Bifurcation Analysis of Chen's Equation

Ueta, Tetsushi; Chen, Guanrong

doi:10.1142/s0218127400001183

Cited by 384 publications

(143 citation statements)

References 4 publications

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“…1(d) for b ¼ 2.9). 38 First, we study the system (9) with a fixed b ¼ 1.5 and a varying noise intensity e. Recall that an analysis of the stochastic cycles of the Chen system was already reported earlier in Ref. 39.…”

Section: Analysis Of Noise-induced Chaos In the Chen Systemmentioning

confidence: 99%

Analysis of noise-induced transitions from regular to chaotic oscillations in the Chen system

Bashkirtseva¹,

Chen

Ryashko³

2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The stochastically perturbed Chen system is studied within the parameter region which permits both regular and chaotic oscillations. As noise intensity increases and passes some threshold value, noiseinduced hopping between close portions of the stochastic cycle can be observed. Through these transitions, the stochastic cycle is deformed to be a stochastic attractor that looks like chaotic. In this paper for investigation of these transitions, a constructive method based on the stochastic sensitivity function technique with confidence ellipses is suggested and discussed in detail. Analyzing a mutual arrangement of these ellipses, we estimate the threshold noise intensity corresponding to chaotization of the stochastic attractor. Capabilities of this geometric method for detailed analysis of the noiseinduced hopping which generates chaos are demonstrated on the stochastic Chen system. Noise-induced chaos in stochastically perturbed nonlinear deterministic systems with regular attractors has been studied extensively. This phenomenon has received a great deal of interest among researchers from various fields of physics, life sciences, and engineering. Nonlinearity, even in a non-chaotic regime, can imply a non-uniformity of phase portraits, diverse forms of coexicting regular attractors with a complicated geometry of basins of attraction. Under random disturbances, a phase trajectory can cross separatices and induce stochastic hopping between both coexisting regular attractors and their different but close portions. As a consequence of this hopping, the random trajectories with a high probability fall within zones of divergency (local instability), and the dynamics of the perturbed system as a whole becomes chaotic. The positivity of the largest Lyapunov exponent can be used as a measure that this new noise-induced regime is chaotic.In this paper, for the constructive prediction of the noise-induced transitions from regular to chaotic oscillations in a dynamical system, a new stochastic sensitivity function technique is developed and applied. This technique enables to find dispersion ellipses of the random states for the stochastically perturbed dynamical system around deterministic attractors. The sizes of these ellipses are enlarged as noise intensity increases. The constructive method of the dispersion ellipses allows to estimate the threshold of the noise intensity corresponding to the deformation of a regular stochastic attractor to a chaotic one. The effectiveness of this approach is demonstrated by the stochastically perturbed Chen system.

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Section: Analysis Of Noise-induced Chaos In the Chen Systemmentioning

confidence: 99%

Analysis of noise-induced transitions from regular to chaotic oscillations in the Chen system

Bashkirtseva¹,

Chen

Ryashko³

2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…This nonlinear model is based on the chaotic Chen's system [14], and it follows from its bifurcation analysis [15] that this interval system remains to be chaotic when c is fixed to be 28. It has been widely experienced that this chaotic system is relatively difficult to control as compared to the Lorenz system and Chua's system, which are all topologically non-equivalent, due to the prominent three-dimensional and complex topological features of its attractor, especially its rapid change in velocity in the x 3 -direction.…”

Section: Synthesis Of the Simplex Control Law Via Chaos Optimizationmentioning

confidence: 99%

“…Let the parameters of system (12) be a ¼ 45, b ¼ 1:5 and c ¼ 28, so that system (12) generates a periodic solution [15]. Starting from the initial state x r ð0Þ ¼ ½À1:7570; À1:9648; 7:9743 T , the three-dimensional phase portrait and the time-domain response of the reference x r ðtÞ are shown in Figs.…”

Section: Simulation Studymentioning

confidence: 99%

Simplex sliding mode control for nonlinear uncertain systems via chaos optimization

Shieh

Chen

et al. 2005

Chaos, Solitons & Fractals

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“…Since a description of the general case is rather messy and uneasy, at least notationally, to facilitate the description and discussion, we use a specific chaotic system as an example. This system is still not too special, in the sense that it unifies both the Lorenz [2] and the Chen systems [9,10], where the latter is the dual system of the former in a sense defined in [11]: The Lorenz system satisfies the condition a 12 a 21 > 0 while the Chen system satisfies a 12 a 21 < 0, where A ¼ ½a ij 3Â3 are the matrix of the linear part of the chaotic systems. Recently, L€ u u et al [12][13][14] found a unified chaotic system that not only includes the case of a 12 a 21 ¼ 0 [12] but also the Lorenz and Chen systems as two extreme cases.…”

Section: The Problem Formulationmentioning

confidence: 99%