Fixed point index for iterations of maps, topological horseshoe and chaos
Fixed point index for iterations of maps, topological horseshoe and chaos
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地图、拓扑马蹄形和混沌迭代的定点索引
DOI：
10.12775/tmna.1996.026
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发表时间：
1996
影响因子：
0.7
通讯作者：
P. Zgliczyński
中科院分区：
文献类型：
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作者：
P. Zgliczyński
有许多复杂或混沌动力学的例子，但混沌已经严格证明的例子是相当少的。在大多数情况下，混沌动力学已被证明，该策略涉及分析一个简单的奇异映射或可积问题，然后扰动的结果（见[2]，[5]）。这通常需要对所考虑的映射的导数进行一些估计。另一种解决此类问题的策略是适当地将给定系统同伦到一个模型问题，其中一些代数不变量可以显式计算，并表明这些不变量保持不变。代数不变量的非平凡性提供了对系统动力学复杂性的最小描述。在[3]、[4]中，借助于[6]中引入的离散Conley指标，将这一策略应用于Henon映射和Lorenz方程。在将这种策略应用于具体问题时，我们必须回答三个密切相关的问题：我们将使用什么样的代数不变量，什么是模型映射，什么是适当的同伦。
There are many examples of complicated or chaotic dynamics, but the set of examples for which chaos has been rigorously demonstrated is quite small. In most cases where chaotic dynamics has been proven, the strategy has involved analysing a simple singular map or integrable problem and then perturbing the results (see [2], [5]). This usually required some estimates on the derivatives of mappings under consideration. Another strategy to tackle such problems is to appropriately homotope the given system to a model problem for which some algebraic invariants could be explicitly computed and show that these invariants remain unchanged. Nontriviality of the algebraic invariant provides a minimal description of the complexity of the dynamics of the system. In [3], [4] with the help of the discrete Conley index introduced in [6], this strategy has been applied to the Henon map and the Lorenz equations. In applying this strategy to a concrete problem we must answer three closely related questions: what algebraic invariants we will use, what is the model map, what are the appropriate homotopies.