Institute of Physics.

Dynamical systems in the theory of nonlinear waves with allowance for nonlocal interactions (the Whitham-Benjamin equation).

In this paper we briefly present a general approach to the description of the nonlinear and nonlocal Whitham-Benjamin model, based on the introduction of a system of auxiliary fields that interact locally with the initial nonlinear field. In the case of stationary waves a corresponding dynamical system is defined that admits of a Hamiltonian representation. Some results are presented of a qualitative and numerical analysis of the stationary solitary waves of the Whitham-Benjamin equation with a rapidly decreasing oscillatory kernel. An investigation is made into a phenomenon related to the loss of smoothness of the solution of the original equation and the noncontinuability of these solutions when the structural parameters of the system are changed (this phenomenon is analogous to the formation of limiting Stokes waves).

Bifurcations of the trajectories at the saddle level in a Hamiltonian system generated by two coupled Schrodinger equations.

Bifurcations of the complex homoclinic loops of an equilibrium saddle point in a Hamiltonian dynamical system with two degrees of freedom are studied. It arises to pick out the stationary solutions in a system of two coupled nonlinear Schrodinger equations. Their relation to bifurcations of hyperbolic and elliptic periodic orbits at the saddle level is studied for varying structural parameters of the system. Series of complex loops are described whose existence is related to periodic orbits.

Homoclinical structures in nonautonomous systems: Nonautonomous chaos.

For general nonautonomous systems, integral sets similar to homoclinic structures of an autonomous system are introduced. A description of integral curves near such a set is given.