The dynamics of physical, chemical, biological, or fluid systems generally must be described by nonlinear models, whose detailed mathematical solutions are not obtainable. To understand some aspects of such dynamics, various complementary methods and viewpoints are of crucial importance. In this book the perspectives generated by analytical, topological and computational methods, and interplays between them, are developed in a variety of contexts. This book is a comprehensive introduction to this field, suited to a broad readership, and reflecting a wide range of applications. Some of the concepts considered are: topological equivalence; embeddings; dimensions and fractals; PoincarГ© maps and map-dynamics; empirical computational sciences vis-ГЎ-vis mathematics; Ulam's synergetics; Turing's instability and dissipative structures; chaos; dynamic entropies; Lorenz and Rossler models; predator-prey and replicator models; FPU and KAM phenomena; solitons and nonsolitons; coupled maps and pattern dynamics; cellular automata. The areas in which these concepts appear include optics, geophysics, meteorology, hydrodynamics, plasma physics, accelerators, astrophysics, chemical dynamics, lattice dynamics, ecology, mathematical biology, electrical and mechanical systems. A number of experimental studies, which employ these theoretical concepts in these fields of research, are also discussed. The presentation and style is intended to stimulate the reader's imagination to apply these methods to a host of problems and situations. The text is complemented by copious references, extensive historical and bibliographic notes, exercises and examples, and appendices giving more details of some mathematical ideas. Each chapter includes an extensive section commentary on the exercises and their solution. Graduate students and research workers in physics, applied mathematics, chemistry, biology, and engineering will welcome this book as the first broad introduction to this important major field of research.
Author(s): E. Atlee Jackson
Publisher: CUP
Year: 1989
Language: English
Pages: 516
Contents of Volume 1......Page 5
Preface......Page 9
Acknowledgements......Page 12
Concepts related to nonlinear dynamics: historical outline......Page 13
1.1 ...there was Poincare......Page 21
1.2 What are `nonlinear phenomena'?; projections, models, and some relationships between linear and nonlinear differential equation......Page 23
1.3 Two myths: a linear and analytic myth......Page 28
1.4 Remarks on modeling: pure mathematics vis a vis `empirical' mathematics......Page 30
1.5 The ordering and organization of ideas......Page 32
1.6 Some thoughts: Albert Einstein, Victor Hugo, A.B. Pippard, Richard Feynman, Henri Poincare......Page 35
Comments on exercises......Page 36
2.1 Dynamic equations; topological orbital equivalence......Page 38
2.2 Existence, uniqueness and constants of the motion......Page 50
2.3 Types of stabilities: Lyapunov, Poincare, Lagrange; Lyapunov exponents; global stability; the Lyapunov function......Page 61
2.4 Integral invariants: the Poincare integral invariants; generalized Liouville theorem; unbounded solutions, Liouville theorem on integral manifolds......Page 64
2.5 More abstract dynamic systems......Page 70
2.6 Dimensions and measures of sets......Page 78
Comments on exercises......Page 86
3.1 Selected dynamic aspects......Page 93
3.2 Control space effects......Page 102
3.3 Structural stability, gradient systems and elementary catastrophe sets......Page 122
3.4 Thom's `universal unfolding' and general theorem (for k 5 5): Brief summary......Page 128
3.5 Catastrophe machines: Poston's (k = 1); Benjamin's (k = 1); Zeeman's (k = 2)......Page 138
3.6 The optical bistability cusp catastrophe set......Page 150
3.7 Some of Rene Thom's perspectives......Page 154
Comments on exercises......Page 156
4.1 General considerations......Page 162
4.2 Two-to-one maps: the logistic map......Page 168
4.3 Universal sequences and scalings......Page 178
4.4 Tangent bifurcations, intermittencies: windows, microcosms, crisis......Page 189
4.5 Characterizing `deterministic chaos': partitioning phase space; correspondence with Bernoulli sequences; Li-Yorke characterization of chaos; other characterizations......Page 192
4.6 Lyapunov exponents: sensitivity to initial condition vs. attractors; a strange attractor concept......Page 202
4.7 The dimensions of `near self-similar' cantor sets......Page 207
4.8 Invariant measures, mixing and ergodicity: the mixed drinks of Arnold, Avez, and Halmos......Page 210
4.9 The circle map: model of coupled oscillators; rotation number, entrainment, Arnold `tongues'; chaotic region......Page 217
4.10 The `suspension' of a tent map......Page 226
4.11 Mathematics, computations and empirical sciences; THE FINITE vs. THE INFINITE; pseudo-orbits, #-shadowing; discrete logistic map, where is the chaos?......Page 230
Comments on exercises......Page 239
5.1 The phase plane......Page 246
5.2 Integrating factors: a few examples......Page 260
5.3 Poincare's index of a curve in a vector field: Brouwer's fixed point theorem......Page 263
*Preview of coming attractions*......Page 271
5.4 The pendulum and polynomial oscillators: elliptic functions, frequency shift, heteroclinic and homoclinic orbits......Page 273
5.5 The averaging method of Krylov-Bogoliubov-Mitropolsky (KBM): autonomous systems; eliminating secular terms, the Duffing equation (passive oscillator)......Page 284
5.6 The Rayleigh and van der Pol equations: Andronov-Hopf bifurcation: self-exciting oscillator; limit cycles; the Poincare-Bendixson theorem......Page 291
5.7 The Lotka-Volterra and chemical reaction equations: predator-prey equations, structurally unstable; one generalization; Lyapunov function......Page 303
5.8 Relaxation oscillations; singular perturbations: Violin strings, Floppy buckets, discharge tubes, neurons, Lienard's phase plane, piecewise linearizations......Page 308
5.9 Global bifurcations (homoclinics galore!): saddle connection; homoclinic orbit......Page 320
5.10 Periodically forced passive oscillators: a cusp catastrophe resonance and hysteresis effect......Page 328
5.11 Harmonic excitations: extended phase space: ultraharmonic, subharmonic, and ultrasubharmonic excitations......Page 334
5.12 Averaging method for nonautonomous systems (KBM)......Page 339
5.13 Forced van der Pol equations -frequency entrainment: van der Pol variables, heterodyning; entrainments of the heat, piano strings, and physiological circadean pacemakers......Page 342
5.14 Nonperturbative forced oscillators......Page 349
5.15 Experimental Poincare (stroboscopic) maps of forced passive oscillators......Page 376
Comments on exercises......Page 385
A A brief glossary of mathematical terms and notation......Page 396
B Notes on topology, dimensions, measures, embeddings and homotopy......Page 402
C Integral invariants......Page 413
D The Schwarzian derivative......Page 416
E The digraph method......Page 420
F Elliptic integrals and elliptic functions......Page 424
G The Poincarb-Bendixson theorem and Birkhoffs a and 0)-limit sets......Page 429
H A modified fourth-order Runge-Kutta iteration method......Page 434
I The Stoker-Haag model of relaxation oscillations......Page 436
Bibliography......Page 441
References by topics......Page 473
References added at 1991 reprinting......Page 509
Index......Page 511
