The Authors’ Preface

Contents

1 Nonlinear Oscillations

1.1 Nonlinear Oscillations of a Conservative Single-Degree-of-Freedom System

1.1.1 Qualitative Description of Motion by the Phase Plane Method

1.2 Oscillations of a Mathematical Pendulum. Elliptic Functions

1.3 Small-Amplitude Oscillations of a Conservative Single-Degree-of-Freedom System

1.3.1 Straightforward Expansion

1.3.2 The Method of Multiple Scales

1.3.3 The Method of Averaging: The Van der Pol Equation

1.3.4 The Generalized Method of Averaging. The Krylov– Bogolyubov Approach

1.4 Forced Oscillations of an Anharmonic Oscillator

1.4.1 Straightforward Expansion

1.4.2 A Secondary Resonance at 9 ˜ ±3

1.4.3 A Primary Resonance: Amplitude–Frequency Response

1.5 Self-Oscillations: Limit Cycles

1.5.1 An Analytical Solution of the Van der Pol Equation for Small Nonlinearity Parameter Values

1.5.2 An approximate solution of the Van der Pol equation for large nonlinearity parameter values

1.6 External Synchronization of Self-Oscillating Systems

1.7 Parametric Resonance

1.7.1 The Floquet Theory

1.7.2 An Analytical Solution of the Mathieu Equation for Small Nonlinearity Parameter Values

2 IntegrableSystems

2.1 Equations of Motion for a Rigid Body

2.1.1 Euler’s Angles

2.1.2 Euler’s Kinematic Equations

2.1.3 Moment of Inertia of a Rigid Body

2.1.4 Euler’s Dynamic Equations

2.1.5 S.V. Kovalevskaya’s Algorithm for Integrating Equations of Motion for a Rigid Body about a Fixed Point

2.2 The Painlevé Property for Differential Equations

2.2.1 A Brief Overview of the Analytic Theory of Differential Equations

2.2.2 A Modern Algorithm of Analysis of Integrable Systems

2.2.3 Integrability of the Generalized Henon–Heiles Model

2.2.4 The Linearization Method for Constructing Particular Solutions of a Nonlinear Model

2.3 Dynamics of Particles in the Toda Lattice: Integration by the Method of the Inverse Scattering Problem

2.3.1 Lax’s Representation

2.3.2 The Direct Scattering Problem

2.3.3 The inverse scattering transform

2.3.4 N-Soliton Solutions

2.3.5 The Inverse Scattering Problem and the Riemann Problem

2.3.6 Solitons as Elementary Excitations of Nonlinear Integrable Systems

2.3.7 The Darboux–Backlund Transformations

2.3.8 Multiplication of Integrable Equations: The modified Toda Lattice

3 Stability of Motion and Structural Stability

3.1 Stability of Motion

3.1.1 Stability of Fixed Points and Trajectories

3.1.2 Succession Mapping or the Poincare Map

3.1.3 Theorem about the Volume of a Phase Drop

3.1.4 Poincare–Bendixson Theorem and Topology of the Phase Plane

3.1.5 The Lyapunov Exponents

3.2 Structural Stability

3.2.1 Topological Reconstruction of the Phase Portrait

3.2.2 Coarse Systems

3.2.3 Cusp Catastrophe

3.2.4 Catastrophe Theory

4 Chaos in Conservative Systems

4.1 Determinism and Irreversibility

4.2 Simple Models with Unstable Dynamics

4.2.1 Homoclinic Structure

4.2.2 The Anosov Map

4.2.3 The Tent Map

4.2.4 The Bernoulli Shift

4.3 Dynamics of Hamiltonian Systems Close to Integrable

4.3.1 Perturbed Motion and Nonlinear Resonance

4.3.2 The Zaslavsky–Chirikov Map

4.3.3 Chaos and Kolmogorov–Arnold–Moser Theory

5 Chaos and Fractal Attractors in Dissipative Systems

5.1 On the Nature of Turbulence

5.2 Dynamics of the Lorenz Model

5.2.1 Dissipativity of the Lorenz Model

5.2.2 Boundedness of the Region of Stationary Motion

5.2.3 Stationary Points

5.2.4 The Lorenz Model’s Dynamic Regimes as a Result of Bifurcations

5.2.5 Motion on a Strange Attractor

5.2.6 Hypothesis About the Structure of a Strange Attractor

5.2.7 The Lorenz Model and the Tent Map

5.2.8 Lyapunov Exponents

5.3 Elements of Cantor Set Theory

5.3.1 Potential and Actual Infinity

5.3.2 Cantor’s Theorem and Cardinal Numbers

5.3.3 Cantor sets

5.4 Cantor Structure of Attractors in Two-Dimensional Mappings

5.4.1 The Henon Map

5.4.2 The Ikeda Map

5.4.3 An Analytical Theory of the Cantor Structure of Attractors

5.5 Mathematical Models of Fractal Structures

5.5.1 Massive Cantor Set

5.5.2 A binomial multiplicative process

5.5.3 The Spectrum of Fractal Dimensions

5.5.4 The Lyapunov Dimension

5.5.5 A Relationship Between the Mass Exponent and the Spectral Function

5.5.6 The Mass Exponent of the Multiplicative Binomial Process

5.5.7 A Multiplicative Binomial Process on a Fractal Carrier

5.5.8 A Temporal Data Sequence as a Source of Information About an Attractor

5.6 Universality and Scaling in the Dynamics of One-Dimensional Maps

5.6.1 General Regularities of a Period-Doubling Process

5.6.2 The Feigenbaum–Cvitanovic Equation

5.6.3 A Universal Regularity in the Arrangement of Cycles: AUniversal Power Spectrum

5.7 Synchronization of Chaotic Oscillations

5.7.1 Synchronization in a System of Two Coupled Maps

5.7.2 Types and Criteria of Synchronization

Conclusion

References

Index