Periodic Solutions of a Singularly Perturbed Delay Differential Equation with Two State-Dependent Delays

Humphries, A. R.; Bernucci, D.A.; Calleja, Renato C.; Homayounfar, Namdar; Snarski, Michael

doi:10.1007/s10884-015-9484-4

Cited by 11 publications

(11 citation statements)

References 33 publications

(93 reference statements)

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“…Mallet-Paret and Nussbaum investigate the existence and form of the slowly oscillating periodic solutions of a singularly perturbed version of (1.2) in detail in [35] and use it as an illustrative example for more general problems in [32,33,34]. This DDE ia also studied in [3,18,19,24,31].…”

Section: Introductionmentioning

confidence: 99%

Lyapunov-Razumikhin techniques for state-dependent delay differential equations

Humphries,

Magpantay

2015

Preprint

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We present Lyapunov stability and asymptotic stability theorems for steady state solutions of general state-dependent delay differential equations (DDEs) using Lyapunov-Razumikhin methods. Our results apply to DDEs with multiple discrete state-dependent delays, which may be nonautonomous for the Lyapunov stability result, but autonomous (or periodically forced) for the asymptotic stability result. Our main technique is to replace the DDE by a nonautonomous ordinary differential equation (ODE) where the delayed terms become source terms in the ODE. The asymptotic stability result and its proof are entirely new, and based on a contradiction argument together with the Arzelà-Ascoli theorem. This approach alleviates the need to construct auxiliary functions to ensure the asymptotic contraction, which is a feature of all other Lyapunov-Razumikhin asymptotic stability results of which we are aware.We apply our results to a state-dependent model equation which includes Hayes equation as a special case, to directly establish asymptotic stability in parts of the stability domain along with lower bounds on the size of the basin of attraction.

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Section: Introductionmentioning

confidence: 99%

Lyapunov-Razumikhin techniques for state-dependent delay differential equations

Humphries,

Magpantay

2015

Preprint

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“…(1). It has been shown [4,13,14,37] that state dependence alone is capable of generating a wealth of dynamical phenomena, including resonant and multi-frequency behavior, in a scalar DDE with two or more state-dependent feedback terms. The special case of system (3) with b = 0 was introduced by Mallet-Paret and Nussbaum.…”

Section: Introductionmentioning

confidence: 99%

“…operator A, given in (13), is a row vector of length 2 of the form 12) with α = 1 (see ( 13)) onto the basis B is…”

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confidence: 99%

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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“…Mallet-Paret and Nussbaum investigate the existence and form of the slowly oscillating periodic solutions of a singularly perturbed version of (2) in detail in [21,22,23,24]. This DDE is also known as the sawtooth equation and has also been studied in [13,14,15,18].…”

Section: Introductionmentioning

confidence: 99%

Generalised Lyapunov-Razumikhin techniques for scalar state-dependent delay differential equations

Magpantay¹,

Humphries²

2020

Discrete &Amp; Continuous Dynamical Systems - S

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We present generalised Lyapunov-Razumikhin techniques for establishing global asymptotic stability of steady-state solutions of scalar delay differential equations. When global asymptotic stability cannot be established, the technique can be used to derive bounds on the persistent dynamics. The method is applicable to constant and variable delay problems, and we illustrate the method by applying it to the state-dependent delay differential equation known as the sawtooth equation, to find parameter regions for which the steady-state solution is globally asymptotically stable. We also establish bounds on the periodic orbits that arise when the steady-state is unstable. This technique can be readily extended to apply to other scalar delay differential equations with negative feedback.

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Periodic Solutions of a Singularly Perturbed Delay Differential Equation with Two State-Dependent Delays

Cited by 11 publications

References 33 publications

Lyapunov-Razumikhin techniques for state-dependent delay differential equations

Lyapunov-Razumikhin techniques for state-dependent delay differential equations

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

Generalised Lyapunov-Razumikhin techniques for scalar state-dependent delay differential equations

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