{"title":"Bifurcation of Multiple Periodic Solutions for a Class of Nonlinear Dynamical Systems in $$(m+4)$$ -Dimension","authors":"","doi":"10.1007/s44198-024-00181-5","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>In this paper, we introduce a curvilinear coordinate transformation to study the bifurcation of periodic solutions from a 2-degree-of-freedom Hamiltonian system, when it is perturbed in <span> <span>\\({\\textbf{R}}^{m+4}\\)</span> </span>, where <em>m</em> represents any positive integer. The extended Melnikov function is obtained by constructing a Poincaré map on the curvilinear coordinate frame of the trajectory of the unperturbed system. Then the criteria for bifurcation of periodic solutions of these Hamiltonian systems under isochronous and non-isochronous conditions are obtained. As for its application, we study the number of periodic solutions of a composite piezoelectric cantilever rectangular plate system whose averaged equation can be transformed into a <span> <span>\\((2+4)\\)</span> </span>-dimensional dynamical system. Furthermore, under the two resonance conditions of 1:1 and 1:2, we obtain the periodic solution numbers of this system with the variation of parametric excitation coefficient <span> <span>\\(p_1.\\)</span> </span></p>","PeriodicalId":48904,"journal":{"name":"Journal of Nonlinear Mathematical Physics","volume":"22 1","pages":""},"PeriodicalIF":1.4000,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Nonlinear Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1007/s44198-024-00181-5","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we introduce a curvilinear coordinate transformation to study the bifurcation of periodic solutions from a 2-degree-of-freedom Hamiltonian system, when it is perturbed in \({\textbf{R}}^{m+4}\), where m represents any positive integer. The extended Melnikov function is obtained by constructing a Poincaré map on the curvilinear coordinate frame of the trajectory of the unperturbed system. Then the criteria for bifurcation of periodic solutions of these Hamiltonian systems under isochronous and non-isochronous conditions are obtained. As for its application, we study the number of periodic solutions of a composite piezoelectric cantilever rectangular plate system whose averaged equation can be transformed into a \((2+4)\)-dimensional dynamical system. Furthermore, under the two resonance conditions of 1:1 and 1:2, we obtain the periodic solution numbers of this system with the variation of parametric excitation coefficient \(p_1.\)
期刊介绍：
Journal of Nonlinear Mathematical Physics (JNMP) publishes research papers on fundamental mathematical and computational methods in mathematical physics in the form of Letters, Articles, and Review Articles.
Journal of Nonlinear Mathematical Physics is a mathematical journal devoted to the publication of research papers concerned with the description, solution, and applications of nonlinear problems in physics and mathematics.
The main subjects are:
-Nonlinear Equations of Mathematical Physics-
Quantum Algebras and Integrability-
Discrete Integrable Systems and Discrete Geometry-
Applications of Lie Group Theory and Lie Algebras-
Non-Commutative Geometry-
Super Geometry and Super Integrable System-
Integrability and Nonintegrability, Painleve Analysis-
Inverse Scattering Method-
Geometry of Soliton Equations and Applications of Twistor Theory-
Classical and Quantum Many Body Problems-
Deformation and Geometric Quantization-
Instanton, Monopoles and Gauge Theory-
Differential Geometry and Mathematical Physics
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