Preface 

ix  


1  (14) 

Chaos, Fractals, and Dynamics 


1  (1) 

Capsule History of Dynamics 


2  (2) 

The Importance of Being Nonlinear 


4  (5) 

A Dynamical View of the World 


9  (6) 
Part I. OneDimensional Flows 



15  (29) 


15  (1) 

A Geometric Way of Thinking 


16  (2) 

Fixed Points and Stability 


18  (3) 


21  (3) 

Linear Stability Analysis 


24  (2) 


26  (2) 

Impossibility of Oscillations 


28  (2) 


30  (2) 

Solving Equations on the Computer 


32  (12) 


36  (8) 


44  (49) 


44  (1) 


45  (5) 

Transcritical Bifurcation 


50  (3) 


53  (2) 


55  (6) 

Overdamped Bead on a Rotating Hoop 


61  (8) 

Imperfect Bifurcations and Catastrophes 


69  (4) 


73  (20) 


79  (14) 


93  (30) 


93  (1) 


93  (2) 


95  (1) 


96  (5) 


101  (2) 


103  (3) 

Superconducting Josephson Junctions 


106  (17) 


113  (10) 
Part II. TwoDimensional Flows 



123  (22) 


123  (1) 


123  (6) 

Classification of Linear Systems 


129  (9) 


138  (7) 


140  (5) 


145  (51) 


145  (1) 


145  (3) 

Existence, Uniqueness, and Topological Consequences 


148  (2) 

Fixed Points and Linearization 


150  (5) 


155  (4) 


159  (4) 


163  (5) 


168  (6) 


174  (22) 


181  (15) 


196  (45) 


196  (1) 


197  (2) 


199  (4) 

PoincareBendixson Theorem 


203  (7) 


210  (1) 


211  (4) 

Weakly Nonlinear Oscillators 


215  (26) 


227  (14) 


241  (60) 


241  (1) 

SaddleNode, Transcritical, and Pitchfork Bifurcations 


241  (7) 


248  (6) 

Oscillating Chemical Reactions 


254  (6) 

Global Bifurcations of Cycles 


260  (5) 

Hysteresis in the Driven Pendulum and Josephson Junction 


265  (8) 

Coupled Oscillators and Quasiperiodicity 


273  (5) 


278  (23) 


284  (17) 
Part III. Chaos 



301  (47) 


301  (1) 


302  (9) 

Simple Properties of the Lorenz Equations 


311  (6) 

Chaos on a Strange Attractor 


317  (9) 


326  (4) 

Exploring Parameter Space 


330  (5) 

Using Chaos to Send Secret Messages 


335  (13) 


341  (7) 


348  (50) 


348  (1) 


349  (4) 


353  (4) 


357  (4) 


361  (5) 


366  (3) 

Universality and Experiments 


369  (10) 


379  (19) 


388  (10) 


398  (25) 


398  (1) 

Countable and Uncountable Sets 


399  (2) 


401  (3) 

Dimension of SelfSimilar Fractals 


404  (5) 


409  (2) 

Pointwise and Correlation Dimensions 


411  (12) 


416  (7) 


423  (32) 


423  (1) 


423  (6) 


429  (5) 


434  (3) 

Chemical Chaos and Attractor Reconstruction 


437  (4) 

Forced DoubleWell Oscillator 


441  (14) 


448  (7) 
Answers to Selected Exercises 

455  (10) 
References 

465  (10) 
Author Index 

475  (3) 
Subject Index 

478  