Numerical Periodic Normalization for Codim 1 Bifurcations of Limit Cycles

Explicit computational formulas for the coefficients of the periodic normal forms for all codim 1 bifurcations of limit cycles in generic autonomous ODEs are derived. They include second-order coefficients for the fold (limit point) bifurcation, as well as third-order coefficients for the flip (period-doubling) and Neimark--Sacker (torus) bifurcations. 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 sub- and supercritical bifurcations, in agreement with earlier asymptotic expansions of the bifurcating solutions. 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. We include three numerical examples, in which codim 2 singularities are detected along branches of codim 1 bifurcations of limit cycles as zeros of the periodic normal form coefficients.