Encyclopedia of dynamical systems

• Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia

Among participants of this encyclopedia are D. Anosov (Anosov Diffeomorphism), L. Bunimovich (Dynamical Billiards), B. Chirikov (Chirikov standard map), N. Fenichel (Normal Hyperbolicity), R. FitzHugh (FitzHugh-Nagumo Model), J. Guckenheimer (5 articles on codim-2 local bifurcations), H. Haken (Self-Organization and Synergetics), M. Hirsch (Monotone Dynamics), P. Holmes (History of Dynamical Systems, Stability, with Shea-Brown), K. Ito (Ito Calculus), A. Katok (Ergodic Theory and Invariant Measure), Y. Kuramoto (Kuramoto Model), Yu.A. Kuznetsov (7 articles on local bifurcations), E.N. Lorenz (Butterfly Effect), B. Mandelbrot (Fractals and Mandelbrot Set), J. Milnor (Attractor), D. Ornstein (Ornstein Theorem), E. Ott (Attractor Dimension, Crises, Controlling Chaos, Basin of Attraction), M.M. Peixoto (Structural Stability), O. Rossler (Rossler Attractor), A.N. Sharkovsky (Sharkovsky Ordering), A. Shilnikov (Shilnikov Bifurcation), Y. Sinai (SRB Measure and Kolmogorov-Sinai Entropy), S. Smale (Smale Horseshoe), J. Yorke (Chaos, with T. Sauer), and many others.

Presently, only curators of Scholarpedia (who are invited by the Editor-in-Chief or other curators) or scholars with high Scholar Index can author and curate articles in Scholarpedia. If you want to reserve an article, contact [1].

There is an ongoing election of authors for such articles as Relaxation Oscillator, Resonance, and many others.

See instructions to register and reserve your article. Most of the articles are reserved and will be written within next few months. Many articles are finished.

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History of Dynamical Systems

Mathematical Theory Differential Equations Ordinary Differential Equations Boundary Value Problem Initial Value Problems Wronskian Sturm-Liouville Equation Monotone Dynamics Lyapunov Function Hartman-Grobman Theorem Normal Forms Partial Differential Equations Green Function Characteristics Separation of Variables Inertial Manifolds Variational Methods Spectral Theory and Eigenvalue Problems for PDE Equivariant Dynamical Systems Piecewise Smooth Dynamical Systems Stiff Equations Stiff Delay Equations Implicit Differential Equations Infinite Order Differential Equations Integro-Differential Equations Delay-Differential Equations Neutral Delay Equations Delay Partial Differential Equations Differential-Algebraic Equations Differential Equations in Abstract Spaces Differential Equations with Impulses Differential Inclusions Differential Inequalities Discontinuous Differential Equations Fractional Differential Equations Wave Equation Methods Asymptotic Methods Averaging Contraction Theory Ill Posed Problem Multiple Scale Analysis Nonstandard Analysis Perturbation Methods Renormalization in Dynamics Pseudospectrum Singular Perturbation Theory Singular Value Decomposition Spectral Analysis Well-posedness Numerical Analysis Editor: Skip Thompson Backward Differentiation Formulas Boundary Value Problem General Linear Methods Initial Value Problems Kreiss Matrix Theory Numerical Analysis Runge-Kutta Methods Stiff Systems Symplectic Integration Methods Smooth Dynamics Attractor Basin of Attraction Bistability Dynamical Systems Equilibrium Equivariant Dynamical Systems Heteroclinic Bifurcation Heteroclinic Cycles Holomorphic Dynamics Hysteresis Invariant Manifold Invariant Set Kupka-Smale Theorem Noninvertibility Normal Hyperbolicity Periodic Orbit Phase Space Phase Portrait Piecewise Smooth Dynamical Systems Seifert Conjecture Separatrix Shadowing Stability Stability of Equilibria Topological Equivalence Vector-Field and Flow Bifurcation Theory Andronov-Hopf Bifurcation Andronov-Leontovich Theorem AUTO Bautin Bifurcation Bifurcations Blue-Sky Catastrophe Bogdanov-Takens Bifurcation Bursting Canards Catastrophe Theory Center Manifold Cusp Bifurcation Delayed Loss of Stability Equivariant Bifurcation Theory Feigenbaum Constant Flip Bifurcation Fold-Hopf Bifurcation Fold Limit Cycle Bifurcation Global Bifurcations Hartman-Grobman Theorem Heteroclinic Bifurcation Homoclinic Bifurcation Homoclinic Orbit Hopf-Hopf Bifurcation Lyapunov-Schmidt Reduction MATCONT Melnikov Method Neimark-Sacker Bifurcation Normal Forms Normal Hyperbolicity Numerical Continuation Period Doubling Phaser Scientific Software Pitchfork Bifurcation Saddle-Node Bifurcation Saddle-Node Bifurcation on Invariant Circle Shilnikov Bifurcation Shoshitaishvili Theorem Singularity Theory Stability of Equilibria Structural Stability Swallowtail Catastrophe Transcritical Bifurcation Transition to Turbulence Torus Breakdown Unfoldings XPPAUT Topological Dynamics Editor: Dr. Jorge Lewowicz Conley Index Combinatorial Dynamics Expansiveness Holomorphic Dynamics Index Theory Rotation Theory Sofic System Subshift Of Finite Type Symbolic Dynamics Smale Horseshoe Topological Dynamics Topological Entropy Topological Exactness Topological transitivity Ergodic Theory Bernoulli Shift Birkhoff Ergodic Theorem Bowen-Margulis Measure Dynamical Billiards Entropy Entropy of Ergodic System Ergodic Theory Ergodicity Exactness Invariant Measure Jakobson Theorem Kolmogorov-Sinai Entropy Lasota-Yorke Theorem Markov Shift Mixing Non-standard Ergodic Theorems Ornstein Theorem Oseledets Theorem Pirogov-Sinai Theory Pressure and equilibrium states Ratner Theory Sinai-Ruelle-Bowen Measure Hyperbolic Dynamics Anosov Diffeomorphism Axiom A Systems Dynamical Billiards Expanding Maps Hartman-Grobman Theorem Hyperbolic Dynamics Morse-Smale Systems Partially Hyperbolic Systems Mappings (Maps) Editor: James D. Meiss Attractors in Two Dimensional Mappings Aubry-Mather Theory Chirikov standard map Circle Map Poincaré-Denjoy Theory Coupled Maps Difference Equations Fatou Domains Fixed Point Hénon Map Herman Rings Ikeda Map Iterated Map Interval Maps Jakobson Theorem Julia Sets Kneading Theory Logistic Map Mandelbrot Set Mappings Cobweb Diagram Maps with Vanishing Denominators Noninvertible Maps Nontwist Map Pseudo-Anosov Maps Quadratic Maps Poincare Map Renormalization Siegel Disks Stability of Symplectic Maps Topological transitivity Twist Maps Universality Zaslavsky Map Zaslavsky Web Map

Nonlinear Phenomena Oscillators Almost Periodic Function Bursting Discrete Models of Bursting Canards Central Pattern Generation Chaotic Oscillators Feedback Control Floquet Theory Isochron Parametric Oscillator Periodic Orbit Phase Model Pendulum Period Doubling Radial Clock Rotator Relaxation Oscillator Routes into Bursting Quasiperiodic Oscillations Linear Oscillator Weakly Nonlinear Oscillator Synchronization Arnold Tongues Burst Synchronization Chain of Oscillators Desynchronization Fast Threshold Modulation Gait Control Generalized Synchronization Isochron Kuramoto Model Malkin Theorem Mean-Field Synchronization Neuronal Synchronization Resonance Phase Response Curve Pulse Coupled Oscillators Slowly Coupled Oscillators Synchronization Synchronization of Chaotic Oscillators Synchrony Measures Weakly Coupled Oscillators Pattern Formation Breathers Cellular Automata Complex Systems Dissipative Systems Drift of Spiral Waves Evans Function Excitable Media Fisher-Kolmogorov-Petrovski-Piskunov Model Ginzburg-Landau Equation Gradient System Neural Fields Kuramoto-Sivashinsky Equation Pattern Formation Reaction-Diffusion Systems Sandpiles Scroll Wave Scroll Wave Turbulence Self-Organization Self-Organization of Brain Function Spiral Breakup Spiral Waves Stripe Formation Symmetry Breaking Symmetry Breaking in Reaction-Diffusion Systems Synergetics Traveling Waves Turing Instability Zeldovich-Frank-Kamenetsky Model Chaos Anti-integrable Limit Attractor Dimension Attractor Embedding Attractor Reconstruction Bubbling Transition Butterfly Effect Chaos Chaos in Optics Chaotic Advection Chua Circuit Crises Controlling Chaos Dynamical Coherence Entropy in Chaotic Dynamics Feigenbaum Constant Fractals Grassberger-Procaccia Algorithm Henon Map Hyperchaos Ikeda Map Intermittency Iterated Functions System Julia Sets Lorenz Attractor Lyapunov Exponent Mandelbrot Set Multifractal Analysis Period Doubling Plykin Attractor Quantum Chaos Renormalization Renormalization in Dynamics Rossler Attractor Sharkovsky Ordering Shilnikov Bifurcation Smale-Williams Attractor Strange Attractor Strange Nonchaotic Attractors Symbolic Dynamics Bernoulli Shift Smale Horseshoe Synchronization of Chaotic Oscillators Transient Chaos Transition to Turbulence Turbulence Ueda Attractor Universality Unstable Periodic Orbits Random Dynamical Systems 1/F Noise Coherence Resonance Deterministic Walk in Random Environment Disordered Systems First Passage Time Fokker-Planck Equation Ising Model Ito Calculus Markov Process Metropolis Method Monte-Carlo Method Ornstein-Uhlenbeck Process Percolation Theory Random Walk Brownian Motion Self-Organized Criticality Avalanche Sandpiles Signal to Noise Ratio Simulated Annealing Stratanovich Calculus Stochastic Dynamical Systems Stochastic Process Stochastic Resonance Control Theory Adaptive Control Control Theory H-infinity Control Hierarchical Control Multivariable Control Nonlinear Control Observability and Controllability Optimal Control Predictive Control Robust Control Stochastic Control Theory System Identification

Applications Neuroscience Binding by Synchrony Bursting Central Pattern Generation Chaos in Nerve Membranes Chaotic Spiking Oscillators Compartmental Models Computational Intelligence Conductance-Based Models Discrete Models of Bursting Dynamic Clamp Ermentrout-Kopell Canonical Model Excitability Firing Rate Model Fitzhugh-Nagumo Model Haken-Kelso-Bunz Model Hallucination Hindmarsh-Rose Model Hodgkin-Huxley Model Integrate-and-Fire Neuron Large-Scale Brain Models Mckean Model Mechanoreceptors and Stochastic Resonance Models of Calcium Waves Morris-Lecar Model Multi-stability in Neuronal Models Neural Networks Neural Oscillator Neuronal Noise Neuronal Synchronization Pinsky-Rinzel Model Plant model Rall Model Resonate-and-Fire Neuron Young Model Routes into Bursting Self-Organization of Brain Function Spike-Response Model Statistical Mechanics of Neocortex Stochastic Channel Model Subthreshold Oscillation Synergies VCON Wilson-Cowan Model Physiology and Medicine Activator-Inhibitor Artificial Life Biofluid Dynamics Biophysics Chay Model Circadian Rhythms Ultradian Rhythm Fluid Dynamics of Swimming and Flying Human Movement Science Mitotic Cycle Models of Calcium Waves Models of Calcium Oscillations Molecular Motors Plant Model Morphogenesis Cellular Mechanisms of Morphogenesis Chemotactic Orientation Gastrulation Gierer-Meinhardt Model Morphogenesis Stripe Formation Cardiac Dynamics Editor: Vadim N. Biktashev Aliev-Panfilov Model Alternans Barkley Model Beeler-Reuter Model Bidomain Model Cardiac Arrhythmia Drift of Spiral Waves Excitable Media FitzHugh-Nagumo Model Initiation of Excitation Waves Kinematics of Excitation Waves Luo-Rudy Models McKean Model Noble Model Meander of Spiral Waves Models of Cardiac Cell Models of Heart Reaction-Diffusion Systems Restitution Scroll Wave Scroll Wave Turbulence Spiral Breakup Spiral Waves Symmetry Breaking in Reaction-Diffusion Systems Traveling Waves Vulnerability of Cardiac Dynamics Chemistry Belousov-Zhabotinsky Reaction Brusselator Chemical Reaction Kinetics Chemical Oscillators Formal Reaction Kinetics Gas Phase Reactions Kinetic Equations Oregonator Pharmakinetics Reaction-Diffusion Systems Population Dynamics Epidemics Evolution Lotka-Volterra Model Mathematical Biology Modern Predator-Prey Models Population Dynamics Predator-Prey Model Replicator Dynamics Traffic Flow Verhulst Equation Engineering Controllability Hysteresis Josephson Junction Lienard Oscillator RC Circuit van der Pol Oscillator Fluid Dynamics Editor: Dr. Oleg Schilling ABC Flow Auto-resonance in Nonlinear systems Biofluid Dynamics Bose Einstein Condensate Dynamics Burgers Equation and One-dimensional Turbulence Cascade and Scaling Chaos vs Turbulence Compactons Contour Dynamics Dispersive Shock Waves DNA Dynamics and Fluctuations Fermi-Pasta-Ulam Nonlinear Lattice Oscillations Fluid Dynamics of Swimming and Flying Integrable Systems Inverse Scattering Transform Method Korteweg-de Vries Equation and its Inverse Scattering Method Linear Instability of Fluid Flows Linear and Nonlinear Waves Magnetohydrodynamics Magnetorotational Instability MHD Reconnection MHD Turbulence Nonlinear Optics and Photonics: Solitons and Localized Structures Nonlinear Schrodinger Systems:Continuous and Discrete Painleve Analysis Rayleigh-Benard Convection Rayleigh-Taylor Instability and Mixing Richtmyer-Meshkov Instability and Re-accelerated Inhomogeneous Flows Shock Waves Solitons Taylor-Couette Flow Transition to Turbulence Turbulence Turbulence: Elastic Turbulence: Two-Dimensional Vesicle Dynamics: Experiments Vesicle Dynamics: Theory and Simulation Vortex Dynamics Mechanics Arnold Diffusion Ballistics Celestial Mechanics Computational Celestial Mechanics Duffing Oscillator Equation of Motion Hamiltonian Normal Forms Hamiltonian Systems Hénon-Heiles System Kolmogorov-Arnold-Moser Theory Lagrangian N-Body Problem Pendulum Stability of Hamiltonian Flows Three Body Problem Optics Chaos in Optics Lang-Koboyashi Model Laser Nonlinear Optics Physics Glassy Dynamics Ising Model Nonlinear Acoustics Quantum Dynamics Plasma Statistical Mechanics Schrodinger Equation Semiconductor Oscillators Sine-Gordon Equations Superconductors Computational Astrophysics Editor: Søren Bertil F. Dorch ABC Flow Accretion Discs Archontis Dynamo Celestial Mechanics Computational Celestial Mechanics Computational Cosmology Core-collapse Supernova Simulations Fast Dynamos Flux Transport Dynamos Flux Tubes Galactic Magnetic Fields Hydromagnetic Dynamo Theory KAM Theory in Celestial Mechanics Magneto-Convection Magnetohydrodynamics Magnetorotational Instability MHD Reconnection MHD Turbulence N-body Simulations Nonlinear Dynamos Shock Structures Solar Dynamo Solar Flare Simulations Solar Granulation Stellar Wind Models Supernova Type Ia Simulations Sunspots Radiative Transfer Radiation Hydrodynamics Theoretical Helioseismology

Sponsored by: Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia

Sponsored by: Dr. John Guckenheimer, Cornell University, Ithaca, NY, USA

Sponsored by: Prof. Yuri A. Kuznetsov, Department of Mathematics, Utrecht University, The Netherlands

Reviewed by: Dr. John Guckenheimer, Cornell University, Ithaca, NY, USA

Reviewed by: Prof. Yuri A. Kuznetsov, Department of Mathematics, Utrecht University, The Netherlands

Accepted on: 2011-10-21 06:06:51 GMT