Chaotic Hhenomena and Their Engineering Relevance

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操作有关的现象。所使用的方法是数学分析和数值计算的结合，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，并且代表了广泛的应用。这项工作始于带有势井的Duffing型受迫振荡器，重点是逃逸过光滑的势垒，例如，适用于船只在波涛汹涌的海面上倾覆和用激光破坏分子键的问题。进一步的努力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。再次强调了导致稳定性丧失和逃逸的分叉，以及si…在对时滞微分方程的研究中，完成了一个示意性的分岔图，该图总结了计算机模拟的结果，并展示了我们所熟悉的低维分叉现象，它解释了迄今观察到的结果。耦合摆动方程的研究继续：稳定运行的盆地的瞬变时间画像显示盆地内部没有混沌结构，因此只有先前观察到的分形盆地边界是相关的。为了寻找潜在的同宿结构，人们进行了大量的努力，并开发了一种基于长瞬时STADE轨道的新方法，但仍然难以取得成功。回顾已知的类属分叉导致了更广泛和更精细的分类方案，包括结果不确定的重要现象，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一个主要出版物即将完成。与工程控制系统有关的微分差分方程式，即具有滞后性的微分方程式的联合工作已经展开。所研究的特殊方程是Minorsky首次提出的与船舶横摇运动镇定有关的问题。这类问题严格地具有无限维空间，但它有望被大的但有限维的空间所近似。在我们的研究中，我们已经在超过100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、不会失败的启动盆地。了解盆地结构的关键在于边界内的基本集合。最成功的方法是跨轨道方法来定位它们，一个特别重要的结果是，这些不吸引人的集合采取了加倍周期的路径走向混沌。较少

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