Contents Dynamical Systems Stability of Dynamical Systems: Linear Approach

Transcription

1 Contents 1 Dynamical Systems Introduction DynamicalSystems andmathematical Models Kinematic Interpretation of a System of Differential Equations Definition of a Dynamical System: Classification Phase Portraits of Typical Oscillatory Systems Conservative Oscillator Damped Linear Oscillator Self-Sustained Oscillatory Systems Regular andchaotic Attractors Discrete-Time Systems: Return Maps StretchingMap Logistic Map Sine Map HenonMap Lozi Map Summary References Stability of Dynamical Systems: Linear Approach Introduction Definition of Stability Linear Analysis of Stability Stability of Solutions of a First-Order DifferentialEquation Stability of a Dynamical System in R N Stability of Phase Trajectories in Discrete-Time Systems Summary References ix

2 x Contents 3 Bifurcations of Dynamical Systems Introduction Double Equilibrium Bifurcation Soft and Hard Bifurcations: Catastrophes Triple Equilibrium Bifurcation Andronov HopfBifurcation Bifurcationsof Limit Cycles Saddle-NodeBifurcation Period-DoublingBifurcation Two-Dimensional Torus Birth (Death) Bifurcation(Neimark SakerBifurcation) Symmetry-BreakingBifurcation Nonlocal Bifurcations: Homoclinic Trajectories and Structures Separatrix Loop of a Saddle Equilibrium Point Saddle-NodeSeparatrixLoop Homoclinic Trajectory Appearance of a Saddle Limit Cycle Summary References Dynamical Systems with One Degree of Freedom Introduction Limit Sets and Attractors in the Phase Plane: The Andronov PoincaréLimit Cycle Structural Stability of Systems in the Phase Plane: Andronov PontryaginSystems Definition of Robustness of a Dynamical System Definition of Structural Stability of a DynamicalSystem Andronov PontryaginTheorem Oscillators with One Degree of Freedom Froude Pendulum FastenedWeight on a MovingBelt RC-Oscillator with Wien Bridge Oscillatory Circuit with Active Nonlinear Element Analysis of the van der Pol Equation: Onset of Self-Sustained Oscillations Amplitude and Phase Equations for the Self-Sustained Oscillator Oscillator with Hard Excitation of Self-Sustained Oscillations Analysis of the Stability of Equilibrium States Truncated Equations for the Amplitude and Phase for the Oscillator with Hard Excitation... 70

3 Contents xi Bifurcation Diagram of the Oscillator with Hard Excitation Summary References Systems with Phase Space Dimension N 3: Deterministic Chaos Introduction Determinismand Chaos for Beginners Determinism Chaos Stability and Instability Nonlinearity Instability and Nonlinear Restriction DeterministicChaos Mixing and Probabilistic Properties of Deterministic Systems Is Deterministic Chaos a Mathematical Oddity or a TypicalPropertyof the Material World? Strange Chaotic Attractors Strange Nonchaoticand Chaotic NonstrangeAttractors Chaotic NonstrangeAttractors StrangeNonchaoticAttractors GeometricCharacteristics of SNAs LCE Spectrumof SNAs SpectrumandAutocorrelationFunction Summary References From Order to Chaos: Bifurcation Scenarios (Part I) Introduction Transition to Chaos via a Cascade of Period-Doubling Bifurcations: Feigenbaum Universality Crisis and Intermittency From Order to Chaos: Bifurcation Scenarios (Part II) Route to Chaos via Two-DimensionalTorusDestruction Two-DimensionalTorus BreakdownTheorem Circle Map: Universal Regularities of Soft Transition from Quasiperiodicity to Chaos Route to Chaos via Ergodic Torus Destruction: Chaotic NonstrangeAttractors Summary References Robust and Nonrobust Dynamical Systems: Classification of Attractor Types Introduction Homoclinicand HeteroclinicCurves

4 xii Contents 8.3 Structurally Stable Systems in R N, N 3: Hyperbolicity Morse SmaleSystems HyperbolicSets AnosovSystems Smale Systems with Nontrivial Hyperbolicity: StrangeAttractors StructurallyUnstable DynamicalSystems Quasihyperbolic Attractors: Lorenz-Type Attractors Quasihyperbolic Attractor in the Lozi Map The LorenzAttractor Nonhyperbolic Attractors and Their Properties Nonhyperbolic Attractor in the Henon Map Nonhyperbolic Attractor in the Oscillator with Inertial Nonlinearity Summary References Characteristics of Poincaré Recurrences Introduction Local Approach Kac s Lemma Exponential Law for Distribution of First RecurrenceTimes NumericalExamples Global Approach: Afraimovich Pesin Dimension of RecurrenceTimes Afraimovich Pesin Dimension and Lyapunov Exponents Summary Reference Fractals in Nonlinear Dynamics Introduction Definition of a Fractal: Classic Examples of Fractal Sets The Nature of Fractality in Dynamical Systems Fractal Dimensions of Sets The Hausdorff BesicovitchDimension Capacity D C Information Dimension D I Correlation Dimension D cor Generalized Dimension D q Lyapunov Dimension D L Relationship Between DifferentDimensions Summary References

5 Contents xiii 11 The Anishchenko Astakhov Oscillator of Chaotic Self-Sustained Oscillations Introduction Theodorchik s Oscillator Modification of the Oscillator with Inertial Nonlinearity: The Anishchenko Astakhov Oscillator Periodic Regimes of Self-Sustained Oscillations and Their Bifurcations Period-Doubling Bifurcations: Feigenbaum Universality Chaotic Attractor and Homoclinic Trajectories in the Oscillator Summary References Quasiperiodic Oscillator with Two Independent Frequencies Introduction Methods for Realizing Two-Frequency Oscillations and TheirProperties Statement of Oscillator Equations Bifurcation Diagram of the Quasiperiodic Oscillator Two-DimensionalTorus-DoublingBifurcation Summary Synchronization of Periodic Self-Sustained Oscillations Introduction Forced Synchronization of the van der Pol Oscillator: TruncatedEquationsfor the Amplitudeand Phase Analysis of Synchronization in the Phase Approximation Bifurcational Analysis of the System of TruncatedEquations Bifurcational Analysis of the Nonautonomous van der Pol Oscillator Mutual Synchronization: Effect of Oscillation Death in Dissipatively Coupled van der Pol Oscillators Summary References Synchronization of Two-Frequency Self-Sustained Oscillations Introduction Influence of an External Periodic Force on a Resonant Limit Cycle in a System of Coupled Oscillators Basic Bifurcations of Quasiperiodic Regimes When Synchronizinga Resonant Limit Cycle Peculiarities in the Synchronization of ResonantLimit Cycles

6 xiv Contents Phase Synchronization of a System of Coupled van der Pol Oscillators by an ExternalHarmonicSignal Bifurcations of Equilibrium States Bifurcationsof InvariantCurves Synchronization of Two-Frequency Oscillations in a Self-Sustained Quasiperiodic Oscillator Summary Synchronization of Chaotic Oscillations Introduction Phase Frequency Synchronization of Chaotic Self-Sustained Oscillations Experimental Investigation of Forced Synchronization of an Oscillator with Spiral Chaos Complete Synchronizationof InteractingChaotic Systems Quantitative Characteristics of the Degree of Synchronization of Chaotic Self-Sustained Oscillations Summary References

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