New Methods for Chaotic Dynamics

The following sections are included:

• Preface

• Contents

The following sections are included:

• Basic Definitions and Theorems
  • Fields of directions and their integral curves
  • Vector fields, differential equations, integral and phase curves
  • Theorems of existence and uniqueness of solutions
  • Differentiable dependence of solutions from initial conditions and parameters, the equations in variations
  • Dissipative and conservative systems of differential equations
  • Numerical methods for solution of systems of ordinary differential equations
  • Ill-posedness of numerical methods in solution of systems of ordinary differential equations

• Singular Points and Their Invariant Manifolds
  • Singular points of systems of ordinary differential equations
  • Stability of singular points and stationary solutions
  • Invariant manifolds
  • Singular points of linear vector fields
  • Separatrices of singular points, homoclinic and heteroclinic trajectories, separatrix contours

• Periodic and Nonperiodic Solutions, Limit Cycles and Invariant Tori
  • Periodic solutions
  • Limit cycles
  • Poincare map
  • Invariant tori
  • Nonperiodic solutions, Lyapunov exponents

• Attractors of Dissipative Systems of Ordinary Differential Equations
  • Basic definitions
  • Classical regular attractors of dissipative systems of ordinary differential equations
  • Classical irregular attractors of dissipative dynamical systems
  • Dimension of attractors, fractals

The following sections are included:

• Structural Stability and Bifurcations
  • Structural stability
  • Bifurcations

• One-Parametrical Local Bifurcations
  • Bifurcations of stable singular points
    • Transcritical (exchange of stability) bifurcation
    • Saddle–node bifurcation
    • Pitchfork type bifurcation
    • Andronov–Hopf bifurcation (cycle birth bifurcation)
  • Bifurcations of stable limit cycles
    • Bifurcation of birth of a pair of stable closed trajectories
    • Transcritical (exchange of stability between cycles) bifurcation
    • Bifurcation of disappearance (appearance) of a pair of closed trajectories
    • Bifurcation of birth(destruction)of two-dimensional torus
    • Period doubling bifurcation of a cycle
  • Bifurcations of stable two-dimensional tori
    • Bifurcation of birth of pair of stable two-dimensional tori
    • Transcritical (exchange of stability between tori) bifurcation
    • Bifurcation of disappearance (appearance) of a pair of two-dimensional tori
    • Bifurcation of birth (destruction) of three-dimensional torus
    • Period doubling bifurcation of two-dimensional torus

• The Simplest Two-Parametrical Local Bifurcations
  • The normal form of a fold
  • The normal form of an assembly

• Nonlocal Bifurcations
  • Bifurcations of homoclinic separatrix contours
    • Separatrix loop of a saddle type singular point
    • Separatrix loop of a degenerate two-dimensional saddle–node
    • Separatrix loop of a saddle–node
    • Separatrix loop of a degenerate saddle–node
    • Homoclinic butterfly
    • Separatrix loop of a saddle–focus
    • Separatrix loop of a singular saddle cycle
  • Bifurcations of heteroclinic separatrix contours
    • Separatrix going from a saddle into a saddle
    • Separatrix contour connecting a saddle–node with a saddle–focus
    • Separatrix contour connecting a saddle–focus with a saddle–focus
    • Point–cycle bifurcation
  • Approximate method for finding bifurcation points of separatrix contours of singular points
    • Heteroclinic contours of saddle–nodes and saddle–focuses
    • Homoclinic separatrix loop of a saddle–focus
    • Homoclinic separatrix loop of a saddle–node
  • Cascades of bifurcations, scenarios of transition to chaos
    • Cascade of period doubling bifurcations, Feigenbaum scenario
    • Subharmonic cascade of bifurcations of stable cycles, Sharkovskii scenario
    • Homoclinic cascade of bifurcations, Magnit-skii scenario
    • Subharmonic cascade of bifurcations of stable two-dimensional tori
    • Cascade of Andronov–Hopf bifurcations, scenarios of Landau and Ruelle–Takens
    • Transition to chaos through an intermittency, the Pomeau–Manneville scenario
  • Bifurcations of irregular (singular) attractors
    • Bifurcation of formation of a regular attractor (a stable cycle)
    • Bifurcation of formation of a regular attractor (a stable torus)
    • Bifurcation of formation of singular attractor
    • Bifurcation of coherence
    • Bifurcation of unification of singular attractors

The following sections are included:

• System of the Lorenz Equations
  • Classical scenario of birth of the Lorenz attractor
  • Scenario of birth of the Lorenz attractor through an incomplete double homoclinic cascade of bifurcations
    • Method for investigating the Lorenz attractor
    • Scenario of transition to chaos
  • Scenario of birth of a complete double homoclinic attractor in the Lorenz system
  • Bifurcations of homoclinic and heteroclinic contours in the Lorenz system
    • Heteroclinic contours connecting the saddle–node with the saddle–focus
    • Homoclinic saddle–focus separatrix loop
    • Homoclinic saddle–node separatrix loop
    • Heteroclinic contour connecting saddle–node with two saddle–focuses
  • Diagrams of nonlocal bifurcations in the Lorenz system

• The Complex System of Lorenz Equations
  • Scenario of transition to chaos

• Systems of the Rossler Equations

• The Chua System

• Other Chaotic Systems of Ordinary Differential Equations
  • The Vallis systems
  • The Rikitaki system
  • The "Simple" system
  • The Rabinovich–Fabrikant system

• Final Remarks and Conclusions

The following sections are included:

• Theory of One-Dimensional Smooth Mappings
  • Monotonic invertible mappings
  • Nonmonotonic mappings

• Feigenbaum Cascade of Period Doubling Bifurcations of Cycles of One-Dimensional Mappings
  • Logistic mapping
  • Period doubling operator
  • Feigenbaum universality
  • Dimension of the Feigenbaum attractor

• Sharkovskii Subharmonic Cascade of Bifurcations of Cycles of One-Dimensional Mappings
  • The Sharkovskii's theorem
  • Behind the Feigenbaum cascade

• Dynamical Chaos in Two-Dimensional Non-Autonomous Systems of Differential Equations
  • Rotor type singular points
  • Scenario of transition to chaos
  • Dynamical chaos in some classical two-dimensional non-autonomous systems
    • The Duffing–Holmes equation
    • The Matie equation
    • The Croquette equation
    • The Krasnoschekov equation

• Dynamical Chaos in Three-Dimensional Autonomous Systems of Differential Equations
  • Singular cycles of three-dimensional autonomous systems
  • Singular attractors of three-dimensional autonomous systems
  • Some examples of three-dimensional autonomous systems with singular attractors

• Final Remarks and Conclusions

The following sections are included:

• Regular Dynamics and Diffusion Chaos in Reaction–Diffusion Systems
  • Turing and Andronov–Hopf bifurcations in the Brus-selator model
  • Diffusion chaos for the Brusselator in a ring
  • Diffusion chaos in the Brusselator on a segment

• Transition to Spatio-Temporal Chaos in the Kuramoto–Tsuzuki Equation
  • Scenario of transition to chaos in system of few-mode approximations
  • Transition to chaos in the space of Fourier coefficients
  • Scenario of transition to chaos in the phase space of the Kuramoto–Tsuzuki equation

• Dynamical Chaos in Differential Equations with Delay Argument

• Cycles and Chaos in Distributed Economic Systems
  • Description of the model of self-developing market economy
  • Behavior of macroeconomic variables
  • Behavior of economic variables in the presence of diffusion of capital and consumer demand

The following sections are included:

• Ott–Grebogi–Yorke and Pyragas methods
  • The OGY-method
  • The Pyragas method

• The Magnitskii Method
  • Localization and stabilization of unstable fixed points and unstable cycles of chaotic mappings
    • Case of one-dimensional mapping
    • Case of two-dimensional mapping
  • Localization and stabilization of unstable fixed points of chaotic dynamical systems
  • Localization and stabilization of unstable cycles of chaotic dynamical systems
  • Chaos control in equations with delay argument
  • Stabilization of a thermodynamic branch in reaction–diffusion systems of equations

• Reconstruction of Dynamical System on Trajectory of Irregular Attractor

The following sections are included:

• Bibliography

• Index