THE DOUBLE SCROLL FAMILY .1. RIGOROUS PROOF OF CHAOS
THE DOUBLE SCROLL FAMILY .1. RIGOROUS PROOF OF CHAOS
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DOI：
10.1109/tcs.1986.1085869
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发表时间：
1986-11-01
期刊：
影响因子：
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通讯作者：
MATSUMOTO, T
中科院分区：
文献类型：
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作者：
CHUA, LO;KOMURO, M;MATSUMOTO, T
本文给出了双涡卷确实是混沌的严格数学证明。我们的方法是推导出一个线性等价类的分段线性微分方程，其中包括作为特殊情况的双涡卷。两个这样的分段线性向量场线性等价的充分必要条件是它们各自的特征值是彼此的缩放版本。在它们相同的特殊情况下，我们在线性共轭意义下具有精确等价性。在全局分岔的背景下，一个显式的正规形方程推导和参数化的特征值。然后导出各种Poincaré映射的解析表达式，并用于表征双涡卷的诞生和死亡，以及导出解析形式的近似一维映射，这对于进一步的分叉分析是有用的。特别是，分析表达式表征各种半返回映射相关的庞加莱映射被用于在一个关键的方式来证明存在的Shilnikov型同宿轨道，从而建立严格的混沌性质的双涡卷。这些解析表达式也是我们深入分析双卷的诞生（双卷的开始）和死亡（混沌的消亡）的基础。贯穿本文的统一主题是分析双涡卷系统作为一个大家庭的分段线性向量场展开。使用这种方法，我们能够证明，双涡卷的混沌动力学是很常见的，是强大的，因为从Shilnikov定理预测的相关马蹄铁结构稳定。事实上，它表现为一个大家族（事实上，无限多linearlyequivalent电路）的向量场，其相关的分段线性微分方程承担彼此没有相似之处。因此，值得注意的是，作为局部概念的归一化特征值完全决定了系统的全局定性行为。
This paper provides a rigorous mathematical proof that the double scroll is indeed chaotic. Our approach is to derive a linearly equivalent class of piecewise-linear differential equations which includes the double scroll as a special case. A necessary and sufficient condition for two such piecewise-linear vector fields to be linearly equivalent is that their respective eigenvalues be a scaled version of each other. In the special case where they are identical, we have exact equivalence in the sense of linear conjugacy. An explicit normalform equation in the context of global bifurcation is derived and parametrized by their eigenvalues. Analytical expressions for various Poincaré maps are then derived and used to characterize the birth and the death of the double scroll, as well as to derive an approximate one-dimensional map in analytic form which is useful for further bifurcation analysis. In particular, the analytical expressions characterizing various half-return maps associated with the Poincaré map are used in a crucial way to prove the existence of a Shilnikov-type homoclinic orbit, thereby establishing rigorously the chaotic nature of the double scroll. These analytical expressions are also fundamental in our in-depth analysis of the birth (onset of the double scroll) and death (extinction of chaos) of the double scroll. The unifying theme throughout this paper is to analyze the double scroll system as an unfolding of a large family of piecewise-linear vector fields in. Using this approach, we were able to prove that the chaotic dynamics of the double scroll is quite common, and is robust because the associated horseshoes predicted from Shilnikov's theorem are structurally stable. In fact, it is exhibited by a large family (in fact, infinitely many linearlyequivalent circuits) of vector fields whose associated piecewise-linear differential equations bear no resemblance to each other. It is therefore remarkable that the normalized eigenvalues, which is a local concept, completely determine the system's global qualitative behavior.