Numerical periodic normalization for codim 2 bifurcations of limit cycles with center manifold of dimension higher than 3

TL;DR

This paper derives explicit formulas for analyzing complex codim 2 bifurcations of limit cycles with high-dimensional center manifolds, enabling detailed bifurcation scenario classification in various models.

Contribution

It provides dimension-independent formulas for normal form coefficients of codim 2 bifurcations with center manifolds of dimension 4 or 5, facilitating bifurcation analysis.

Findings

Formulas distinguish bifurcation scenarios involving tori and chaos.

Application to models from laser physics, biology, and mechanics.

Numerical Lyapunov exponents confirm theoretical predictions.

Abstract

Explicit computational formulas for coefficients of the periodic normal forms of the three most complex codim 2 bifurcations of limit cycles with dimension of the center manifold equal to 4 or to 5 in generic autonomous ODEs are derived. The resulting formulas are independent of the dimension of the phase space and involve solutions of certain boundary-value problems as well as multilinear functions from the Taylor expansion of the ODE right-hand side near the cycle. The formulas allow one to distinguish between the complicated bifurcation scenarios which can happen near these codim 2 bifurcations of limit cycles, 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 one for 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 and chaos.