Conditional Lyapunov exponent criteria in terms of ergodic theory

Shintani, Masaru; Umeno, Ken

doi:10.1093/ptep/ptx168

Cited by 6 publications

(2 citation statements)

References 22 publications

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“…In Ref. [5] the generalized Boole transform with a particular parameter was used as a toy model for clarifying mechanism of a class of synchronizations. Since the generalized Boole transform family has a mixing property, the long-time limit of distribution functions can be estimated.…”

Section: Introductionmentioning

confidence: 99%

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Maps on statistical manifolds exactly reduced from the Perron-Frobenius equations for solvable chaotic maps

Goto

Umeno

2018

Journal of Mathematical Physics

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Maps on a parameter space for expressing distribution functions are exactly derived from the Perron-Frobenius equations for a generalized Boole transform family. Here the generalized Boole transform family is a one-parameter family of maps where it is defined on a subset of the real line and its probability distribution function is the Cauchy distribution with some parameters. With this reduction, some relations between the statistical picture and the orbital one are shown. From the viewpoint of information geometry, the parameter space can be identified with a statistical manifold, and then it is shown that the derived maps can be characterized. Also, with an induced symplectic structure from a statistical structure, symplectic and information geometric aspects of the derived maps are discussed.

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Section: Introductionmentioning

confidence: 99%

“…When α = 1/2, one has (ν, γ) = (0, 1) (see Ref. [5]). This fixed point corresponds to the standard Cauchy distribution.…”

Section: As a Notational Convenience Introducementioning

confidence: 99%

Maps on statistical manifolds exactly reduced from the Perron-Frobenius equations for solvable chaotic maps

Goto

Umeno

2018

Journal of Mathematical Physics

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Chaotic synchronization of mutually coupled systems–arbitrary proportional linear relations

Kano

Umeno

2022

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Considering a system combining two generalized Boolean transformations, we found that depending on the parameters, we can generate generalized synchronization such that the two chaotic orbits have arbitrary proportional linear relations. We rigorously determined its synchronization conditions by the explicit computing conditional Lyapunov exponent using the ergodic property and stable property of the Cauchy distribution. We found that a phenomenon similar to chaotic synchronization occurs even when the synchronization conditions are not strictly satisfied, which exhibits some degree of structural stability of chaotic synchronization. Our model can be further extended to systems with more degrees of freedom and, in the future, can be applied to reservoir computing.

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Evaluation of high-stability optical beats in laser chaos by plasmonic photomixing

et al. 2020

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The stability of optical beats in a chaotically oscillating laser is compared to that of a free-running continuous-wave laser using a highly efficient plasmonic photomixer. Using a chaotically oscillating laser diode, stable optical beats are observed over an operation current range of 60-90 mA. The optical spectra are stable even with frequent mode hopping. In contrast, optical beats in a free-running continuous-wave laser are not stable compared to those of a chaotically oscillating laser, because of intermittent hopping of the laser modes. The high stability of chaotically oscillating lasers makes these lasers promising candidates for optical pump sources in terahertz time-domain spectroscopy systems.

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Conditional Lyapunov exponent criteria in terms of ergodic theory

Cited by 6 publications

References 22 publications

Maps on statistical manifolds exactly reduced from the Perron-Frobenius equations for solvable chaotic maps

Maps on statistical manifolds exactly reduced from the Perron-Frobenius equations for solvable chaotic maps

Chaotic synchronization of mutually coupled systems–arbitrary proportional linear relations

Evaluation of high-stability optical beats in laser chaos by plasmonic photomixing

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