Switching to Nonhyperbolic Cycles from Codimension Two Bifurcations of Equilibria of Delay Differential Equations

Bosschaert, Maikel M.; Janssens, Sebastiaan G.; Kuznetsov, Yuri A.

doi:10.1137/19m1243993

Cited by 17 publications

(48 citation statements)

References 62 publications

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“…As far as we know, there are as yet no theoretical results for the pseudospectral approximation considered here. The (numerical) bifurcation theory of delay equations is well developed, see for instance [7] and the references given there. Our analysis of the Hopf bifurcation can be seen as a proof of principle that pseudospectral approximation yields a reliable bifurcation diagram, a reliable 'picture'.…”

Section: Discussionmentioning

confidence: 99%

Pseudospectral approximation of Hopf bifurcation for delay differential equations

de Wolff,

Scarabel,

Lunel

et al. 2020

Preprint

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Pseudospectral approximation reduces DDE (delay differential equations) to ODE (ordinary differential equations). Next one can use ODE tools to perform a numerical bifurcation analysis. By way of an example we show that this yields an efficient and reliable method to qualitatively as well as quantitatively analyse certain DDE. To substantiate the method, we next show that the structure of the approximating ODE is reminiscent of the structure of the generator of translation along solutions of the DDE. Concentrating on the Hopf bifurcation, we then exploit this similarity to reveal the connection between DDE and ODE bifurcation coefficients and to prove the convergence of the latter to the former when the dimension approaches infinity.

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Section: Discussionmentioning

confidence: 99%

Pseudospectral approximation of Hopf bifurcation for delay differential equations

de Wolff,

Scarabel,

Lunel

et al. 2020

Preprint

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“…where q 1 , q 2 , q 3 , q 4 ∈ N 0 , and A q 1 q 2 q 3 q 4 = A (1) q 1 q 2 q 3 q 4 , A (2) q 1 q 2 q 3 q 4 T .…”

Section: Calculation Of Gmentioning

confidence: 99%

“…The complex dynamics arising form double Hopf bifurcation has been recently studied by many authors for various dynamical systems, refering to [20,22,33,44,49] for ordinary differential equations, to [2,3,6,7,14,18,27,31,45,46] for delay differential equations. More recently, based on the theory of normal forms for partial functional differential equations developed by Faria [12], the double Hopf bifurcation in the reaction-diffusion system with delay has attracted the attention of the researchers [4,8,9,24].…”

Section: Introductionmentioning

confidence: 99%

Double Hopf bifurcation analysis in the memory-based diffusion system

Song¹,

Peng²,

Zhang³

2022

Preprint

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In this paper, we derive the algorithm for calculating the normal form of the double Hopf bifurcation that appears in a memory-based diffusion system via taking memory-based diffusion coefficient and the memory delay as the perturbation parameters. Using the obtained theoretical results, we study the dynamical classification near the double Hopf bifurcation point in a predator-prey system with Holling type II functional response. We show the existence of different kinds of stable spatially inhomogeneous periodic solutions, the transition from one kind to the other as well as the coexistence of two types of periodic solutions with different spatial profiles by varying the memory-based diffusion coefficient and the memory delay.

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“…Their capabilities include the continuation of steady states, periodic orbits and their codimensionone bifurcations; see [27,50] for background information. The present version of DDE-BIFTOOL [43] is able to compute center manifold expansions of bifurcations of steady states of codimension up to two [2]. Importantly for this work, the formulation of the DDE in the package DDE-BIFTOOL allows for state-dependent delays, so that the same suite of capabilities is available for state-dependent DDEs [27,43].…”

Section: Introductionmentioning

confidence: 99%

“…The remaining equations are enforced by the orthogonality condition ( 23) on h j κ (θ). This is in contrast to the approach taken in [30] and [2], where the coefficients A j c are forced to be in an a-priori known normal form for a range of standard bifurcations. Since there is no well-known normal form established for the degenerate point DZGH, we keep the terms h j (θ)(x c ; p; q) j orthogonal to the linear spectral projection B † .…”

mentioning

confidence: 95%

Nonlinear effects of instantaneous and delayed state dependence in a delayed feedback loop

Humphries

Krauskopf

Ruschel

et al. 2022

DCDS-B

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<p style='text-indent:20px;'>We study a scalar, first-order delay differential equation (DDE) with instantaneous and state-dependent delayed feedback, which itself may be delayed. The state dependence introduces nonlinearity into an otherwise linear system. We investigate the ensuing nonlinear dynamics with the case of instantaneous state dependence as our starting point. We present the bifurcation diagram in the parameter plane of the two feedback strengths showing how periodic orbits bifurcate from a curve of Hopf bifurcations and disappear along a curve where both period and amplitude grow beyond bound as the orbits become saw-tooth shaped. We then 'switch on' the delay within the state-dependent feedback term, reflected by a parameter <inline-formula><tex-math id="M1">\begin{document}$ b>0 $\end{document}</tex-math></inline-formula>. Our main conclusion is that the new parameter <inline-formula><tex-math id="M2">\begin{document}$ b $\end{document}</tex-math></inline-formula> has an immediate effect: as soon as <inline-formula><tex-math id="M3">\begin{document}$ b>0 $\end{document}</tex-math></inline-formula> the bifurcation diagram for <inline-formula><tex-math id="M4">\begin{document}$ b = 0 $\end{document}</tex-math></inline-formula> changes qualitatively and, specifically, the nature of the limiting saw-tooth shaped periodic orbits changes. Moreover, we show — numerically and through center manifold analysis — that a degeneracy at <inline-formula><tex-math id="M5">\begin{document}$ b = 1/3 $\end{document}</tex-math></inline-formula> of an equilibrium with a double real eigenvalue zero leads to a further qualitative change and acts as an organizing center for the bifurcation diagram. Our results demonstrate that state dependence in delayed feedback terms may give rise to new dynamics and, moreover, that the observed dynamics may change significantly when the state-dependent feedback depends on past states of the system. This is expected to have implications for models arising in different application contexts, such as models of human balancing and conceptual climate models of delayed action oscillator type.</p>

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Switching to Nonhyperbolic Cycles from Codimension Two Bifurcations of Equilibria of Delay Differential Equations

Cited by 17 publications

References 62 publications

Pseudospectral approximation of Hopf bifurcation for delay differential equations

Pseudospectral approximation of Hopf bifurcation for delay differential equations

Double Hopf bifurcation analysis in the memory-based diffusion system

Nonlinear effects of instantaneous and delayed state dependence in a delayed feedback loop

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