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

Lentjes, Bram; Spek, Len; Bosschaert, Maikel M.; Kuznetsov, Yu. А.

doi:10.1007/s10884-023-10289-9

J Dyn Diff Equat

2023

DOI: 10.1007/s10884-023-10289-9

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

Bram Lentjes

Len Spek

Maikel M. Bosschaert

et al.

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Cited by 2 publications

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Period-Doubling Bifurcation of Cycles in Retarded Functional Differential Equations

Záthurecký

2023

J Dyn Diff Equat

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A rigorous description of a period-doubling bifurcation of limit cycles in retarded functional differential equations based on tools of functional analysis and singularity theory is presented. Particularly, sufficient conditions for its occurrence and its normal form coefficients are expressed in terms of derivatives of the operator defining given equations. We also prove the exchange of stability in the case of a non-degenerate period-doubling bifurcation. The approach concerns Fredholm operators, Lyapunov–Schmidt reduction and recognition problem for pitchfork bifurcation.

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Period-Doubling Bifurcation of Cycles in Retarded Functional Differential Equations

Záthurecký

2023

J Dyn Diff Equat

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Periodic Center Manifolds for Nonhyperbolic Limit Cycles in ODEs

Lentjes,

Windmolders,

Kuznetsov

2023

Int. J. Bifurcation Chaos

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In this paper, we deal with a classical object, namely, a nonhyperbolic limit cycle in a system of smooth autonomous ordinary differential equations. While the existence of a center manifold near such a cycle was assumed in several studies on cycle bifurcations based on periodic normal forms, no proofs were available in the literature until recently. The main goal of this paper is to give an elementary proof of the existence of a periodic smooth locally invariant center manifold near a nonhyperbolic cycle in finite-dimensional ordinary differential equations by using the Lyapunov–Perron method. In addition, we provide several explicit examples of analytic vector fields admitting (non)-unique, (non)-[Formula: see text]-smooth and (non)-analytic periodic center manifolds.

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

Cited by 2 publications

References 29 publications

Period-Doubling Bifurcation of Cycles in Retarded Functional Differential Equations

Period-Doubling Bifurcation of Cycles in Retarded Functional Differential Equations

Periodic Center Manifolds for Nonhyperbolic Limit Cycles in ODEs

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