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Stability, Instability and Chaos : An Introduction to the Theory of Nonlinear Differential Equations - Paul Glendinning

Stability, Instability and Chaos

An Introduction to the Theory of Nonlinear Differential Equations

Paperback Published: 6th February 1995
ISBN: 9780521425667
Number Of Pages: 404

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By providing an introduction to nonlinear differential equations, Dr. Glendinning aims to equip the student with the mathematical know-how needed to appreciate stability theory and bifurcations. His approach is readable and covers material both old and new to undergraduate courses. Included are treatments of the Poincare-Bendixson theorem, the Hopf bifurcation and chaotic systems.

Industry Reviews

'I have rarely read an introductory matematical book with such pleasure ... Those new graduate students who will use any branch of nonlinear systems theory in their studies, and who have not had the advantage of attending Dr Glendinning's final year undergraduate lectures, should sacrifice their bread and beer for the means to rush out and buy this book. More eminent and senior scientists would equally find it worth the sacrifice of a bottle or two of their favorite claret ... The book is full of excellent and appropriate examples and virtually empty of errors.' J. Brindley, Bulletin of the Institute of Mathematics 'The book has a vigorous style. Readers will also appreciate Glendinning's efforts to make it clear from the start where his discussions are going and what the important results will be ... Exercises for students, provided in each chapter, are of graded difficulty and nicely cover the material ... This book is likely to become a standard undergraduate mathematics text in non-linear differential equations.' Edward Ott, Nature 'This introduction to non-linear differential equations will prove a very useful addition to the JFM reader's library, and will play an important role in the education of future graduate students.' Journal of Fluid Mechanics 'The author writes clearly and carefully, weaving together general results with a steady supply of simple examples and excercises for the reader. Pick a section of the book to read at random and one is completely confident that effort is going to be rewarded; that the author is really explaining rather than simply recounting.' Times Higher Education Supplement

Introductionp. 1
Solving differential equationsp. 6
Existence and uniqueness theoremsp. 12
Phase space and flowsp. 13
Limit sets and trajectoriesp. 19
Exercises 1p. 23
Stabilityp. 25
Definitions of stabilityp. 27
Liapounov functionsp. 32
Strong linear stabilityp. 40
Orbital stabilityp. 46
Bounding functionsp. 47
Non-autonomous equationsp. 50
Exercises 2p. 51
Linear Differential Equationsp. 54
Autonomous linear differential equationsp. 55
Normal formsp. 57
Invariant manifoldsp. 66
Geometry of phase spacep. 69
Floquet Theoryp. 70
Exercises 3p. 73
Linearization and Hyperbolicityp. 77
Poincare's Linearization Theoremp. 78
Hyperbolic stationary points and the stable manifold theoremp. 83
Persistence of hyperbolic stationary pointsp. 89
Structural stabilityp. 90
Nonlinear sinksp. 92
The proof of the stable manifold theoremp. 96
Exercises 4p. 100
Two Dimensional Dynamicsp. 102
Linear systems in R[superscript 2]p. 102
The effect of nonlinear termsp. 109
Rabbits and sheepp. 119
Trivial linearizationp. 124
The Poincare indexp. 126
Dulac's criterionp. 129
Pike and eelsp. 130
The Poincare-Bendixson Theoremp. 132
Decay of large amplitudesp. 137
Exercises 5p. 141
Periodic Orbitsp. 145
Linear and nonlinear mapsp. 146
Return mapsp. 150
Floquet Theory revisitedp. 151
Periodically forced differential equationsp. 156
Normal forms for maps in R[superscript 2]p. 157
Exercises 6p. 158
Perturbation Theoryp. 161
Asymptotic expansionsp. 162
The method of multiple scalesp. 165
Multiple scales for forced oscillators: complex notationp. 170
Higher order termsp. 172
Interludep. 172
Nearly Hamiltonian systemsp. 176
Resonance in the Mathieu equationp. 179
Parametric excitation in Mathieu equationsp. 180
Frequency lockingp. 181
Hysteresisp. 184
Asynchronous quenchingp. 188
Subharmonic resonancep. 190
Relaxation oscillatorsp. 194
Exercises 7p. 196
Bifurcation Theory I: Stationary Pointsp. 199
Centre manifoldsp. 201
Local bifurcationsp. 206
The saddlenode bifurcationp. 208
The transcritical bifurcationp. 213
The pitchfork bifurcationp. 215
An examplep. 219
The Implicit Function Theoremp. 222
The Hopf bifurcationp. 224
Calculation of the stability coefficient, ReA(0)p. 238
A canardp. 244
Exercises 8p. 246
Bifurcation Theory II: Periodic Orbits and Mapsp. 249
A simple eigenvalue of +1p. 250
Period-doubling bifurcationsp. 255
The Hopf bifurcation for mapsp. 259
Arnol'd (resonant) tonguesp. 267
Exercises 9p. 271
Bifurcational Miscellanyp. 274
Unfolding degenerate singularitiesp. 274
Imperfection theoryp. 276
Isolasp. 278
Periodic orbits in Lotka-Volterra modelsp. 279
Subharmonic resonance revisitedp. 285
Chaosp. 291
Characterizing chaosp. 293
Period three implies chaosp. 300
Unimodal maps I: an overviewp. 302
Tent mapsp. 307
Unimodal maps II: the non-chaotic casep. 313
Quantitative universality and scalingp. 317
Intermittencyp. 324
Partitions, graphs and Sharkovskii's Theoremp. 327
Exercises 11p. 335
Global Bifurcation Theoryp. 338
An examplep. 339
Homoclinic orbits and the saddle indexp. 344
Planar homoclinic bifurcationsp. 346
Homoclinic bifurcations to a saddle-focusp. 351
Lorenz-like equationsp. 359
Cascades of homoclinic bifurcationsp. 367
Exercises 12p. 373
Notes and further readingp. 377
Bibliographyp. 382
Indexp. 386
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521425667
ISBN-10: 0521425662
Series: Cambridge Texts in Applied Mathematics
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 404
Published: 6th February 1995
Country of Publication: GB
Dimensions (cm): 22.76 x 15.49  x 2.46
Weight (kg): 0.58

Earn 180 Qantas Points
on this Book