Study of double Hopf bifurcation and chaos for an oscillator with time delayed feedback

Yu, Pei; Yuan, Yuejin; Xu, Jian

doi:10.1016/s1007-5704(02)00007-2

Cited by 79 publications

(37 citation statements)

References 5 publications

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“…Such quasiperiodicity in differential-delay equations is well established and has also been studied by e.g. Yu et al [10,11] by investigating Poincaré maps. They also show that chaos naturally evolves via the breakup of tori in the phase space.…”

Section: Introductionmentioning

confidence: 88%

Asymptotic Analysis of the Hopf-Hopf Bifurcation in a Time-delay System

Heckman

Rand²

2012

JAND

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Section: Introductionmentioning

confidence: 88%

Asymptotic Analysis of the Hopf-Hopf Bifurcation in a Time-delay System

Heckman

Rand²

2012

JAND

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“…Let λ=µ+iω be a root of characteristic equation (11). It is easy to see that λ is a root of (11) if and only if λ satisfies…”

Section: Local Analysis and Double Hopf Bifurcationmentioning

confidence: 99%

“…In order to reach a deep and clear understanding of the dynamics of such models, most researchers have limited their study to the of models with a single delay [5,11]. In some papers, multiple delays are considered but there are no self-connection terms and moreover the systems with two delays have been generally investigated [8][9].…”

Section: Introductionmentioning

confidence: 99%

Double Hopf Bifurcation for an Hopfield Eural Network Model With Time Delayed Feedback

Moslehi¹,

Mohammadinejad²,

Motlagh³

2015

Int. J. of Pure and Appl. Math.

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In this paper, we consider a system of delay differential equations which represents the general model of a Hopfield neural networks type. We focus on the case that the corresponding linear system has two pairs of purely imaginary eigenvalues at the trivial equilibrium, giving rise to double Hopf bifurcations. An analytical approach is used to find the explicit expressions for the critical values of the system parameters at which nonresonant or resonant double Hopf bifurcations may occur. We also investigate the occurrence of an double Hopf bifurcation about the trivial equilibrium.

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“…The bifurcation sets consisting of the oscillation behaviors with the different frequencies and quasi-periodic state are obtained in the delay parameter plane in terms of the central manifold reduction and normal form method (Orosz and Stepan 2004;Dombovari et al 2008), which was introduced firstly by Faria and Magalhaes (1995a, b). The double Hopf bifurcation analysis for the differential equation with one single delay can be found in some existing works, such as (Yu et al 2002;Xu et al 2007;Xu and Pei 2008). This approach provides a convenient tool to compute a relatively simple form of the original differential equation, which can be used to analyze the system dynamic behaviors.…”

Section: Introductionmentioning

confidence: 99%

Stability switches and double Hopf bifurcation in a two-neural network system with multiple delays

Song

2013

Cogn Neurodyn

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Time delay is an inevitable factor in neural networks due to the finite propagation velocity and switching speed. Neural system may lose its stability even for very small delay. In this paper, a two-neural network system with the different types of delays involved in self-and neighborconnection has been investigated. The local asymptotic stability of the equilibrium point is studied by analyzing the corresponding characteristic equation. It is found that the multiple delays can lead the system dynamic behavior to exhibit stability switches. The delay-dependent stability regions are illustrated in the delay-parameter plane, followed which the double Hopf bifurcation points can be obtained from the intersection points of the first and second Hopf bifurcation, i.e., the corresponding characteristic equation has two pairs of imaginary eigenvalues. Taking the delays as the bifurcation parameters, the classification and bifurcation sets are obtained in terms of the central manifold reduction and normal form method. The dynamical behavior of system may exhibit the quasi-periodic solutions due to the NeimarkSacker bifurcation. Finally, numerical simulations are made to verify the theoretical results.

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Study of double Hopf bifurcation and chaos for an oscillator with time delayed feedback

Cited by 79 publications

References 5 publications

Asymptotic Analysis of the Hopf-Hopf Bifurcation in a Time-delay System

Asymptotic Analysis of the Hopf-Hopf Bifurcation in a Time-delay System

Double Hopf Bifurcation for an Hopfield Eural Network Model With Time Delayed Feedback

Stability switches and double Hopf bifurcation in a two-neural network system with multiple delays

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