Introduction to Chotic Dynamical Systems

Preface vii Part One: One-Dimensional Dynamics 1(158) Examples of dynamical systems 2(6) Preliminaries from calculus 8(9) Elementary definitions 17(7) Hyperbolicity 24(7) An example: the quadratic family 31(8) Symbolic dynamics 39(5) Topological conjugacy 44(4) Chaos 48(5) Structural stability 53(7) Sarkovskii's theorem 60(9) The Schwarzian derivative 69(11) Bifurcation theory 80(13) Another view of period three 93(9) Maps of the circle 102(12) Morse-Smale diffeomorphisms 114(8) Homoclinic points and bifurcations 122(8) The period-doubling route to chaos 130(9) The kneading theory 139(8) Genealogy of periodic points 147(12) Part Two: Higher Dimensional Dynamics 159(101) Preliminaries from linear algebra and advanced calculus 161(12) The dynamics of linear maps: two and three dimensions 173(8) The horseshoe map 181(9) Hyperbolic toral automorphisms 190(11) Attractors 201(13) The stable and unstable manifold theorem 214(18) Global results and hyperbolic sets 232(8) The Hopf bifurcation 240(11) The Henon map 251(9) Part Three: Complex Analytic Dynamics 260(69) Preliminaries from complex analysis 261(7) Quadratic maps revisited 268(4) Normal families and exceptional points 272(4) Periodic points 276(7) The Julia set 283(6) The geometry of Julia sets 289(11) Neutral periodic points 300(11) The Mandelbrot set 311(8) An example: the exponential function 319(10) Color Plates 329(3) Index 332