Organization of spatially periodic solutions of the steady Kuramoto–Sivashinsky equation

Dong, Chengwei; Lan, Yueheng

doi:10.1016/j.cnsns.2013.09.040

Cited by 22 publications

(9 citation statements)

References 32 publications

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“…In the previous work, we used the variational method effectively in calculating the periodic orbits in low-and high-dimensional systems, such as the Kuramoto-Sivashinsky equation and its steady-state solutions, [22,23] the Rössler flow, [24] and the Rydberg atom in crossed electromagnetic fields. [25] It is obvious that this method is applicable to the Lorenz flow.…”

Section: Variational Methods For Finding Periodic Orbits In General Flowmentioning

confidence: 99%

Topological classification of periodic orbits in Lorenz system

Dong

2018

Chinese Phys. B

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We systematically investigate the periodic orbits of the Lorenz flow up to certain topological length. As an alternative to Poincaré section map analysis, we propose a new approach for establishing one-dimensional symbolic dynamics based on the topological structure of the orbit. A newly designed variational method is stable numerically for cycle searching, and two orbital fragments can be used as basic building blocks for initialization. The topological classification based on the entire orbital structure is revealed to be effective. The deformation of periodic orbits with the change of parameters provides a chart to the periods of cycles. The current research may provide a methodology for finding and systematically classifying periodic orbits in other similar chaotic flows.

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Section: Variational Methods For Finding Periodic Orbits In General Flowmentioning

confidence: 99%

Topological classification of periodic orbits in Lorenz system

Dong

2018

Chinese Phys. B

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“…We calculated the unstable periodic orbits for parameter values of (a, b) � (2, 6.7) for the Rucklidge system. In the application of the variational method, there are many ways to initialize the problem [44]. First, a conjecture loop that is suitable for periodic orbit calculations is a prerequisite.…”

Section: Symbolic Encoding Of Periodic Orbits With Two Letters Formentioning

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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“…The only significant modification that we have implemented is that the loop derivative operator d/dt is approximated using an eight-order accurate finite-difference stencil (instead of fourth order), enhancing the overall accuracy/cost ratio and allowing longer orbits to be found. The same high-order discretisation is used for the solution of the tangent problem (24). Initial guesses are obtained from near recurrences of the chaotic flow.…”

Section: A Search Of Long Periodic Orbitsmentioning

confidence: 99%

“…Obtaining all short cycles up to a given topological length, i.e. low-period orbits identified by a short symbol se-quence [20,24], may be practical for low-dimensional systems (see e.g. Ref.…”

Section: Introductionmentioning

confidence: 99%

Sensitivity of long periodic orbits of chaotic systems

Lasagna

2020

Phys. Rev. E

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The properties of long, numerically-determined periodic orbits of two low-dimensional chaotic systems, the Lorenz equations and the Kuramoto-Sivashinsky system in a minimal-domain configuration, are examined. The primary question is to establish whether the sensitivity of period averaged quantities with respect to parameter perturbations computed over long orbits can be used as a sufficiently good proxy for the response of the chaotic state to finite-amplitude parameter perturbations. To address this question, an inventory of thousands of orbits at least two-order of magnitude longer than the shortest admissible cycles is constructed. The expectation of period averages, Floquet exponents and sensitivities over such set is then obtained. It is shown that all these quantities converge to a limiting value as the orbit period is increased. However, while period averages and Floquet exponents appear to converge to analogous quantities computed from chaotic trajectories, the limiting value of the sensitivity is not necessarily consistent with the response of the chaotic state, similar to observations made with other shadowing algorithms.

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Organization of spatially periodic solutions of the steady Kuramoto–Sivashinsky equation

Cited by 22 publications

References 32 publications

Topological classification of periodic orbits in Lorenz system

Topological classification of periodic orbits in Lorenz system

Symbolic Encoding of Periodic Orbits and Chaos in the Rucklidge System

Sensitivity of long periodic orbits of chaotic systems

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