Numerical Periodic Normalization for Codim 1 Bifurcations of Limit Cycles

Abstract. 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 to distinguish between suband super-critical 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 zeroes of the periodic normal form coefficients.