An Introduction to Chaotic Dynamical Systems
The study of nonlinear dynamical systems has exploded in the past 25 years, and Robert L. Devaney has made these advanced research developments accessible to undergraduate and graduate mathematics students as well as researchers in other disciplines with the introduction of this widely praised book.In this second edition of his best-selling text, Devaney includes new material on the orbit diagram fro maps of the interval and the Mandelbrot set, as well as striking color photos illustrating both Julia and Mandelbrot sets.This book assumes no prior acquaintance with advanced mathematical topics such as measure theory, topology, and differential geometry, Assuming only a knowledge of calculus, Devaney introduces many of the basic concepts of modern dynamical systems theory and leads the reader to the point of current research in several areas. The first two chapters introduce the reader to a broad spectrum of fundamental topics in dynamics: hyperbolicity, symbolic dynamics, structural stability, stable and unstable manifolds and bifurcation theory. Readers familiar with linear algebra and complex analysis will be led to the brink of contemporary research in the book’s concluding chapter, but for anyone with a background in calculus, Devaney provides a comprehensive exploration into the mathematics of chaos.
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Complex Analytic Dynamics
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analytic maps assume attracting fixed point attracting periodic orbit attractor basin of attraction behavior bifurcation diagram Cantor set chain recurrent chapter converges critical point define Definition denote dense depicted in Fig diffeomorphism disk dynamical systems eigenvalues equation example Exercise exists finite fix(l follows function given graph Hence higher dimensional homeomorphism homoclinic point horizontal hyperbolic set hyperbolic toral automorphism infinitely integer invariant inverse irrational rotation itinerary Julia set Lemma linear map Mandelbrot set matrix neighborhood Note open interval parameter period-doubling bifurcation periodic points phase portrait plane point for F points of period polynomial preimage proof Proposition Prove quadratic map real line remark repelling fixed point repelling periodic points result rotation number Sarkovskii's Theorem satisfies Schwarzian derivative sequence stable and unstable stable set structurally stable symbolic dynamics topological conjugacy topologically conjugate torus unimodal map unstable manifolds unstable set vector