A Codimension-3 Bifurcation Problem

In this paper, a bifurcation phenomenon which arises from the interaction of static and dynamic modes in the vicinity of a compound critical point of a nonlinear autonomous system is considered. The critical point is characterized by a double zero and a pair of pure imaginary eigenvalues of the Jacobian. A set of simplified differential equations is obtained by applying the unification technique coupled with the intrinsic harmonic balancing procedure. Based on the simplified equations, the equilibrium solutions, Hopf bifurcation solutions and quasi-periodic motions lying on 2-D tori, as well as the associated stability conditions are explored. An example drawn from electrical circuits is analyzed to demonstrate the direct applicability of the analytic results. INTRODUCTION Systems with a high codimension are likely to exhibit secondary bifurcations, and sequences of bifurcations into tori associated with quasi-periodic motions, and the phenomena have been considered by a number of authors (e.g. see [1-6]). It has also been observed that such a system may exhibit chaotic behaviour [7,8]. Attention in this paper is focused on the local dynamics, in particular, bifurcation and stability properties, of an autonomous system in the vicinity of a critical point a t which the Jacobian has a double zero of index one and a pair of pure imaginary eigenvalues. A special case of the system t o be studied has been analyzed in (61 by applying the unification technique [4,5,6] which is based on the in t r insic harmonic balancing [9] perturbation procedure. It is assumed in (61, however, tha t the system under consideration has a trivial equilibrium surface, which is not the mathematically most generic case. The developed methodology will be applied to the generic case in this paper. Based on a set of simplified rate equations which are derived systematically by using the method, the solutions for the equilibrium states, Hopf bifurcations, sequences of bifurcations into 2-D tori are obtained in terms of the system coefficients, which facilitates applications to specific problems. An example which is quite typical in electrical circuits is considered to illustrate the practical significance of the theoretical and analytical results. It also demonstrates tha t electrical circuits may exhibit plenty of different bifurcation patterns. * This work is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). FORMULATION For simplicity, consider a 4-dimensional nonlinear autonomous system involving the critical state variables only as dz'/dt = Z;( zJ ; q 3 ) ( i , j = 1,2,3,4; 17 = 1,2,3), (1) where the z' are the components of the state vector z, and the tl3 are three certain independent parameters. It is assumed that the functions Z, are analytic, at least in the region of interest. Introduce the transformations t' = 2: + TI, yJ and q3 = 0," + p3, (2) where the subscript c indicates a critical point, to (1) to obtain a new system dy'/dt = Y,(yJ; /13) ( i , j = 1,2,3,4; = 1,2,3), (3) such that its Jacobian evaluated a t c is in the canonical form [ o 1 0 0 1 ISCAS'88 255 CH2458-8/88/0000-0255$1 .OO