On the existence of separatrix loops in four-dimensional systems similar to the integrable hamiltonian systems

Abstract

A method analogous to the V.K. Mel'nikov method /1/ is used to derive the conditions of existence of separatrix loops of the saddle-focus type singularity, for the systems similar to the integrable Hamiltonian systems. Many important and interesting effects appearing in the dynamic systems are connected with the presence of loops in the separatrices of the singularities. For example, a system with a loop of a saddle-focus separatrix with the positive saddle value has an enumerable set of periodic saddle-type motions /2/; the appearance of the saddle separatrix loops can represent the first of a sequence of bifurcations leading to the appearance of an attractor /3/. The existence of a separatrix loop in a finite-dimensional system describing the travelling wave-type solutions of partial differential equations implies the presence of a soliton solution in the latter system /4/.