Numerical periodic normalization for codim 2 bifurcations of limit cycles with center manifold of dimension higher than 3
This work provides a practical computational tool for analyzing complex bifurcations in high-dimensional dynamical systems, relevant to applied mathematicians and engineers studying nonlinear oscillations.
The authors derive explicit formulas for periodic normal form coefficients for three codim 2 bifurcations of limit cycles with center manifold dimension 4 or 5, enabling analysis of bifurcation scenarios involving 3-tori and 4-tori. They apply the method to three models (laser, population biology, mechanical vibrations) and confirm results via Lyapunov exponents.
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.