An Introduction to Dynamical Systems

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Paperback | 8 October 1990

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432 Pages

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In recent years there has been an explosion of research centred on the appearance of so-called 'chaotic behaviour'. This book provides a largely self contained introduction to the mathematical structures underlying models of systems whose state changes with time, and which therefore may exhibit this sort of behaviour. The early part of this book is based on lectures given at the University of London and covers the background to dynamical systems, the fundamental properties of such systems, the local bifurcation theory of flows and diffeomorphisms, Anosov automorphism, the horseshoe diffeomorphism and the logistic map and area preserving planar maps . The authors then go on to consider current research in this field such as the perturbation of area-preserving maps of the plane and the cylinder. This book, which has a great number of worked examples and exercises, many with hints, and over 200 figures, will be a valuable first textbook to both senior undergraduates and postgraduate students in mathematics, physics, engineering, and other areas in which the notions of qualitative dynamics are employed.

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' ... a book which can be recommended unreservedly ... the many exercises are particularly useful.' International Mathematical News

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Non-Fiction Science Physics Classical Mathematics Dynamics & Statics Engineering & Technology Mechanical Engineering & Materials Materials Science Mechanics of Solids Dynamics & Vibration Mathematics Calculus & Mathematical Analysis

Differential Calculus & Equations Applied Mathematics Chaos Theory Number Theory

Preface
Diffeomorphisms and flows	p. 1
Introduction	p. 1
Elementary dynamics of diffeomorphisms	p. 5
Definitions	p. 5
Diffeomorphisms of the circle	p. 6
Flows and differential equations	p. 11
Invariant sets	p. 16
Conjugacy	p. 20
Equivalence of flows	p. 28
Poincare maps and suspensions	p. 33
Periodic non-autonomous systems	p. 38
Hamiltonian flows and Poincare maps	p. 42
Exercises	p. 56
Local properties of flows and diffeomorphisms	p. 64
Hyperbolic linear diffeomorphisms and flows	p. 64
Hyperbolic non-linear fixed points	p. 67
Diffeomorphisms	p. 68
Flows	p. 69
Normal forms for vector fields	p. 72
Non-hyperbolic singular points of vector fields	p. 79
Normal forms for diffeomorphisms	p. 83
Time-dependent normal forms	p. 89
Centre manifolds	p. 93
Blowing-up techniques on R[superscript 2]	p. 102
Polar blowing-up	p. 102
Directional blowing-up	p. 105
Exercises	p. 108
Structural stability, hyperbolicity and homoclinic points	p. 119
Structural stability of linear systems	p. 120
Local structural stability	p. 123
Flows on two-dimensional manifolds	p. 125
Anosov diffeomorphisms	p. 132
Horseshoe diffeomorphisms	p. 138
The canonical example	p. 139
Dynamics on symbol sequences	p. 147
Symbolic dynamics for the horseshoe diffeomorphism	p. 149
Hyperbolic structure and basic sets	p. 154
Homoclinic points	p. 164
The Melnikov function	p. 170
Exercises	p. 180
Local bifurcations I: planar vector fields and diffeomorphisms on R	p. 190
Introduction	p. 190
Saddle-node and Hopf bifurcations	p. 199
Saddle-node bifurcation	p. 199
Hopf bifurcation	p. 203
Cusp and generalised Hopf bifurcations	p. 206
Cusp bifurcation	p. 206
Generalised Hopf bifurcations	p. 211
Diffeomorphisms on R	p. 215
D[subscript x]f(0) = +1: the fold bifurcation	p. 218
D[subscript x]f(0) = -1: the flip bifurcation	p. 221
The logistic map	p. 226
Exercises	p. 234
Local bifurcations II: diffeomorphisms on R[superscript 2]	p. 245
Introduction	p. 245
Arnold's circle map	p. 248
Irrational rotations	p. 253
Rational rotations and weak resonance	p. 258
Vector field approximations	p. 262
Irrational [beta]	p. 262
Rational [beta] = p/q, q [greater than or equal] 3	p. 264
Rational [beta] = p/q, q = 1, 2	p. 268
Equivariant versal unfoldings for vector field approximations	p. 271
q = 2	p. 272
q = 3	p. 275
q = 4	p. 276
q [greater than or equal] 5	p. 282
Unfoldings of rotations and shears	p. 286
Exercises	p. 291
Area-preserving maps and their perturbations	p. 302
Introduction	p. 302
Rational rotation numbers and Birkhoff periodic points	p. 309
The Poincare-Birkhoff Theorem	p. 309
Vector field approximations and island chains	p. 310
Irrational rotation numbers and the KAM Theorem	p. 319
The Aubry-Mather Theorem	p. 332
Invariant Cantor sets for homeomorphisms on S[superscript 1]	p. 332
Twist homeomorphisms and Mather sets	p. 335
Generic elliptic points	p. 338
Weakly dissipative systems and Birkhoff attractors	p. 345
Birkhoff periodic orbits and Hopf bifurcations	p. 355
Double invariant circle bifurcations in planar maps	p. 368
Exercises	p. 379
Hints for exercises	p. 394
References	p. 413
Index	p. 417
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An Introduction to Dynamical Systems

At a Glance

Paperback

Industry Reviews

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How to return your order

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