Invariant tori arising at a general critical point of codimension three

Abstract

In this paper the stability, instability, and bifurcation behavior of a nonlinear autonomous system in the vicinity of a generic compound critical point of codimension three are studied in detail. The critical point considered is characterized by a simple zero and two pairs of pure imaginary eigenvalues of the Jacobian, and the system contains three independent parameters. The equilibrium solutions, dynamic bifurcations, and quasi-periodic motions resulting from the interactions of the bifurcation modes and the associated invariant tori are analyzed via the multiple-scale intrinsic harmonic balancing procedure and the unification technique. This approach leads to a set of simplified equivalent differential equations, which in turn yield explicit asymptotic results concerning stationary solutions, periodic and nonresonant quasi-periodic motions that take place on invariant tori. The criteria leading to various bifurcations and the associated stability conditions are derived. An electrical network is analyzed to demonstrate the applicability of the analytic results obtained in this paper.