On Periodic Motion and Bifurcations in Three-Dimensional Nonlinear Systems

Abstract

We present geometric conditions for the existence of a closed trajectory with symmetry in threedimensional nonlinear systems. A generator with quadratic nonlinearity and a Chua circuit are considered as examples. Preliminary Results. Statement of the ProblemThe oscillations and stability of nonlinear multidimensional systems are used in the problems of mechanics [1] and radiophysics [2]. The problem of existence of periodic solutions in three-dimensional autonomous systems is studied in [3] with the use of the principle of torus formulated earlier. In the present paper, we consider three-dimensional systems with certain symmetry conditions. This simplifies the solution of the problem, which is reduced to the problem of existence of closed integral curves. The dynamics of three-dimensional nonlinear systems is connected with bifurcation processes and the appearance of both periodic motions and strange attractors (see [2,[4][5][6] and the references therein).The solution of the posed problem is based on the following results:(i) a procedure of detection of the bifurcation processes;(ii) a principle of symmetry for two-dimensional systems;(iii) a principle of comparison used to confirm the instability of solutions of the original system in the neighborhood of the origin.The bifurcations leading to changes in the qualitative behavior of the system can be studied by using the variational equations [6]. In the present paper, the variational equations differ from the known equations by the dependence of the coefficients of equations not on time but on the partial solutions of the system of differential equations [7].Consider a systemwhere n D 3; F .x; p/ is a smooth function and R m is the space of parameters. We introduce a small deviation in the neighborhood of partial solutions N x i ; i D 1; 2; : : : ; n; namely, ıx i D x i .t / N x i .t /; and consider ıx i as new coordinates. The linear system corresponding to system (1) in the coordinates ıx i d ıx dt D A. N x/ıx; ıx 2 R n ;