Stability of the Scenarios Towards Chaos
For the last ten years, the study of chaotic behaviors generated by deterministic dynamical systems has received an increasing interest among Physicists and Mathematicians. A fascinating aspect of this field of research is its interdisciplinarity. The deep reason for this universality lies in the very nature of the subject: The Mathematics of Time . The study of simple dynamical systems, such as the iterations of mappings or the ordinary differential equations thus appears as a study of these Mathematics of time. Two kinds of problems can be easily identified. The first one deals with the transition towards complex behaviors. The second one is concerned by the chaotic behaviors themselves. In this paper, we will only discuss some general aspects of the first class of problems. In fact, these problems are in some sense simpler to study. They generally lead to universal scenarios  by which a periodic or quasiperiodic signal progressively looses its regularity, as an external constraint varies. These problems of transition bear a striking analogy with critical phenomena. The common analytical tool is the renormalization group .
KeywordsChaotic Behavior Critical Phenomenon Stable Manifold Period Doubling Invariant Torus
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