Spatially extended dislocations produced by the dispersive Swift-Hohenberg equation

Balch, Brenden; Shipman, Patrick D.; Bradley, R. Mark

doi:10.1103/physreve.107.044214

Phys. Rev. E

2023

DOI: 10.1103/physreve.107.044214

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Spatially extended dislocations produced by the dispersive Swift-Hohenberg equation

Brenden Balch

Patrick D. Shipman

R. Mark Bradley

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2024

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Cited by 2 publications

References 26 publications

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Generalization of the Gauss map: A jump into chaos with universal features

Beck,

Tirnakli,

Tsallis

2024

Phys. Rev. E

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The Gauss map (or continued fraction map) is an important dissipative one-dimensional discrete-time dynamical system that exhibits chaotic behavior, and it generates a symbolic dynamics consisting of infinitely many different symbols. Here we introduce a generalization of the Gauss map, which is given by xt+1=1xtα−1xtα where α≥0 is a parameter and xt∈[0,1] (t=0,1,2,3,...). The symbol [⋯] denotes the integer part. This map reduces to the ordinary Gauss map for α=1. The system exhibits a sudden “jump into chaos” at the critical parameter value α=αc≡0.241485141808811⋯ which we analyze in detail in this paper. Several analytical and numerical results are established for this new map as a function of the parameter α. In particular, we show that, at the critical point, the invariant density approaches a q-Gaussian with q=2 (i.e., the Cauchy distribution), which becomes infinitely narrow as α→αc+. Moreover, in the chaotic region for large values of the parameter α we analytically derive approximate formulas for the invariant density, by solving the corresponding Perron-Frobenius equation. For α→∞ the uniform density is approached. We provide arguments that some features of this transition scenario are universal and are relevant for other, more general systems as well. Published by the American Physical Society 2024

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Generalization of the Gauss map: A jump into chaos with universal features

Beck,

Tirnakli,

Tsallis

2024

Phys. Rev. E

View full text Add to dashboard Cite

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Complex Ginzburg–Landau equation for time‐varying anisotropic media

Van Gorder

2024

Stud Appl Math

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When extending the complex Ginzburg–Landau equation (CGLE) to more than one spatial dimension, there is an underlying question of whether one is capturing all the interesting physics inherent in these higher dimensions. Although spatial anisotropy is far less studied than its isotropic counterpart, anisotropy is fundamental in applications to superconductors, plasma physics, and geology, to name just a few examples. We first formulate the CGLE on anisotropic, time‐varying media, with this time variation permitting a degree of control of the anisotropy over time, focusing on how time‐varying anisotropy influences diffusion and dispersion within both bounded and unbounded space domains. From here, we construct a variety of exact dissipative nonlinear wave solutions, including analogs of wavetrains, solitons, breathers, and rogue waves, before outlining the construction of more general solutions via a dissipative, nonautonomous generalization of the variational method. We finally consider the problem of modulational instability within anisotropic, time‐varying media, obtaining generalizations to the Benjamin–Feir instability mechanism. We apply this framework to study the emergence and control of anisotropic spatiotemporal chaos in rectangular and curved domains. Our theoretical framework and specific solutions all point to time‐varying anisotropy being a potentially valuable feature for the manipulation and control of waves in anisotropic media.

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Spatially extended dislocations produced by the dispersive Swift-Hohenberg equation

Cited by 2 publications

References 26 publications

Generalization of the Gauss map: A jump into chaos with universal features

Generalization of the Gauss map: A jump into chaos with universal features

Complex Ginzburg–Landau equation for time‐varying anisotropic media

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