arXiv subjects

B. Lentjes

Publications and source records attributed to B. Lentjes.

3 recordsLinked to original sources

arXiv2025

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

Recent work in [53, 54] by the authors 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 discrete DDEs. The actual implementation is described in detail and its effectiveness is demonstrated on various models.

math.DS↗

arXiv2023

Periodic Center Manifolds for DDEs in the Light of Suns and Stars

In this paper we prove the existence of a periodic smooth finite-dimensional center manifold near a nonhyperbolic cycle in classical delay differential equations by using the Lyapunov-Perron method. The results are based on the rigorous functional analytic perturbation framework for dual semigroups (sun-star calculus). The generality of the dual perturbation framework shows that the results extend to a much broader class of evolution equations.

math.DS↗

arXiv2023

Periodic Normal Forms for Bifurcations of Limit Cycles in DDEs

A recent work by the authors on the existence of a periodic smooth finite-dimensional center manifold near a nonhyperbolic cycle in delay differential equations motivates the derivation of periodic normal forms. In this paper, we prove the existence of a special coordinate system on the center manifold that will allow us to describe the local dynamics on the center manifold near the cycle in terms of these periodic normal forms. To construct the linear part of this coordinate system, we prove the existence of time periodic smooth Jordan chains for the original and adjoint system. Moreover, we establish duality and spectral relations between both systems by using tools from the theory of delay equations and Volterra integral equations, dual perturbation theory, duality theory and evolution semigroups.

math.DS↗