Elements of Differentiable Dynamics and Bifurcation Theory (David Ruelle)
نویسنده
چکیده
This book is a thin gem having two main parts: Part mDifferential Dynamical Systems and Part 2--Bifurcations. The approach to these subjects is a thorough grounding in dynamics on manifolds in finite-dimensional and Banach spaces, invariant sets and attractors, especially stable, unstable, and center manifolds for maps and semiflows; however, there is no mention of inertial manifolds, which are current objects of intensive study. The bifurcations treated include those of fixed points on maps and periodic orbits ofmaps and semiflows: the saddle-node, flip, and Hopf bifurcations for maps, the Hopf bifurcation for semiflows and their Poincar6 maps, Feigenbaum cascades, and the bifurcations arising from homoclinic and heteroclinic intersections. These sections are jewels of lucidity. For the most part, what is going on is explained without detailed proofs. For example, the Hopfbifurcation of a semiflow is related to a possible subsequent Hopf bifurcation of its Poincar6 map that yields a thin torus. A lovely digression to hydrodynamic turbulence and chaos describes the three routes ofMannville and Pommeau (the intermittent-burst scenario), Feigenbaum (period-doubling cascade), and Ruelle and Takens (the quasiperiodic route). There are interesting sections on diffeomorphisms ofthe circle, including Herman’s famous theorem:
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ورودعنوان ژورنال:
- SIAM Review
دوره 32 شماره
صفحات -
تاریخ انتشار 1990