Analysis of bifurcations of limit cycles with Lyapunov exponents and numerical normal forms

Abstract

In this paper we focus on the combination of normal form and Lyapunov exponent computations in the numerical study of the three codim 2 bifurcations of limit cycles with dimension of the center manifold equal to 4 or to 5 in generic autonomous ODEs. The normal form formulas are independent of the dimension of the phase space and involve solutions of certain linear boundary-value problems. The formulas allow one to distinguish between the complicated bifurcation scenarios which can happen near these codim 2 bifurcations, where 3-tori and 4-tori can be present. We apply our techniques to the study of a known laser model, a novel model from population biology, and a model of mechanical vibrations. These models exhibit Limit Point–Neimark–Sacker, Period-Doubling–Neimark–Sacker, and double Neimark–Sacker bifurcations. Lyapunov exponents are computed to numerically confirm the results of the normal form analysis, in particular with respect to the existence of stable invariant tori of various dimensions. Conversely, the normal forms are essential to understand the significance of the Lyapunov exponents.