Numerical Periodic Normalization for Codim 2 Bifurcations of Limit Cycles
This work provides a practical numerical framework for analyzing codim 2 bifurcations of limit cycles, which is important for researchers studying dynamical systems and bifurcation theory.
The authors derived periodic normal forms and computational formulas for codim 2 bifurcations of limit cycles in generic autonomous ODEs, enabling robust numerical algorithms based on orthogonal collocation. The method is demonstrated with numerical examples.
Periodic normal forms for the codim 2 bifurcations of limit cycles up to a 3-dimensional center manifold in generic autonomous ODEs and computational formulas for their coefficients are derived. The formulas are independent of the dimension of the phase space and involve solutions of certain boundary-value problems on the interval [0, T ], where T is the period of the critical cycle, as well as multilinear functions from the Taylor expansion of the right-hand sides near the cycle. The formulas allow us to distinguish between various bifurcation scenarios near codim 2 bifurcations. Our formulation makes it possible to use robust numerical boundary-value algorithms based on orthogonal collocation, rather than shooting techniques, which greatly expands its applicability. The actual implementation is described in detail with numerical examples.