Continuation and bifurcations of periodic orbits and symbolic dynamics in the Swift–Hohenberg equation

IF 3.4 2区数学 Q1 MATHEMATICS, APPLIED

Communications in Nonlinear Science and Numerical Simulation Pub Date : 2024-12-28 DOI:10.1016/j.cnsns.2024.108569

Jakub Czwórnóg , Daniel Wilczak

引用次数: 0

Abstract

Steady states of the Swift–Hohenberg (Swift and Hohenberg, 1977) equation are studied. For the associated four-dimensional ODE we prove that on the energy level

E = 0

two smooth branches of even periodic solutions are created through the saddle–node bifurcation. We also show that these orbits satisfy certain geometric properties, which implies that the system has positive topological entropy for an explicit and wide range of parameter values of the system.

The proof is computer-assisted and it uses rigorous computation of bounds on certain Poincaré map and its higher order derivatives.

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Swift-Hohenberg方程的周期轨道延延分岔与符号动力学

研究了Swift - Hohenberg （Swift and Hohenberg, 1977）方程的稳态。对于相关的四维ODE，我们证明了在能级E=0上通过鞍节点分岔产生了两个偶周期解的光滑分支。我们还证明了这些轨道满足一定的几何性质，这意味着系统具有正的拓扑熵，对于系统的显式和大范围的参数值。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Communications in Nonlinear Science and Numerical Simulation MATHEMATICS, APPLIED-MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

CiteScore

6.80

自引率

7.70%

发文量

378

审稿时长

78 days

期刊介绍： The journal publishes original research findings on experimental observation, mathematical modeling, theoretical analysis and numerical simulation, for more accurate description, better prediction or novel application, of nonlinear phenomena in science and engineering. It offers a venue for researchers to make rapid exchange of ideas and techniques in nonlinear science and complexity. The submission of manuscripts with cross-disciplinary approaches in nonlinear science and complexity is particularly encouraged. Topics of interest: Nonlinear differential or delay equations, Lie group analysis and asymptotic methods, Discontinuous systems, Fractals, Fractional calculus and dynamics, Nonlinear effects in quantum mechanics, Nonlinear stochastic processes, Experimental nonlinear science, Time-series and signal analysis, Computational methods and simulations in nonlinear science and engineering, Control of dynamical systems, Synchronization, Lyapunov analysis, High-dimensional chaos and turbulence, Chaos in Hamiltonian systems, Integrable systems and solitons, Collective behavior in many-body systems, Biological physics and networks, Nonlinear mechanical systems, Complex systems and complexity. No length limitation for contributions is set, but only concisely written manuscripts are published. Brief papers are published on the basis of Rapid Communications. Discussions of previously published papers are welcome.