On phase-locked motions associated with strong resonance

Abstract

Abstract. This paper is concerned with the stability and bifurcation behaviour of a nonlinear autonomous system in the vicinity of a compound critical point characterized by two pairs of pure imaginary eigenvalues of the Jacobian. Attention is focused on the local dynamics of the system near-to-resonance. The methodology developed earlier for the bifurcation analysis into periodip and quasi-periodic motions (unification technique coupled with the intrinsic harmonic balancing) is extended to consider the stability and bifurcations of resonant cases. A set of simplified rate equations characterizing the local dynamics of the system is derived. These equations differ from those associated with nonresonant cases in that they are phase-coupled. Furthermore, the stability conditions of the phase-locked periodic bifurcation solutions are presented. All the results are expressed in explicit forms.1. Introduction. It is well known that a nonautonomous system may quite often exhibit periodic as well as quasi-periodic bifurcation solutions. Nonlinear systems with periodic driving forces have been considered in detail by, for example, Nayfeh and Mook [1], Iooss and Joseph [2], who used perturbation techniques and operator function theory to obtain the first-order approximations of bifurcation solutions and stability conditions. In a nonlinear autonomous system, interactions of static and dynamic bifurcation modes in the vicinity of a compound critical point may lead to invariant tori, which has been investigated by many authors (e.g., see [3][4][5][6][7]). Recently, such phenomena are studied by using the unification technique [7][8][9][10][11] coupled with the intrinsic harmonic balancing [12] and its generalization, multiple-scale intrinsic harmonic balancing [11]. The approach enables one to obtain systematically a set of simplified rate equations which govern the local dynamics of the system. The solutions for static bifurcations, Hopf bifurcations, and quasi-periodic motions lying on two-and higher-dimensional invariant tori, as well as the associated stability conditions, are expressed in terms of the system coefficients.

Symbolic Computation of Normal Forms for Resonant Double Hopf Bifurcations Using a Perturbation Technique

“…/2. This indicates that the results obtained in this paper using the multiple scales are the same as those given by a di!erent method such as the IHB technique [16].…”

“…Thus, we may assume that A "1 and A "2. This example was studied before using the IHB technique [16]. Since our interest here is focused on the computation of the normal form, we may ignore the perturbation parameter (or simply set "0).…”

“…Since our interest here is focused on the computation of the normal form, we may ignore the perturbation parameter (or simply set "0). The general normal form up to second order was obtained using the IHB method in reference [16] as…”

Analysis of Nonlinear Dynamics and Bifurcations for a Rectangular Symmetric Cross-Ply Laminated Composite Plate with a Parametric Excitation

“…solution bifurcate into a family of periodic solutions, i.e., H.B. solutions (11). Now, Consider the stability of the S.B.…”

“…Langford [8] as well as Yu and Huseyin [9] devoted to the systems characterized by a critical point with a simple zero and a pair of pure imaginary eigenvalues. Yu and Huseyin [10,11] focused on the systems characterized by two pairs of pure imaginary eigenvalues. Later, many physical or engineering systems have been studied to present some complex dynamical behaviors.…”