Note: Intro -- 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.