Chaotic Hhenomena and Their Engineering Relevance

混沌现象及其工程相关性

• 批准号: 04044084
• 负责人: UEDA Yoshisuke
• 金额: $ 1.28万
• 依托单位: Kyoto University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for international Scientific Research
• 财政年份: 1992
• 资助国家: 日本
• 起止时间: 1992 至 无数据
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-04044084/
• 关键词: Dynamical systems; Ship capsize; Phase-locked loop; Stability of electric generators; Bifurcations causing loss of stability and escape; 脱出現象; 電力系統; 発電機の安定問題

The aim of this project is to investigate mathematical models of dynamical systems which exhibit chaotic behavior, with particular emphasis on phenomena which may be relevant to the operation of engineering systems. The methods used are a combination of mathematical analysis and numerical computation, especially computer graphics displays of phase space structures. Simple mathematical models are chosen which have the most important nonlinear properties, and which represent a wide range of applications. Work began with forced oscillators of Duffing type with a potential well, with emphasis on escape over a smooth potential barrier, applicable to problems of ship capsize in rough seas and breaking of moleculor bonds with lasers, for example. Further efforts involve a differential-delay equation describing a phase-locked loop, and swing equations which model the stability of electric generators in a power grid. Again emphasis is on bifurcations causing loss of stability and escape, and si … More tuations in which chaotic motions appear just prior to escape.In the study of the delay-differential equation, a schematic bifurcation diagram was completed which summarizes the results of numerois computer simulations, and demonstrates that familiar low-dimensionals, bifurcations explain results observed so far.A manuscript was completed for publication in a referred Journal. Investigation of coupled swing equations continued : transient time portraits of the basin of stable operation revealed no chaotic structure in the basin interior, so that only the previously observed fractal basin boundaries are relevant. An intensive effort to find underlying homoclinic structures was pursued, and a new approach based on long transient staddle orbits was developed, but success is still elusive.Review of known generic bifurcations led to a more extensive and refined classification scheme, including the important phenomena of indeterminate outcome, and the regular or chaotic structure of saddle-type structures which cause instability. A major publication on these fundamental results is nearlying complete.Joint work has been carried out on a differential-difference equation, namely a differential equation with delay, relevant to engineering control systems. The particular equation studied was fist introduced by Minorsky in connection with the stabilization of ship rolling motions.Such a problem has strictly an infinite-dimensional space, but this can hopefully be approximated by a large but finite-dimensional space. In our studies we have worked in a space of over a hundred dimensions. An important problem is the location of basins of attraction, and in particular the basin of safe, non-failing starts. The key to understanding the structure of the basins lies with the basic sets within the boundaries. The straddle-orbit method has been used most successfully to locate these, and a particularly significant result is that these non-attracting sets take a period-doubling route to chaos. Less

这个项目的目的是研究表现出混沌行为的动力系统的数学模型，特别强调可能与工程系统运行相关的现象。所采用的方法是数学分析与数值计算相结合，特别是相空间结构的计算机图形显示。选择简单的数学模型，这些模型具有最重要的非线性性质，并且具有广泛的应用范围。工作开始于Duffing型的强制振荡器，带有一个势阱，重点是在光滑的势垒上逃脱，适用于波涛汹涌的海洋中船舶倾覆和激光破坏分子键的问题。进一步的研究包括描述锁相环的微分延迟方程，以及模拟电网中发电机稳定性的摆动方程。再次强调分叉会导致稳定性和逃逸的丧失，还有更多逃逸之前出现的混沌运动。在对时滞微分方程的研究中，完成了一个示意图的分岔图，它总结了大量计算机模拟的结果，并证明了常见的低维分岔可以解释迄今为止观察到的结果。一篇稿子已完成，准备在某杂志上发表。耦合摆动方程的研究继续：稳定运行的盆地的瞬态时间画像没有显示出盆地内部的混沌结构，因此只有先前观测到的分形盆地边界相关。人们努力寻找潜在的同斜结构，并开发了一种基于长瞬态鞍轨的新方法，但成功仍然是难以捉摸的。对已知的一般分岔的回顾导致了一个更广泛和精细的分类方案，包括不确定结果的重要现象，以及导致不稳定的鞍型结构的规则或混沌结构。关于这些基本结果的主要出版物即将完成。联合研究了与工程控制系统有关的微分-差分方程，即带时滞的微分方程。所研究的特殊方程是由Minorsky首先引入的，与船舶横摇运动的稳定有关。这样的问题有严格的无限维空间，但它可以近似于一个大的有限维空间。在我们的研究中，我们在一百多个维度的空间中工作。一个重要的问题是吸引盆地的位置，特别是安全的、不失败的启动盆地。认识盆地构造的关键在于边界内的基本套。用跨轨方法对这些天体进行定位是最成功的，一个特别重要的结果是，这些非吸引集合以周期加倍的方式走向混沌。少

项目成果

Y.Ueda,H.Ohata and H.B.Stewart: "Bifurcations in a system described by a nonlinear differential equation with delay" 未定.

Y.Ueda、H.Ohata 和 H.B.Stewart：“由具有延迟的非线性微分方程描述的系统中的分岔”TBA。

Y.Ueda,T.Enomoto and H.B.Stewart: "Chaotic transients and fractal str ucture governing coupled swing dynamics" Applied Chaos,Ed.by Jong Kim and John Stringer,John Wiley. 207-218 (1992)

Y.Ueda、T.Enomoto 和 H.B.Stewart：“控制耦合摆动动力学的混沌瞬态和分形结构”Applied Chaos，编者：Jong Kim 和 John Stringer、John Wiley。

J.M.T.Thompson,H.B.Stewart and Y.Ueda: "Safe,explosive,and dangerous bifurcations in dissipative dynamical systems" 未定.

J.M.T.Thompson、H.B.Stewart 和 Y.Ueda：“耗散动力系统中的安全、爆炸性和危险的分岔”待定。

T. Mitsui, Y.Ueda and J.M.T. Thompson: "Analysis of a differential-difference equation by applying staddle orbit method" Technical Report of Inst. Elec. Inf. Comm. Engrs. NLP-93. (1993)

T. Mitsui、Y.Ueda 和 J.M.T.

J.M.T.Thompson: "Global unpredictability in nonlinear dynamics: capture, dispersal and the indeterminate bifurcations" Physica D. 58. 260-272 (1992)

J.M.T.Thompson：“非线性动力学中的全局不可预测性：捕获、扩散和不确定分岔”Physica D. 58. 260-272 (1992)