Topological classification of periodic orbits in the Kuramoto–Sivashinsky equation

In this paper, we systematically research periodic orbits of the Kuramoto–Sivashinsky equation (KSe). In order to overcome the difficulties in the establishment of one-dimensional symbolic dynamics in the nonlinear system, two basic periodic orbits can be used as basic building blocks to initialize cycle searching, and we use the variational method to numerically determine all the periodic orbits under parameter [Formula: see text] = 0.02991. The symbolic dynamics based on trajectory topology are very successful for classifying all short periodic orbits in the KSe. The current research can be conveniently adapted to the identification and classification of periodic orbits in other chaotic systems.

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Organization of the periodic orbits in the Rössler flow

International Journal of Modern Physics B ◽

10.1142/s0217979218502272 ◽

2018 ◽

Vol 32 (21) ◽

pp. 1850227 ◽

Cited By ~ 5

Author(s):

Chengwei Dong

Keyword(s):

Periodic Orbits ◽

Symbolic Dynamics ◽

Building Blocks ◽

Topological Classification ◽

Systematic Classification ◽

New Approach ◽

One Dimensional ◽

Surface Of Section ◽

Chaotic Flow

In this paper, we systematically investigate the periodic solutions of the Rössler equations up to certain topological length. To overcome the difficulties for a return map that is multivalued and non-invertible in the nonlinear system, we propose a new approach that establishes one-dimensional symbolic dynamics based on the topological structure of the orbit. A newly designed variational method is numerically stable for cycle searching, and two-orbit fragments can be used as basic building blocks to initialize the system. The topological classification based on the whole orbit structure seems more effective than partitioning the Poincaré surface of section. The current research supplies an interesting framework for a systematic classification of periodic orbits in a chaotic flow.

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Topological classification of periodic orbits in the generalized Lorenz-type system with diverse symbolic dynamics

Chaos Solitons & Fractals ◽

10.1016/j.chaos.2021.111686 ◽

2022 ◽

Vol 154 ◽

pp. 111686

Author(s):

Chengwei Dong ◽

Huihui Liu ◽

Qi Jie ◽

Hantao Li

Keyword(s):

Periodic Orbits ◽

Symbolic Dynamics ◽

Type System ◽

Topological Classification

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Unstable cycles for the Burke–Shaw system via variational approach

International Journal of Modern Physics B ◽

10.1142/s0217979219502400 ◽

2019 ◽

Vol 33 (21) ◽

pp. 1950240

Author(s):

Chengwei Dong ◽

Huihui Liu

Keyword(s):

Variational Method ◽

Periodic Orbits ◽

Symbolic Dynamics ◽

Topological Structure ◽

Variational Approach ◽

Chaotic Systems ◽

Dissipative System ◽

Section Method ◽

Varied Structure ◽

Low Dimensional

In this paper, the systematical calculations of the unstable cycles for the Burke–Shaw system (BSS) are presented. In contrast to the Poincaré section method used in previous studies, we used the variational method for the cycle search and established appropriate symbolic dynamics on the basis of the topological structure of the cycles. The variational approach made it easy to continuously track the periodic orbits when the parameters were varied. Structure of the whole cycle in the dissipative system demonstrated that the methodology could be effective in most low-dimensional chaotic systems.

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Invariants of weak equivalence in primitive matrices

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700000316 ◽

2000 ◽

Vol 20 (2) ◽

pp. 611-626 ◽

Cited By ~ 7

Author(s):

RICHARD SWANSON ◽

HANS VOLKMER

Keyword(s):

Inverse Limit ◽

Direct Application ◽

Topological Classification ◽

Weak Equivalence ◽

Transition Matrices ◽

Inverse Limit Spaces ◽

Positive Dimension ◽

One Dimensional ◽

Dimension Group

Weak equivalence of primitive matrices is a known invariant arising naturally from the study of inverse limit spaces. Several new invariants for weak equivalence are described. It is proved that a positive dimension group isomorphism is a complete invariant for weak equivalence. For the transition matrices corresponding to periodic kneading sequences, the discriminant is proved to be an invariant when the characteristic polynomial is irreducible. The results have direct application to the topological classification of one-dimensional inverse limit spaces.

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Map Dynamics Study of the Lorenz Equations

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127497000248 ◽

1997 ◽

Vol 07 (02) ◽

pp. 373-382 ◽

Cited By ~ 8

Author(s):

Olivier Michielin ◽

Paul E. Phillipson

Keyword(s):

Periodic Orbits ◽

Symbolic Dynamics ◽

Strange Attractor ◽

Good Accuracy ◽

Simple Solution ◽

Lorenz Equations ◽

Computer Solution ◽

One Dimensional

The Lorenz equations [Lorenz, 1963], in addition to a strange attractor, display sequences of periodic and aperiodic orbits. Approximate one-dimensional map solutions are heuristically constructed, supplementing previous symbolic dynamics studies, which closely reproduce these sequences. A relatively simple solution reproduces the sequence topology to good accuracy. A second more refined solution reproduces to higher accuracy both the topology and scale of the attractor. The second solution is sufficiently accurate to predict periodic orbits not previously observed and difficult to extract directly from computer solution of the Lorenz equations.

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Periodic orbits analysis for the Zhou system via variational approach

Modern Physics Letters B ◽

10.1142/s0217984919502129 ◽

2019 ◽

Vol 33 (19) ◽

pp. 1950212 ◽

Cited By ~ 2

Author(s):

Chengwei Dong ◽

Lian Jia

Keyword(s):

Fixed Points ◽

Periodic Orbits ◽

Symbolic Dynamics ◽

Variational Approach ◽

Topological Classification ◽

Dissipative Systems ◽

Systematic Calculation ◽

Low Dimensional ◽

General Method

We proposed a general method for the systematic calculation of unstable cycles in the Zhou system. The variational approach is employed for the cycle search, and we establish interesting symbolic dynamics successfully based on the orbits circuiting property with respect to different fixed points. Upon the defined symbolic rule, cycles with topological length up to five are sought and ordered. Further, upon parameter changes, the homotopy evolution of certain selected cycles are investigated. The topological classification methodology could be widely utilized in other low-dimensional dissipative systems.

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