Chaos near resonance
Chaos near resonance
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DOI：
10.1007/978-1-4612-1508-0
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发表时间：
1999
期刊：
影响因子：
--
通讯作者：
G. Haller
中科院分区：
文献类型：
--
作者：
G. Haller
共振在具有多个自由度的动力系统中普遍存在。它们的基本作用是在进化系统中引入慢速行为，再加上不稳定性，可能会导致高度不规则的行为。本书对共振问题进行了统一的处理，特别强调了最近发现的同宿跳跃现象。在对必要的背景进行调查之后，发展了同宿跳跃的一般有限维理论并举例说明。在耗散和哈密顿量背景下讨论了共振附近混沌的主要机制。描述了先前未发表的关于共振附近的通用同宿分岔以及多脉冲 Silnikov 流形的新结果。研究结果适用于各种不同的问题，包括梁振荡、表面波动力学、非线性光学、大气科学和流体力学的应用。该理论进一步用于研究哈密顿系统中的共振，并应用于分子动力学和刚体运动。最后一章包含有限维理论的无限维扩展，并将其应用于扰动非线性薛定谔方程和耦合 NLS 方程。
Resonances are ubiquitous in dynamical systems with many degrees of freedom. They have the basic effect of introducing slow-fast behavior in an evolutionary system which, coupled with instabilities, can result in highly irregular behavior. This book gives a unified treatment of resonant problems with special emphasis on the recently discovered phenomenon of homoclinic jumping. After a survey of the necessary background, a general finite dimensional theory of homoclinic jumping is developed and illustrated with examples. The main mechanism of chaos near resonances is discussed in both the dissipative and the Hamiltonian context. Previously unpublished new results on universal homoclinic bifurcations near resonances, as well as on multi-pulse Silnikov manifolds are described. The results are applied to a variety of different problems, which include applications from beam oscillations, surface wave dynamics, nonlinear optics, atmospheric science and fluid mechanics. The theory is further used to study resonances in Hamiltonian systems with applications to molecular dynamics and rigid body motion. The final chapter contains an infinite dimensional extension of the finite dimensional theory, with application to the perturbed nonlinear Schrödinger equation and coupled NLS equations.