Invariant Sets for Windows

By: A. D. Morozov

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Hardcover | 1 December 1998

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272 Pages

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22.23 x 13.97 x 2.54

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This book deals with the visualization and exploration of invariant sets (fractals, strange attractors, resonance structures, patterns etc.) for various kinds of nonlinear dynamical systems. The authors have created a special Windows 96 application called WInSet, which allows one to visualize the invariant sets. A WInSet installation disk is enclosed with the book. The book consists of two parts. Part I contains a description of WlnSet and a list of the built-in invariant sets which can be plotted using the program. This part is intended for a wide audience with interests ranging from dynamical systems to computer design. In Part II, the invariant sets presented in Part I are investigated from the theoretical perspective. The invariant sets of dynamical systems with one, one-and-a-half and two degrees of freedom, as well as those of two-dimensional maps, are discussed. The basic models of the diffusion equations are also considered. This part of the book is intended for a more advanced reader, with at least a BSc in Mathematics.

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Introduction	p. 1
Invariant Structures Everywhere	p. 1
Resonance Structures in Celestial Mechanics	p. 2
Cellular, Spiral, Vortex and Crystal Structures	p. 4
Fractals	p. 9
Dynamical Systems	p. 11
Attractors	p. 13
Invariant Tori	p. 15
Discrete Dynamical Systems--Maps	p. 16
Computer-Generated Invariant Sets	p. 19
Description of WInSet Program	p. 21
Installation	p. 21
Basics of WInSet	p. 21
First Run of WInSet	p. 22
Using the Mouse and the Keyboard	p. 23
Your First Invariant Set	p. 24
WInSet Menu	p. 25
Three-Dimensional Objects	p. 35
Diffusion Equations	p. 36
Defining Your Own Equations	p. 41
List of the Built-in Equations, Maps and Fractals of WInSet. Main Invariant Sets of WInSet	p. 45
Maps	p. 45
Cathala Map	p. 45
Chirikov Map	p. 45
Henon Maps	p. 47
Julia Map	p. 47
Mira and Gumowski Maps	p. 48
Zaslavsky Map	p. 52
Fractals	p. 52
Coloring the Fractals	p. 52
Julia Fractals	p. 53
Mandelbrot Fractal	p. 56
Mira Fractals	p. 56
Newton Fractal	p. 57
Ordinary Differential Equations (ODE)	p. 57
Brusselator	p. 57
Chua Equations	p. 58
Duffing Type Equations	p. 58
Hamiltonian Systems on Torus	p. 62
Henon-Heiles Model	p. 62
Henon-Heiles Type Equations	p. 63
Kepler Equation	p. 63
Kolmogorov-Volterra Equations	p. 63
Lorenz Equations	p. 65
Motion of Particle in Gravitation Field	p. 65
Pendulum Equations	p. 67
Equations with Quadratic Nonlinearity	p. 69
Roessler Equations	p. 70
Volterra Equations	p. 70
Diffusion Equations (PDE)	p. 71
Brusselator Model	p. 71
Fitz Hugh-Nagumo Equations	p. 72
Lengyel-Epstein Model (CIMA)	p. 72
Semi-Discrete Equation	p. 73
Numerical Methods Used by WInSet	p. 74
Mathematical Description of Invariant Sets	p. 77
Invariant Sets in Hamiltonian Mechanics	p. 79
Generalities	p. 79
Invariant Sets of Hamiltonian Systems with One Degree of Freedom	p. 82
Invariant Sets of Hamiltonian Systems with 3/2 Degrees of Freedom	p. 90
Poincare Map	p. 90
Analytic Study	p. 93
Duffing Type Equations	p. 98
Pendulum Type Equation	p. 101
Systems on the Torus	p. 104
Kepler Equation	p. 104
Invariant Sets of Hamiltonian Systems with Two Degrees of Freedom	p. 106
Henon-Heiles Type Systems	p. 106
Invariant Sets in the Dynamics of a Solid	p. 107
Area-Preserving Maps	p. 111
Chirikov Map	p. 111
Gumowski and Mira Map	p. 113
Henon Map	p. 113
Zaslavsky Map	p. 118
Non-Conservative Systems	p. 123
Characteristics of Chaotic Dynamics	p. 124
Characteristics which do not Use Measure	p. 125
Measure-Theoretic Characteristics of the Attractor	p. 127
Power Spectrum of an Observable	p. 129
Self-Oscillations	p. 130
Some Technical Transformations	p. 132
Qualitative Behavior of Solutions in an Individual Cell	p. 134
Behavior of Solutions near Separatrices of the Unperturbed System	p. 136
Van der Pole-Duffing Type Equations	p. 137
Pendulum Type Equations	p. 138
Brusselator Equation	p. 140
Three-Dimensional Systems	p. 143
Resonances and Synchronization	p. 145
Theoretical Analysis of Quasi-Hamiltonian Systems with 3/2 Degrees of Freedom	p. 146
Characteristics of Chaotic Dynamics for Systems with 3/2 Degrees of Freedom	p. 160
Theoretical Analysis of Quasi-Hamiltonian Systems with Two Degrees of Freedom	p. 162
Examples	p. 169
Parametric Resonances	p. 177
General Results	p. 178
Example 1	p. 186
Example 2	p. 190
Strange Attractors in Three-Dimensional Systems	p. 198
Lorenz System	p. 198
Roessler System	p. 205
Chua System	p. 205
Non-Conservative Maps	p. 211
One-Dimensional Maps	p. 213
Two-Dimensional Non-Conservative Maps	p. 216
One-Dimensional Complex Rational Endomorphisms	p. 216
Fractals	p. 219
Non-Invertible Mira Maps and their Fractals	p. 226
Henon Map	p. 232
Diffusion Equations	p. 235
Parabolic Equations	p. 236
One-Component Models	p. 236
Two-Component Model	p. 238
Semi-Discrete Approximation	p. 243
Approximation of Equation (8.1)	p. 244
Approximation of the Basic Multi-Component Models	p. 245
Semi-Discrete Diffusion Equations	p. 245
Bibliography	p. 251
Table of Contents provided by Syndetics. All Rights Reserved.

Invariant Sets for Windows

At a Glance

Hardcover

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