Construction of Quasi-periodic Solutions of State-Dependent Delay Differential Equations by the Parameterization Method I: Finitely Differentiable, Hyperbolic Case

He, Xiaolong; Llave, Rafael de la

doi:10.1007/s10884-016-9522-x

Cited by 25 publications

(20 citation statements)

References 22 publications

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“…This possible route to C k regularity is already proposed in [29]. We also mention that results for invariant tori of state dependent DDEs have been derived recently in spaces of smooth and analytic functions; see [30,31].…”

Section: Normal Form At Hopf-hopf Bifurcationmentioning

confidence: 55%

Resonance Phenomena in a Scalar Delay Differential Equation with Two State-Dependent Delays

Calleja¹,

Humphries²,

Krauskopf³

2017

SIAM J. Appl. Dyn. Syst.

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Abstract. We study a scalar DDE with two delayed feedback terms that depend linearly on the state. The associated constant-delay DDE, obtained by freezing the state dependence, is linear and without recurrent dynamics. With state dependent delay terms, on the other hand, the DDE shows very complicated dynamics. To investigate this, we perform a bifurcation analysis of the system and present its bifurcation diagram in the plane of the two feedback strengths. It is organized by Hopf-Hopf bifurcation points that give rise to curves of torus bifurcation and associated two-frequency dynamics in the form of invariant tori and resonance tongues. We numerically determine the type of the Hopf-Hopf bifurcation points by computing the normal form on the center manifold; this requires the expansion of the functional defining the state-dependent DDE in a power series whose terms up to order three only contain constant delays. We implemented this expansion and the computation of the normal form coefficients in Matlab using symbolic differentiation, and the resulting code HHnfDDE is supplied as a supplement to this article. Numerical continuation of the torus bifurcation curves confirms the correctness of our normal form calculations. Moreover, it enables us to compute the curves of torus bifurcations more globally, and to find associated curves of saddle-node bifurcations of periodic orbits that bound the resonance tongues. The tori themselves are computed and visualized in a three-dimensional projection, as well as the planar trace of a suitable Poincaré section. In particular, we compute periodic orbits on locked tori and their associated unstable manifolds (when there is a single unstable Floquet multiplier). This allows us to study transitions through resonance tongues and the breakup of a 1 : 4 locked torus. The work presented here demonstrates that state dependence alone is capable of generating a wealth of dynamical phenomena.Key words. State-dependent delay differential equations, bifurcation analysis, invariant tori, resonance tongues, Hopf-Hopf bifurcation, normal form computation AMS subject classifications. 34K60, 34K18, 37G05, 37M201. Introduction. Time delays arise naturally in numerous areas of application as an unavoidable phenomenon, for example, in balancing and control [8,19,35,39,64,65,66,67], machining [36], laser physics [40,46,54], agent dynamics [52,53,70,73], neuroscience and biology [1,18,20,42,79], and climate modelling [13,41,48]. Important sources of delays are communication times between components of a system, maturation and reaction times, and the processing time of information received. When they are sufficiently large compared to the relevant internal time scales of the system under consideration, then the delays must be incorporated into its mathematical description. This leads to mathematical models in the form of delay differential equations (DDEs). In many situations the relevant delays can be considered to be fixed; examples are the travel time of light between components of a laser syst...

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Section: Normal Form At Hopf-hopf Bifurcationmentioning

confidence: 55%

Resonance Phenomena in a Scalar Delay Differential Equation with Two State-Dependent Delays

Calleja¹,

Humphries²,

Krauskopf³

2017

SIAM J. Appl. Dyn. Syst.

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“…• Invariant tori for state dependent delay equations: C k /hyperbolic case [51], Analytic/KAM case [52].…”

Section: Parameterization Methods For Unstable Manifolds Of Parabolic Pdementioning

confidence: 99%

Fourier–Taylor parameterization of unstable manifolds for parabolic partial differential equations: Formalism, implementation and rigorous validation

Reinhardt

James

2019

Indagationes Mathematicae

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“…We emphasize that in (1.8) and (1.10) both K and ω are unknown. In [20,21] only the simpler case of quasi-periodically forced systems was considered, so that ω was externally fixed. Remark 1.3.…”

Section: Introductionmentioning

confidence: 99%

Expansions in the delay of quasi-periodic solutions for state dependent delay equations

Casal¹,

Corsi²,

Llave³

2020

J. Phys. A: Math. Theor.

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We consider several models of State Dependent Delay Differential Equations (SDDEs), in which the delay is affected by a small parameter. This is a very singular perturbation since the nature of the equation changes.Under some conditions, we construct formal power series, which solve the SDDEs order by order. These series are quasi-periodic functions of time. This is very similar to the Lindstedt procedure in celestial mechanics.Truncations of these power series can be taken as input for a-posteriori theorems, that show that near the approximate solutions there are true solutions. In this way, we hope that one can construct a catalogue of solutions for SDDEs, bypassing the need of a systematic theory of existence and uniqueness for all initial conditions.

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Construction of Quasi-periodic Solutions of State-Dependent Delay Differential Equations by the Parameterization Method I: Finitely Differentiable, Hyperbolic Case

Cited by 25 publications

References 22 publications

Resonance Phenomena in a Scalar Delay Differential Equation with Two State-Dependent Delays

Resonance Phenomena in a Scalar Delay Differential Equation with Two State-Dependent Delays

Fourier–Taylor parameterization of unstable manifolds for parabolic partial differential equations: Formalism, implementation and rigorous validation

Expansions in the delay of quasi-periodic solutions for state dependent delay equations

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