Chaotic Dynamics: An Introduction Based on Classical Mechanics

by Tamás Tél , Márton Gruiz

• ISBN13: 9780521839129
• ISBN10: 0521839122
• Format: Hardcover
• Copyright: 2006-09-18
• Publisher: Cambridge University Press

Summary

Since Newton, a basic principle of natural philosophy has been determinism, the possibility of predicting evolution overtime into the far future, given the governing equations and starting conditions. Our everyday experience often strongly contradicts this expectation. In the past few decades we have come to understand that even motions in simple systems can have complex and surprising properties.

Table of Contents

List of colour plates	ix
Preface	xiii
Acknowledgements	xv
How to read the book	xvi
Part I The phenomenon: complex motion, unusual geometry	1	(48)
Chaotic motion	3	(21)
What is chaos?	3	(1)
Examples of chaotic motion	4	(15)
Phase space	19	(2)
Definition of chaos; a summary	21	(1)
How should chaotic motion be examined?	22	(2)
Box 1.1 Brief history of chaos	23	(1)
Fractal objects	24	(25)
What is a fractal?	24	(8)
Types of fractals	32	(8)
Fractal distributions	40	(5)
Fractals and chaos	45	(4)
Box 2.1 Brief history of fractals	47	(2)
Part II Introductory concepts	49	(62)
Regular motion	51	(39)
Instability and stability	51	(14)
Box 3.1 Instability, randomness and chaos	59	(6)
Stability analysis	65	(2)
Emergence of instability	67	(9)
Box 3.2 How to determine manifolds numerically	73	(3)
Stationary periodic motion: the limit cycle (skiing on a slope)	76	(3)
General phase space	79	(11)
Driven motion	90	(21)
General properties	90	(5)
Harmonically driven motion around a stable state	95	(3)
Harmonically driven motion around an unstable state	98	(2)
Kicked harmonic oscillator	100	(3)
Fixed points and their stability in two-dimensional maps	103	(2)
The area contraction rate	105	(1)
General properties of maps related to differential equations	106	(3)
Box 4.1 The world of non-invertible maps	108	(1)
In what systems can we expect chaotic behaviour?	109	(2)
Part III Investigation of chaotic motion	111	(211)
Chaos in dissipative systems	113	(78)
Baker map	114	(17)
Kicked oscillators	131	(18)
Box 5.1 Henon-type maps	147	(2)
Parameter dependence: the period-doubling cascade	149	(5)
General properties of chaotic motion	154	(17)
Box 5.2 The trap of the `butterfly effect'	159	(9)
Box 5.3 Determinism and chaos	168	(3)
Summary of the properties of dissipative chaos	171	(4)
Box 5.4 What use is numerical simulation?	172	(2)
Box 5.5 Ball bouncing on a vibrating plate	174	(1)
Continuous-time systems	175	(6)
The water-wheel	181	(10)
Box 5.6 The Lorenz model	187	(4)
Transient chaos in dissipative systems	191	(36)
The open baker map	193	(6)
Kicked oscillators	199	(3)
Box 6.1 How do we determine the saddle and its manifolds?	201	(1)
General properties of chaotic transients	202	(8)
Summary of the properties of transient chaos	210	(4)
Box 6.2 Significance of the unstable manifold	211	(2)
Box 6.3 The horseshoe map	213	(1)
Parameter dependence: crisis	214	(3)
Transient chaos in water-wheel dynamics	217	(2)
Other types of crises, periodic windows	219	(2)
Fractal basin boundaries	221	(6)
Box 6.4 Other aspects of chaotic transients	225	(2)
Chaos in conservative systems	227	(37)
Phase space of conservative systems	227	(3)
The area preserving baker map	230	(4)
Box 7.1 The origin of the baker map	233	(1)
Kicked rotator -- the standard map	234	(8)
Box 7.2 Connection between maps and differential equations	236	(3)
Box 7.3 Chaotic diffusion	239	(3)
Autonomous conservative systems	242	(8)
General properties of conservative chaos	250	(9)
Summary of the properties of conservative chaos	259	(1)
Homogeneously chaotic systems	260	(4)
Box 7.4 Ergodicity and mixing	261	(1)
Box 7.5 Conservative chaos and irreversibility	262	(2)
Chaotic scattering	264	(15)
The scattering function	265	(1)
Scattering on discs	266	(8)
Scattering in other systems	274	(3)
Box 8.1 Chemical reactions as chaotic scattering	276	(1)
Summary of the properties of chaotic scattering	277	(2)
Applications of chaos	279	(39)
Spacecraft and planets: the three-body problem	279	(6)
Box 9.1 Chaos in the Solar System	284	(1)
Rotating rigid bodies: the spinning top	285	(8)
Box 9.2 Chaos in engineering practice	292	(1)
Climate variability and climatic change: Lorenz's model of global atmospheric circulation	293	(11)
Box 9.3 Chaos in different sciences	300	(3)
Box 9.4 Controlling chaos	303	(1)
Vortices, advection and pollution: chaos in fluid flows	304	(14)
Box 9.5 Environmental significance of chaotic advection	315	(3)
Epilogue: outlook	318	(4)
Box 10.1 Turbulence and spatio-temporal chaos	320	(2)
Appendix	322	(20)
Deriving stroboscopic maps	322	(3)
Writing equations in dimensionless forms	325	(4)
Numerical solution of ordinary differential equations	329	(3)
Sample programs	332	(5)
Numerical determination of chaos parameters	337	(5)
Solutions to the problems	342	(28)
Bibliography	370	(17)
Index	387