The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors
The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors
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DOI：
10.1007/978-1-4612-5767-7
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发表时间：
1982-12
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影响因子：
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通讯作者：
C. Sparrow
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文献类型：
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作者：
C. Sparrow
洛伦兹方程以艾德·洛伦兹的名字命名，他首先将洛伦兹方程作为二维对流的模型引入[21]，在过去的20年左右的时间里，由于许多不同的原因，洛伦兹方程一直很重要。最初，它们之所以引人注目，仅仅是因为它们是一个简单的三维非线性自治常微分方程系统，表现出混沌行为;尽管现在已知许多这样的系统（如本卷其他地方所述），但在1963年，这样的系统几乎闻所未闻。事实上，尽管洛伦茨的原始论文很美，而且他在理解他的系统行为方面取得了显著的进步，但这篇论文（和思想）在近十年的时间里基本上被忽视了。此外，当然，方程是重要的，因为他们的联系与问题，他们来自：人们希望湍流现象可以用简单的有限维系统来模拟，但这方面也在很大程度上被忽视了。（以及混沌行为的概念被更广泛地接受），洛伦兹方程在许多论文中都有特色。在一些情况下，重要的行为类型首先在洛伦兹方程中进行了研究，尽管这些行为后来被认为是许多系统的典型行为。例如Manneville和Pomeau的工作，对双折射[24]和Henon地图[18]都出现在文件的洛伦兹系统。同时，在小范围的参数值的方程的特定的几何模型被开发。这项工作是由威廉姆斯[38，39]和Guckenheimer [14，17]领导的，尽管所讨论的参数值的范围实际上是洛伦兹在他的原始论文中研究的。这种几何
The Lorenz equations, named after Ed Lorenz who first introduced them as a model of a two-dimensional convection [21], have been important for a number of different reasons at various times in the past 20 years or so. Initially they were remarkable just because they are a simple threedimensional nonlinear system of autonomous ordinary differential equa tions showing chaotic behaviour; though many such systems are now known (as described elsewhere in this volume), in 1963 such systems were almost unheard of. So much so, in fact, that, despite the beauty of Lorenz's original paper, and the remarkable progress he made in understanding the behaviour of his system, the paper (and the ideas) were largely ignored for nearly ten years. Also, of course, the equations were of importance because of their connection with the problem from which they were derived: here was the hope that turbulent phenomena could be modelled by simple finite-dimensional systems, but this aspect too was largely ignored.In the 1970s, with the burgeoning interest in dynamical systems (and the wider acceptance of the notion of chaotic behaviour), the Lorenz equations featured in many papers. In several cases, important types of behaviour were first investigated in the Lorenz equations although these behaviours have since been recognised as typical of many systems. For instance Manneville and Pomeau's work on intermittence [24] and the Henon map [18] both appeared in papers on the Lorenz system. At the same time, a specific geometric model of the equations in a small range of parameter values was developed. This work was led by Williams [38, 39] and Guckenheimer [14, 17], though the range of parameter values in question was actually that studied by Lorenz in his original paper. This geometric