The evolution of a novel four-dimensional autonomous system: Among 3-torus, limit cycle, 2-torus, chaos and hyperchaos

Wu, Wenjuan; Chen, Zengqiang; Yuan, Zhen

doi:10.1016/j.chaos.2007.07.016

Cited by 41 publications

(35 citation statements)

References 37 publications

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“…In 1999, Chen and Ueta presented another well-known three-dimensional chaotic system (Chen et al, 1999), referred to as Chen system that has been proved to be dual and topologically nonequivalent to Lorenz system. Whereafter, Lü system (Lü et al , 2002), a critical system between Lorenz and Chen system, was constructed in 2002, followed by the Lorenz system family (Lü et al, 2003) as a connection of Lorenz, Chen and Lü systems; so far, there are many researchers who studied the chaos theory (Wu et al , 2007). During the last decades dynamic chaos theory has been deeply studied and applied to many fields extensively, such as secure communications, optical system, biology and so forth (Sun, 2004).…”

Section: Introductionmentioning

confidence: 99%

Passive and impulsive synchronization of a new four-dimensional chaotic system

Hamiche¹,

Kemih²,

Ghanes³

et al. 2011

Nonlinear Analysis: Theory, Methods & Applications

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Section: Introductionmentioning

confidence: 99%

Passive and impulsive synchronization of a new four-dimensional chaotic system

Hamiche¹,

Kemih²,

Ghanes³

et al. 2011

Nonlinear Analysis: Theory, Methods & Applications

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“…Recently, a novel four-dimensional hyperchaotic system [20] in which each equation contains a quadratic cross-product term has been constructed. The system can be described by the following non-linear differential equations:…”

Section: Introductionmentioning

confidence: 99%

“…The authors analyzed the evolution processes of the system in [20], where extremely rich dynamical behaviors, including 3-torus, 2-torus, limit cycles, chaotic and hyperchaotic attractors, were explored. In [21], local bifurcations, such as pitchfork bifurcation and Hopf bifurcation, were studied by considering r as the bifurcation parameter.…”

Section: Introductionmentioning

confidence: 99%

3-torus, quasi-periodic bursting, symmetric subHopf/fold-cycle bursting, subHopf/fold-cycle bursting and their relation

Han

Jiang

2010

Nonlinear Dyn

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In this paper, the dynamical behaviors of a perturbed hyperchaotic system is studied. The fast subsystem is examined using local stability and bifurcations, including simple bifurcation, Hopf bifurcation, and fold bifurcation of limit cycle. The results of these analysis are applied to the perturbed hyperchaotic system, where two types of periodic bursting, i.e., symmetric subHopf/fold-cycle bursting and subHopf/fold-cycle bursting, can be observed. In particular, the symmetric subHopf/fold-cycle bursting is new and has not been reported in previous work. With variation of the parameter, subHopf/foldcycle bursting with symmetric structure may bifurcate into two coexisted subHopf/fold-cycle bursting symmetric to each other. Moreover, 3-torus and quasiperiodic bursting (2-torus) are presented. The relation among 3-torus, quasi-periodic bursting, and symmetric subHopf/fold-cycle bursting is discussed, which suggests that 3-torus may develop to quasi-periodic bursting, while quasi-periodic bursting may further evolve to symmetric subHopf/fold-cycle bursting.

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“…Recently, hyperchaos generation and hyperchaotic circuit realization have attracted researchers' increasing attention. Except for Rössler system, many other continuous hyperchaotic systems have been found utilizing one effective method by introducing a feedback controller to a chaotic system [11][12][13][14][15][16][17][18]. These previous works mainly focus on the hyperchaotic phenomenon by analysis of Lyapunov exponents' spectra, bifurcation diagram and phase portraits, but lack of theoretical analysis.…”

Section: Introductionmentioning

confidence: 99%

Hopf bifurcation and intermittent transition to hyperchaos in a novel strong four-dimensional hyperchaotic system

Chen

2009

Nonlinear Dyn

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This paper presents a new four-dimensional autonomous system having complex hyperchaotic dynamics. Basic properties of this new system are analyzed, and the complex dynamical behaviors are investigated by dynamical analysis approaches, such as time series, Lyapunov exponents' spectra, bifurcation diagram, phase portraits. Moreover, when this new system is hyperchaotic, its two positive Lyapunov exponents are much larger than those of hyperchaotic systems reported before, which implies the new system has strong hyperchaotic dynamics in itself. The Kaplan-Yorke dimension, Poincaré sections and the frequency spectra are also utilized to demonstrate the complexity of the hyperchaotic attractor. It is also observed that the system undergoes an intermittent transition from period directly to hyperchaos. The statistical analysis of the intermittency transition process reveals that the mean lifetime of laminar state between bursts obeys the power-law distribution. It is shown that in such four-dimensional continuous system, the occurrence of intermittency may indicate a transition from period to hyperchaos not only to chaos, which provides a possible route to hyperchaos. Besides, the local bifurcation in this system is analyzed and then a Hopf bifurcation is proved to occur when the appropriate bifurcation parameter passes the critical value.All the conditions of Hopf bifurcation are derived by applying center manifold theorem and Poincaré-Andronov-Hopf bifurcation theorem. Numerical simulation results show consistency with our theoretical analysis.

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The evolution of a novel four-dimensional autonomous system: Among 3-torus, limit cycle, 2-torus, chaos and hyperchaos

Cited by 41 publications

References 37 publications

Passive and impulsive synchronization of a new four-dimensional chaotic system

Passive and impulsive synchronization of a new four-dimensional chaotic system

3-torus, quasi-periodic bursting, symmetric subHopf/fold-cycle bursting, subHopf/fold-cycle bursting and their relation

Hopf bifurcation and intermittent transition to hyperchaos in a novel strong four-dimensional hyperchaotic system

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