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An Introduction to Dynamical Systems - D. K. Arrowsmith

An Introduction to Dynamical Systems

Paperback Published: 8th October 1990
ISBN: 9780521316507
Number Of Pages: 432

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Largely self-contained, this is an introduction to the mathematical structures underlying models of systems whose state changes with time, and which therefore may exhibit "chaotic behavior." The first portion of the 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 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. The text contains many worked examples and exercises, many with hints. It will be a valuable first textbook for senior undergraduate and postgraduate students of mathematics, physics, and engineering.

Industry Reviews

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

Diffeomorphisms and flowsp. 1
Introductionp. 1
Elementary dynamics of diffeomorphismsp. 5
Definitionsp. 5
Diffeomorphisms of the circlep. 6
Flows and differential equationsp. 11
Invariant setsp. 16
Conjugacyp. 20
Equivalence of flowsp. 28
Poincare maps and suspensionsp. 33
Periodic non-autonomous systemsp. 38
Hamiltonian flows and Poincare mapsp. 42
Exercisesp. 56
Local properties of flows and diffeomorphismsp. 64
Hyperbolic linear diffeomorphisms and flowsp. 64
Hyperbolic non-linear fixed pointsp. 67
Diffeomorphismsp. 68
Flowsp. 69
Normal forms for vector fieldsp. 72
Non-hyperbolic singular points of vector fieldsp. 79
Normal forms for diffeomorphismsp. 83
Time-dependent normal formsp. 89
Centre manifoldsp. 93
Blowing-up techniques on R[superscript 2]p. 102
Polar blowing-upp. 102
Directional blowing-upp. 105
Exercisesp. 108
Structural stability, hyperbolicity and homoclinic pointsp. 119
Structural stability of linear systemsp. 120
Local structural stabilityp. 123
Flows on two-dimensional manifoldsp. 125
Anosov diffeomorphismsp. 132
Horseshoe diffeomorphismsp. 138
The canonical examplep. 139
Dynamics on symbol sequencesp. 147
Symbolic dynamics for the horseshoe diffeomorphismp. 149
Hyperbolic structure and basic setsp. 154
Homoclinic pointsp. 164
The Melnikov functionp. 170
Exercisesp. 180
Local bifurcations I: planar vector fields and diffeomorphisms on Rp. 190
Introductionp. 190
Saddle-node and Hopf bifurcationsp. 199
Saddle-node bifurcationp. 199
Hopf bifurcationp. 203
Cusp and generalised Hopf bifurcationsp. 206
Cusp bifurcationp. 206
Generalised Hopf bifurcationsp. 211
Diffeomorphisms on Rp. 215
D[subscript x]f(0) = +1: the fold bifurcationp. 218
D[subscript x]f(0) = -1: the flip bifurcationp. 221
The logistic mapp. 226
Exercisesp. 234
Local bifurcations II: diffeomorphisms on R[superscript 2]p. 245
Introductionp. 245
Arnold's circle mapp. 248
Irrational rotationsp. 253
Rational rotations and weak resonancep. 258
Vector field approximationsp. 262
Irrational [beta]p. 262
Rational [beta] = p/q, q [greater than or equal] 3p. 264
Rational [beta] = p/q, q = 1, 2p. 268
Equivariant versal unfoldings for vector field approximationsp. 271
q = 2p. 272
q = 3p. 275
q = 4p. 276
q [greater than or equal] 5p. 282
Unfoldings of rotations and shearsp. 286
Exercisesp. 291
Area-preserving maps and their perturbationsp. 302
Introductionp. 302
Rational rotation numbers and Birkhoff periodic pointsp. 309
The Poincare-Birkhoff Theoremp. 309
Vector field approximations and island chainsp. 310
Irrational rotation numbers and the KAM Theoremp. 319
The Aubry-Mather Theoremp. 332
Invariant Cantor sets for homeomorphisms on S[superscript 1]p. 332
Twist homeomorphisms and Mather setsp. 335
Generic elliptic pointsp. 338
Weakly dissipative systems and Birkhoff attractorsp. 345
Birkhoff periodic orbits and Hopf bifurcationsp. 355
Double invariant circle bifurcations in planar mapsp. 368
Exercisesp. 379
Hints for exercisesp. 394
Referencesp. 413
Indexp. 417
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521316507
ISBN-10: 0521316502
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 432
Published: 8th October 1990
Country of Publication: GB
Dimensions (cm): 23.6 x 18.95  x 2.41
Weight (kg): 0.72