{"title":"具有双谐波参数激励的真正非线性克莱因-戈登链中弛豫节拍波的出现","authors":"Majdi Gzal, Victor Kislovsky, Yuli Starosvetsky","doi":"10.1016/j.chaos.2024.115692","DOIUrl":null,"url":null,"abstract":"<div><div>This paper provides analytical investigation of nonstationary regimes in a strongly anharmonic Klein-Gordon chain subjected to the two-component parametric excitation. We explore the mechanisms of formation and provide a comprehensive analytical characterization of the dynamics of two distinct highly nonstationary beat-wave regimes, namely the weakly- and strongly- modulated beat-waves. To this end, we derive the double parametrically driven discrete p-Schrodinger model in the neighborhood of 2:2:1 parametric resonance. The obtained non-autonomous slow-flow model depicts the low-energy complex amplitude modulations of coupled oscillators in the vicinity of 2:2:1 resonance. Through a special coordinate transformation, we exactly reduce the slow-flow system dynamics to a beat-wave slow invariant manifold governed by three collective coordinates. To study the complex nonstationary dynamics of beat-waves, we further reduce the overall system dynamics onto the super-slow invariant manifold (SSIM) by applying an additional multi-scale procedure to the system of collective coordinates. Analysis of the system dynamics on the SSIM reveals the two types of non-stationary beat-wave regimes. The first type is a weakly modulated beat-wave response, exhibiting super-slow amplitude modulation without amplitude relaxation. The second, more intriguing type is a strongly modulated beat-wave response, which exhibits rapid amplitude relaxations characterized by two distinct behaviors: one involving rapid amplitude decay to the trivial state, and the other manifested by the recurrent relaxation oscillations. We derive analytical approximations that describe the mechanisms of formation and the entire dynamics of these highly nonstationary beat-wave states. Remarkably, the analytical model aligns satisfactorily with numerical simulations for both weakly and strongly modulated beat-wave states.</div></div>","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":"189 ","pages":"Article 115692"},"PeriodicalIF":5.3000,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Emergence of relaxation beat-waves in genuinely nonlinear Klein-Gordon chain with bi-harmonic parametric excitation\",\"authors\":\"Majdi Gzal, Victor Kislovsky, Yuli Starosvetsky\",\"doi\":\"10.1016/j.chaos.2024.115692\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>This paper provides analytical investigation of nonstationary regimes in a strongly anharmonic Klein-Gordon chain subjected to the two-component parametric excitation. We explore the mechanisms of formation and provide a comprehensive analytical characterization of the dynamics of two distinct highly nonstationary beat-wave regimes, namely the weakly- and strongly- modulated beat-waves. To this end, we derive the double parametrically driven discrete p-Schrodinger model in the neighborhood of 2:2:1 parametric resonance. The obtained non-autonomous slow-flow model depicts the low-energy complex amplitude modulations of coupled oscillators in the vicinity of 2:2:1 resonance. Through a special coordinate transformation, we exactly reduce the slow-flow system dynamics to a beat-wave slow invariant manifold governed by three collective coordinates. To study the complex nonstationary dynamics of beat-waves, we further reduce the overall system dynamics onto the super-slow invariant manifold (SSIM) by applying an additional multi-scale procedure to the system of collective coordinates. Analysis of the system dynamics on the SSIM reveals the two types of non-stationary beat-wave regimes. The first type is a weakly modulated beat-wave response, exhibiting super-slow amplitude modulation without amplitude relaxation. The second, more intriguing type is a strongly modulated beat-wave response, which exhibits rapid amplitude relaxations characterized by two distinct behaviors: one involving rapid amplitude decay to the trivial state, and the other manifested by the recurrent relaxation oscillations. We derive analytical approximations that describe the mechanisms of formation and the entire dynamics of these highly nonstationary beat-wave states. Remarkably, the analytical model aligns satisfactorily with numerical simulations for both weakly and strongly modulated beat-wave states.</div></div>\",\"PeriodicalId\":9764,\"journal\":{\"name\":\"Chaos Solitons & Fractals\",\"volume\":\"189 \",\"pages\":\"Article 115692\"},\"PeriodicalIF\":5.3000,\"publicationDate\":\"2024-11-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Chaos Solitons & Fractals\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S096007792401244X\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Chaos Solitons & Fractals","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S096007792401244X","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
摘要
本文对受到双分量参数激励的强非谐克莱因-戈登链中的非稳态进行了分析研究。我们探讨了两种不同的高度非稳态拍波机制(即弱调制拍波和强调制拍波)的形成机制,并对其动力学特性进行了全面的分析。为此,我们推导了 2:2:1 参数共振附近的双参数驱动离散 p 薛定谔模型。得到的非自治慢流模型描述了耦合振荡器在 2:2:1 共振附近的低能复振幅调制。通过特殊的坐标变换,我们将慢流系统动力学精确还原为由三个集合坐标支配的节拍波慢不变流形。为了研究节拍波的复杂非稳态动力学,我们通过对集合坐标系应用额外的多尺度程序,进一步将整个系统动力学还原为超慢不变流形(SSIM)。对 SSIM 上系统动力学的分析揭示了两种非稳态节拍波状态。第一种是弱调制节拍波响应,表现出超慢振幅调制而无振幅松弛。第二种更有趣的类型是强调制节拍波响应,它表现出快速的振幅松弛,有两种不同的行为特征:一种是振幅快速衰减到微不足道的状态,另一种则表现为反复出现的松弛振荡。我们推导出分析近似值,描述了这些高度非稳态节拍波状态的形成机制和整个动力学过程。值得注意的是,分析模型与弱调制和强调制节拍波状态的数值模拟结果都非常吻合。
Emergence of relaxation beat-waves in genuinely nonlinear Klein-Gordon chain with bi-harmonic parametric excitation
This paper provides analytical investigation of nonstationary regimes in a strongly anharmonic Klein-Gordon chain subjected to the two-component parametric excitation. We explore the mechanisms of formation and provide a comprehensive analytical characterization of the dynamics of two distinct highly nonstationary beat-wave regimes, namely the weakly- and strongly- modulated beat-waves. To this end, we derive the double parametrically driven discrete p-Schrodinger model in the neighborhood of 2:2:1 parametric resonance. The obtained non-autonomous slow-flow model depicts the low-energy complex amplitude modulations of coupled oscillators in the vicinity of 2:2:1 resonance. Through a special coordinate transformation, we exactly reduce the slow-flow system dynamics to a beat-wave slow invariant manifold governed by three collective coordinates. To study the complex nonstationary dynamics of beat-waves, we further reduce the overall system dynamics onto the super-slow invariant manifold (SSIM) by applying an additional multi-scale procedure to the system of collective coordinates. Analysis of the system dynamics on the SSIM reveals the two types of non-stationary beat-wave regimes. The first type is a weakly modulated beat-wave response, exhibiting super-slow amplitude modulation without amplitude relaxation. The second, more intriguing type is a strongly modulated beat-wave response, which exhibits rapid amplitude relaxations characterized by two distinct behaviors: one involving rapid amplitude decay to the trivial state, and the other manifested by the recurrent relaxation oscillations. We derive analytical approximations that describe the mechanisms of formation and the entire dynamics of these highly nonstationary beat-wave states. Remarkably, the analytical model aligns satisfactorily with numerical simulations for both weakly and strongly modulated beat-wave states.
期刊介绍:
Chaos, Solitons & Fractals strives to establish itself as a premier journal in the interdisciplinary realm of Nonlinear Science, Non-equilibrium, and Complex Phenomena. It welcomes submissions covering a broad spectrum of topics within this field, including dynamics, non-equilibrium processes in physics, chemistry, and geophysics, complex matter and networks, mathematical models, computational biology, applications to quantum and mesoscopic phenomena, fluctuations and random processes, self-organization, and social phenomena.