Nonlinear Systems

Front Cover
Cambridge University Press, Jun 26, 1992 - Mathematics - 317 pages
The theories of bifurcation, chaos and fractals as well as equilibrium, stability and nonlinear oscillations, are part of the theory of the evolution of solutions of nonlinear equations. A wide range of mathematical tools and ideas are drawn together in the study of these solutions, and the results applied to diverse and countless problems in the natural and social sciences, even philosophy. The text evolves from courses given by the author in the UK and the United States. It introduces the mathematical properties of nonlinear systems, mostly difference and differential equations, as an integrated theory, rather than presenting isolated fashionable topics. Topics are discussed in as concrete a way as possible and worked examples and problems are used to explain, motivate and illustrate the general principles. The essence of these principles, rather than proof or rigour, is emphasized. More advanced parts of the text are denoted by asterisks, and the mathematical prerequisites are limited to knowledge of linear algebra and advanced calculus, thus making it ideally suited to both senior undergraduates and postgraduates from physics, engineering, chemistry, meteorology et cetera as well as mathematics.
 

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Contents

Introduction
1
2 The origin of bifurcation theory
3
3 A turning point
6
4 A transcritical bifurcation
10
5 A pitchfork bifurcation
13
6 A Hopf bifurcation
19
7 Nonlinear oscillations of a conservative system
22
8 Difference equations
28
Ordinary differential equations
149
2 Hamiltonian systems
153
3 The geometry of orbits
155
4 The stability of a periodic solution
157
Further reading
161
Secondorder autonomous differential systems
170
2 Linear systems
172
3 The direct method of Liapounov
178

9 An experiment on statics
35
Further reading
37
Problems
38
Classification of bifurcations of equilibrium points
48
2 Classification of bifurcations in one dimension
49
3 Imperfections
55
4 Classification of bifurcations in higher dimensions
59
Further reading
63
Difference equations
68
2 Periodic solutions and their stability
72
3 Attractors and volume
74
32 Volume
80
4 The logistic equation
81
5 Numerical and computational methods
94
6 Some twodimensional difference equations
96
7 Iterated maps of the complex plane
103
Further reading
109
Some special topics
125
2 Dimension and fractals
127
3 Reoonnalization group theory
132
32 Feigenbaums theory of scaling
135
4 Liapounov exponents
140
Further reading
143
Problems
144
4 The LindstedtPoincaré method
181
5 Limit cycles
186
6 Van der Pols equation
190
Further reading
197
Problems
199
Forced oscillations
214
regular perturbation theory
215
3 Weakly nonlinear oscillations near resonance
219
4 Subharmonics
225
Further reading
229
Problems
230
Chaos
233
2 Duffings equation with negative stiffness
246
Melnikovs method
251
4 Routes to chaos
261
5 Analysis of time series
264
Further reading
277
Some partialdifferential problems
283
Additional problems
290
Answers and hints to selected problems
305
Bibliography and author index
316
Motion picture and video index
324
Subject index
325
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