Synergetic System Analysis for the Delay-Induced Hopf Bifurcation in the Wright Equation

Schanz, Michael; Pelster, Axel

doi:10.1137/s1111111102412802

Cited by 16 publications

(18 citation statements)

References 30 publications

(69 reference statements)

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“…Numerous examples exist where such a situation arise, especially in applications where the dynamics is subject to a delayed feedback component expressing a sigmoidal behavior. It has indeed been shown that the center manifold for a system with quadratic nonlinearity affects the deterministic dynamics [35]. The treatment of the cubic case is analogous to the one we perform here, and is presented elsewhere.…”

Section: Application To the Asymmetrical Transcritical Bifurcationmentioning

confidence: 73%

“…In this section, we will introduce the notation and define the material required to perform the rather standard center manifold reduction of delayed functional differential equations. Reviews and in depth discussions of this method can be found in [30,31,34,35].…”

Section: A Analysis Of Autonomous Ddesmentioning

confidence: 99%

“…The precise point in parameter space where σ U (A) = ∅ is called the instability threshold, and can be quantitatively described by max{Re(λ) = 0|λ ∈ σ(A)}, which occurs in our formulation for = 0. In the neighborhood of a critical point, the center manifold theorem specifies that the stable modes in S are slaved to the dynamics of the unstable modes in U [31,35]. Then…”

Section: B Spectral Analysis and Phase Space Decompositionmentioning

confidence: 99%

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Reduced dynamics for delayed systems with harmonic or stochastic forcing

Lefebvre

Hutt

LeBlanc

et al. 2012

Chaos

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The analysis of nonlinear delay-differential systems (DDE's) subjected to external forcing is difficult due to the infinite dimensionality of the space in which they evolve. To simplify the analysis of such systems, the present work develops a non-homogeneous center manifold (CM) reduction scheme, which allows the derivation of a time-dependent order parameter equation in finite dimensions. This differential equation captures the major dynamical features of the delayed system. The forcing is assumed to be small compared to the amplitude of the autonomous system, in order to cause only small variations of the fixed points and of the autonomous CM. The time-dependent CM is shown to satisfy a non-homogeneous partial differential equation. We first briefly review CM theory for DDE's. Then we show, for the general scalar case, how an ansatz that separates the CM into one for the autonomous problem plus an additional time-dependent order-two correction leads to satisfying results. The paper then details the application to a transcritical bifurcation subjected to single or multiple periodic forcings. The validity limits of the reduction scheme are also highlighted. Finally, we characterize the specific case of additive stochastic driving of the transcritical bifurcation, where additive white noise shifts the mode of the probability density function of the state variable to larger amplitudes. PACS numbers:Delayed feedback systems have a widespread use accross the engineering, physical and biological sciences. In their applications, these are often subjected to temporally varying forcing to implement the active role played by external influences on the dynamics. However, due to the complexity of infinite dimensional systems, the way additive temporal fluctuations interact with retarded dynamics is far from understood. In particular, a statement on the exact role played by non-homogeneous components on the stability of solutions in delayed feedback models is still missing. Here, the authors propose a novel approach to handle that problem using center manifold theory formulated for delay equations, where an explicit time-dependence is taken into account. Their method makes possible to capture the effect of forcing on the stability of an delayed feedback system in the vicinity of a bifurcation, and serves as new strategy to highlight novel non-linear phenomena.

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Section: Application To the Asymmetrical Transcritical Bifurcationmentioning

confidence: 73%

Section: A Analysis Of Autonomous Ddesmentioning

confidence: 99%

Section: B Spectral Analysis and Phase Space Decompositionmentioning

confidence: 99%

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Reduced dynamics for delayed systems with harmonic or stochastic forcing

Lefebvre

Hutt

LeBlanc

et al. 2012

Chaos

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“…Thus our analytical result (27), (29) can be interpreted as the first terms within a spectral representation (33), where the frequency Ω = 2π/T and the Fourier coefficients c 0 , c 1 , c 2 are given by (28) and (30). Analyzing the Hopf bifurcation with a FFT, we numerically determinded Ω, c 0 , c 1 , c 2 as a function of the smallness parameter ε.…”

Section: Numerical Verificationmentioning

confidence: 99%

Analytical and numerical investigations of the phase-locked loop with time delay

Schanz

Pelster

2003

Phys. Rev. E

Self Cite

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We derive the normal form for the delay-induced Hopf bifurcation in the first-order phase-locked loop with time delay by the multiple scaling method. The resulting periodic orbit is confirmed by numerical simulations. Further detailed numerical investigations demonstrate exemplarily that this system reveals a rich dynamical behavior. With phase portraits, Fourier analysis and Lyapunov spectra it is possible to analyze the scaling properties of the control parameter in the period-doubling scenario, both qualitatively and quantitatively. Within the numerical accuracy there is evidence that the scaling constant of the time-delayed phase-locked loop coincides with the Feigenbaum constant δ ≈ 4.669 in one-dimensional discrete systems.

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“…Chimera states, which were initially revealed and investigated in ensembles of coupled oscillators, have also been found in single oscillators with time-delayed feedback [29,30]. It is well-known that in the presence of time delay simple dynamical systems can exhibit complex behavior, such as delay-induced bifurcations [31], delayinduced multistability [32], stabilization of unstable periodic orbits [33] or stationary states [34], to name only a few examples. As noted in [35], there exists an analogy between the behavior of time-delayed systems and the dynamics of ensembles of coupled oscillators or spatially extended systems [36,38].…”

mentioning

confidence: 99%

Delayed-feedback chimera states: Forced multiclusters and stochastic resonance

Семенов

Zakharova

Maistrenko

et al. 2016

EPL

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A nonlinear oscillator model with negative time-delayed feedback is studied numerically under external deterministic and stochastic forcing. It is found that in the unforced system complex partial synchronization patterns like chimera states as well as salt-and-pepper like solitary states arise on the route from regular dynamics to spatio-temporal chaos. The control of the dynamics by external periodic forcing is demonstrated by numerical simulations. It is shown that one-cluster and multi-cluster chimeras can be achieved by adjusting the external forcing frequency to appropriate resonance conditions. If a stochastic component is superimposed to the deterministic external forcing, chimera states can be induced in a way similar to stochastic resonance, they appear, therefore, in regimes where they do not exist without noise.

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Synergetic System Analysis for the Delay-Induced Hopf Bifurcation in the Wright Equation

Cited by 16 publications

References 30 publications

Reduced dynamics for delayed systems with harmonic or stochastic forcing

Reduced dynamics for delayed systems with harmonic or stochastic forcing

Analytical and numerical investigations of the phase-locked loop with time delay

Delayed-feedback chimera states: Forced multiclusters and stochastic resonance

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