Topological classification of periodic orbits in Lorenz system

Dong, Chengwei

doi:10.1088/1674-1056/27/8/080501

Cited by 9 publications

(3 citation statements)

References 30 publications

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“…This method is not only suitable for the determination of periodic orbits but also for the homoclinic and heteroclinic orbits [32]. In the previous work, the periodic orbits in various chaotic systems were calculated efficiently using the variational method [33][34][35][36], which illustrates the practicability of this method in the GLTS.…”

Section: Numerical Implementationmentioning

confidence: 90%

Unstable periodic orbits analysis in the generalized Lorenz-type system

Dong¹,

Liu²,

Li³

2020

J. Stat. Mech.

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In this paper, we investigated the unstable periodic orbits of a nonlinear chaotic generalized Lorenz-type system. By means of the variational method, appropriate symbolic dynamics are put forward, and the homotopy evolution approach, which can be used in the initialization of the cycle search, is introduced. Fourteen short unstable periodic orbits with different topological lengths, under specific parameter values, are calculated. We also explored the continuous deformation for part of the orbits while changing the parameter values, which provides a new approach to observe various bifurcations. The scale transformation of the generalized Lorenz-type system leads to a single parameter system known as the diffusionless Lorenz equations. By systematically calculating the periodic orbits in the diffusionless Lorenz equations, our research shows the efficiency of this topological classification method for the periodic solutions in the variants of a classical Lorenz system.

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Section: Numerical Implementationmentioning

confidence: 90%

Unstable periodic orbits analysis in the generalized Lorenz-type system

Dong¹,

Liu²,

Li³

2020

J. Stat. Mech.

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“…In this case, the topological structures of the unstable periodic orbits in the phase space become more complicated, which is worthy of further study. To calculate the unstable cycles effectively, the variational method is used for the calculations [32], and the applicability of this method is verified by various chaotic systems [36][37][38], including dissipative or Hamilton systems. We will review the variational calculation method in the next section.…”

Section: Dynamics Of the Rucklidge Systemmentioning

confidence: 99%

Symbolic Encoding of Periodic Orbits and Chaos in the Rucklidge System

Dong

Jia

et al. 2021

Complexity

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To describe and analyze the unstable periodic orbits of the Rucklidge system, a so-called symbolic encoding method is introduced, which has been proven to be an efficient tool to explore the topological properties concealed in these periodic orbits. In this work, the unstable periodic orbits up to a certain topological length in the Rucklidge system are systematically investigated via a proposed variational method. The dynamics in the Rucklidge system are explored by using phase portrait analysis, Lyapunov exponents, and Poincaré first return maps. Symbolic encodings of the periodic orbits with two and four letters based on the trajectory topology in the phase space are implemented under two sets of parameter values. Meanwhile, the bifurcations of the periodic orbits are explored, significantly improving the understanding of the dynamics of the Rucklidge system. The multiple-letter symbolic encoding method could also be applicable to other nonlinear dynamical systems.

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“…In the previous work, we used the variational method effectively in calculating the periodic orbits for dynamical systems with fractal attractors or repellers, such as the Kuramoto-Sivashinsky equation and its steady-state solutions [17,18], the Rössler flow [19], and the Lorenz chaotic family [20]. We also used the method successfully in a conservative system, e.g., the Rydberg atom in crossed electromagnetic fields [21].…”

Section: Yangmentioning

confidence: 99%

Topological classification of periodic orbits in the Yang-Chen system

Dong

2018

EPL

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This paper studies the periodic orbits of the Yang-Chen system. As an alternative to the Poincaré section map analysis, a new approach for establishing one-dimensional symbolic dynamics is proposed, and the periodic orbits are systematically calculated with the variational method. Two cycles can be used as basic building blocks for initialization searching, and the topological classification based on the entire orbital structure is revealed to be effective. Furthermore, the deformation of periodic orbits with the change of parameters is discussed, which provides a chart to the periods of cycles. The current research may provide a methodology for systematic calculation and classification of periodic orbits in other similar chaotic flows.

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Topological classification of periodic orbits in Lorenz system

Cited by 9 publications

References 30 publications

Unstable periodic orbits analysis in the generalized Lorenz-type system

Unstable periodic orbits analysis in the generalized Lorenz-type system

Symbolic Encoding of Periodic Orbits and Chaos in the Rucklidge System

Topological classification of periodic orbits in the Yang-Chen system

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