{"title":"Continuation and bifurcations of periodic orbits and symbolic dynamics in the Swift–Hohenberg equation","authors":"Jakub Czwórnóg, Daniel Wilczak","doi":"10.1016/j.cnsns.2024.108569","DOIUrl":null,"url":null,"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 <mml:math altimg=\"si91.svg\" display=\"inline\"><mml:mrow><mml:mi>E</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> 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.","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":"41 1","pages":""},"PeriodicalIF":3.4000,"publicationDate":"2024-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Nonlinear Science and Numerical Simulation","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1016/j.cnsns.2024.108569","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
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.
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.
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