Numerical Periodic Normalization at Codim 1 Bifurcations of Limit Cycles in DDEs

Abstract. Recent work by B. Lentjes, L. Spek, M. M. Bosschaert, and Yu. A. Kuznetsov [ J. Dynam. Differential Equations, 37 (2023), pp. 815–858; and J. Differential Equations, 423 (2025), pp. 631–694] on periodic center manifolds and normal forms for bifurcations of limit cycles in delay differential equations (DDEs) motivates the derivation of explicit computational formulas for the critical normal form coefficients of all codimension one bifurcations of limit cycles. In this paper, we derive such formulas via an application of the periodic normalization method in combination with the functional analytic perturbation framework for dual semigroups (sun-star calculus). The explicit formulas allow us to distinguish between nondegenerate, sub- and supercritical bifurcations. To efficiently apply these formulas, we introduce the characteristic operator as this enables us to use robust numerical boundary-value algorithms based on orthogonal collocation. Although our theoretical results are proven in a more general setting, the software implementation and examples focus on DDEs with discrete delays. The actual implementation is described in detail and its effectiveness is demonstrated on various models.