Heteroclinic orbits in the T and the Lü systems

Tigan, Gheorghe; Constantinescu, Dana

doi:10.1016/j.chaos.2008.10.024

Cited by 52 publications

(44 citation statements)

References 9 publications

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“…One has recently found that the chaos is very useful and has great potentials in many disciplines, such as information and computer science [1], power system protection [2,3], biomedical system analysis [4], encryption and communication [5], and so on. Furthermore, the research on chaotic dynamics has evolved from the traditional trend of analyzing and understanding chaos [3,4,[6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][28][29][30] to the new direction of controlling and utilizing it [2,21,22].…”

Section: Introductionmentioning

confidence: 99%

Dynamical properties and simulation of a new Lorenz-like chaotic system

2010

Nonlinear Dyn

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This paper formulates a new three-dimensional chaotic system that originates from the Lorenz system, which is different from the known Lorenz system, Rössler system, Chen system, and includes Lü systems as its special case. By using the center manifold theorem, the stability character of its nonhyperbolic equilibria is obtained. The Hopf bifurcation and the degenerate pitchfork bifurcation, the local character of stable manifold and unstable manifold, are also in detail shown when the parameters of this system vary in the space of parameters. Corresponding bifurcation cases are illustrated by numerical simulations, too. The existence or non-existence of homoclinic and heteroclinic orbits of this system is also studied by both rigorous theoretical analysis and numerical simulation.

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Section: Introductionmentioning

confidence: 99%

Dynamical properties and simulation of a new Lorenz-like chaotic system

2010

Nonlinear Dyn

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“…If one denotes the generalized Lorenz system by the sum of the linear part and the nonlinear part, i.e.,ẋ = Ax + F (x), where A = (a ij ) 3×3 , then, for the coefficient matrix A, the Lorenz system satisfies the condition a 12 a 21 > 0 while the Chen system satisfies the condition a 12 a 21 < 0 (So, the Chen system is called a dual system of Lorenz system in the sense defined byČelikovský and Vaȇcek [6,7]), and the Lü system [32] satisfies the condition a 12 a 21 = 0, representing the transition between the Lorenz system and the Chen system. The system (1.1), however, correspondingly takes this forṁ…”

Section: Introductionmentioning

confidence: 99%

Hopf Bifurcation of Codimension One and Dynamical Simulation for a 3d Autonomous Chaotic System

Li¹,

Zhou²

2014

Bulletin of the Korean Mathematical Society

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Abstract. In this paper, a 3D autonomous system, which has only stable or non-hyperbolic equilibria but still generates chaos, is presented. This system is topologically non-equivalent to the original Lorenz system and all Lorenz-type systems. This motivates us to further study some of its dynamical behaviors, such as the local stability of equilibrium points, the Lyapunov exponent, the dissipativity, the chaotic waveform in time domain, the continuous frequency spectrum, the Poincaré map and the forming mechanism for compound structure of its special cases. Especially, with the help of the Project Method, its Hopf bifurcation of codimension one is in detailed formulated. Numerical simulation results not only examine the corresponding theoretical analytical results, but also show that this system possesses abundant and complex dynamical properties not solved theoretically, which need further attention.

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“…Shortly thereafter, Chen [12] studied the existence and non-existence of homoclinic orbits of the Lorenz system. Under certain conditions, Li [13] studied the problem of existence of homoclinic and heteroclinic orbits of the Chen system, and Tigan [14] studied ones of the T system and the Lü system. Recently, the literature [15] also studies the existence of homoclinic and heteroclinic orbits of the Yang system.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of the general Lorenz family

Liu

2011

Nonlinear Dyn

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The present work is devoted to investigating the dynamical entities of the general Lorenz family, which contains four independent parameters. The classical Lorenz system, the Chen system, and the Lü system are all contained by the system considered in this paper as special cases. First, the properties of the equilibria, in particular, the stability of the non-hyperbolic equilibrium obtained by using the center manifold theorem and the technique of the polar transformation, the pitchfork bifurcation and the degenerate pitchfork bifurcation, Hopf bifurcations, and the local stable and unstable manifold character, are all analyzed when the parameters are varied in the space of parameters. Based on the theoretic analysis and numerical simulations, the dynamics of the system are discussed subtly under all kind of the critical state. Second, the properties of the existence of homoclinic and heteroclinic orbits for the system are rigorously studied. Finally, the existence of singularly degenerate heteroclinic cycles for a suitable choice of the parameters is investigated.

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Heteroclinic orbits in the T and the Lü systems

Cited by 52 publications

References 9 publications

Dynamical properties and simulation of a new Lorenz-like chaotic system

Dynamical properties and simulation of a new Lorenz-like chaotic system

Hopf Bifurcation of Codimension One and Dynamical Simulation for a 3d Autonomous Chaotic System

Dynamics of the general Lorenz family

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