Center Manifolds and Normal Forms for Nonlinearly Periodically Forced DDEs
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866918468700340224 |
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| author | Lentjes, Bram Daniëls, Seppe Follon, Meinder Kuznetsov, Yuri A. |
| author_facet | Lentjes, Bram Daniëls, Seppe Follon, Meinder Kuznetsov, Yuri A. |
| contents | The aim of this paper is to provide an effective framework for analysing bifurcations of equilibria in nonlinearly periodically forced delay differential equations. First, we establish the existence of a periodic smooth finite-dimensional center manifold near a nonhyperbolic equilibrium using the rigorous functional analytic framework of dual semigroups (sun-star calculus). Second, we construct a center manifold parametrization that allows us to describe the local dynamics on the center manifold near the equilibrium in terms of periodically forced normal forms. Third, we present a normalization method to derive explicit computational formulas for the critical normal form coefficients at a bifurcation of interest. In particular, we obtain such formulas for the periodically forced fold and nonresonant Hopf bifurcation. Several examples and indications from the literature confirm the validity and effectiveness of our approach. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_18918 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Center Manifolds and Normal Forms for Nonlinearly Periodically Forced DDEs Lentjes, Bram Daniëls, Seppe Follon, Meinder Kuznetsov, Yuri A. Dynamical Systems Functional Analysis 34K17, 34K18, 34K19, 34L10 The aim of this paper is to provide an effective framework for analysing bifurcations of equilibria in nonlinearly periodically forced delay differential equations. First, we establish the existence of a periodic smooth finite-dimensional center manifold near a nonhyperbolic equilibrium using the rigorous functional analytic framework of dual semigroups (sun-star calculus). Second, we construct a center manifold parametrization that allows us to describe the local dynamics on the center manifold near the equilibrium in terms of periodically forced normal forms. Third, we present a normalization method to derive explicit computational formulas for the critical normal form coefficients at a bifurcation of interest. In particular, we obtain such formulas for the periodically forced fold and nonresonant Hopf bifurcation. Several examples and indications from the literature confirm the validity and effectiveness of our approach. |
| title | Center Manifolds and Normal Forms for Nonlinearly Periodically Forced DDEs |
| topic | Dynamical Systems Functional Analysis 34K17, 34K18, 34K19, 34L10 |
| url | https://arxiv.org/abs/2601.18918 |