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Center Manifolds and Normal Forms for Nonlinearly Periodically Forced DDEs.

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Title: Center Manifolds and Normal Forms for Nonlinearly Periodically Forced DDEs.
Authors: Lentjes, Bram1 (AUTHOR) bram.lentjes@uhasselt.be, Daniëls, Seppe2 (AUTHOR) seppe.daniels@student.kuleuven.be, Follon, Meinder3 (AUTHOR) m.follon@student.tudelft.nl, Kuznetsov, Yuri A.4,5 (AUTHOR) i.a.kouznetsov@uu.nl
Source: International Journal of Bifurcation & Chaos in Applied Sciences & Engineering. Aug2026, Vol. 36 Issue 10, p1-31. 31p.
Subjects: Center manifolds (Mathematics), Normal forms (Mathematics), Hopf bifurcations, Delay differential equations, Bifurcation theory, Functional analysis
Abstract: The aim of this paper is to provide an effective framework for analyzing 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 semi-groups (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. [ABSTRACT FROM AUTHOR]
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Database: Engineering Source
Description
Abstract:The aim of this paper is to provide an effective framework for analyzing 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 semi-groups (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. [ABSTRACT FROM AUTHOR]
ISSN:02181274
DOI:10.1142/S0218127426501427