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Periodic normal forms for the codim 2 bifurcations of limit cycles up to a3-dimensional center manifold in generic autonomous ODEs and computationalformulas for their coefficients are derived. The formulas are independent ofthe dimension of the phase space and involve solutions of certainboundary-value problems on the interval [0, T ], where T is the period of thecritical cycle, as well as multilinear functions from the Taylor expansion ofthe right-hand sides near the cycle. The formulas allow us to distinguishbetween various bifurcation scenarios near codim 2 bifurcations. Ourformulation makes it possible to use robust numerical boundary-value algorithmsbased on orthogonal collocation, rather than shooting techniques, which greatlyexpands its applicability. The actual implementation is described in detailwith numerical examples.

Numerical Periodic Normalization for Codim 2 Bifurcations of Limit Cycles