Bifurcation of Multiple Periodic Solutions for a Class of Nonlinear Dynamical Systems in $$(m+4)$$ -Dimension

IF 1.4 4区物理与天体物理 Q2 MATHEMATICS, APPLIED Journal of Nonlinear Mathematical Physics Pub Date : 2024-04-03 DOI:10.1007/s44198-024-00181-5

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 $p_1.$

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$$(m+4)$$维非线性动态系统多周期解的分岔

摘要本文引入一种曲线坐标变换来研究二自由度哈密顿系统在 ${\textbf{R}}^{m+4}$ 中受到扰动时周期解的分岔问题，其中 m 代表任意正整数。通过在未扰动系统轨迹的曲线坐标系上构建一个普因卡雷映射，可以得到扩展梅尔尼科夫函数。然后得到了这些哈密顿系统的周期解在等时和非等时条件下的分岔准则。至于其应用，我们研究了一个复合压电悬臂矩形板系统的周期解的数量，该系统的平均方程可以转化为一个（(2+4)\)-维动力系统。此外，在 1:1 和 1:2 两种共振条件下，我们得到了该系统的周期解数随参数激励系数 $p_1.$的变化而变化。

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

Journal of Nonlinear Mathematical Physics PHYSICS, MATHEMATICAL-PHYSICS, MATHEMATICAL

CiteScore

1.60

自引率

0.00%

发文量

审稿时长

3 months

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