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Period-doubling bifurcation

About: Period-doubling bifurcation is a research topic. Over the lifetime, 4773 publications have been published within this topic receiving 91373 citations.


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Book
01 Jan 1984
TL;DR: In this article, the authors present a model for the detection of deterministic chaos in the Lorenz model, which is based on the idea of the Bernoulli shift and the Kicked Quantum Rotator.
Abstract: Preface.Color Plates.1 Introduction.2 Experiments and Simple Models.2.1 Experimental Detection of Deterministic Chaos.2.2 The Periodically Kicked Rotator.3 Piecewise Linear Maps and Deterministic Chaos.3.1 The Bernoulli Shift.3.2 Characterization of Chaotic Motion.3.3 Deterministic Diffusion.4 Universal Behavior of Quadratic Maps.4.1 Parameter Dependence of the Iterates.4.2 Pitchfork Bifurcation and the Doubling Transformation.4.3 Self-Similarity, Universal Power Spectrum, and the Influence of External Noise.4.4 Behavior of the Logistic Map for r ≤ r.4.5 Parallels between Period Doubling and Phase Transitions.4.6 Experimental Support for the Bifurcation Route.5 The Intermittency Route to Chaos.5.1 Mechanisms for Intermittency.5.2 Renormalization-Group Treatment of Intermittency.5.3 Intermittency and 1/f-Noise.5.4 Experimental Observation of the Intermittency Route.6 Strange Attractors in Dissipative Dynamical Systems.6.1 Introduction and Definition of Strange Attractors.6.2 The Kolmogorov Entropy.6.3 Characterization of the Attractor by a Measured Signal.6.4 Pictures of Strange Attractors and Fractal Boundaries.7 The Transition from Quasiperiodicity to Chaos.7.1 Strange Attractors and the Onset of Turbulence.7.2 Universal Properties of the Transition from Quasiperiodicity to Chaos.7.3 Experiments and Circle Maps.7.4 Routes to Chaos.8 Regular and Irregular Motion in Conservative Systems.8.1 Coexistence of Regular and Irregular Motion.8.2 Strongly Irregular Motion and Ergodicity.9 Chaos in Quantum Systems?9.1 The Quantum Cat Map.9.2 A Quantum Particle in a Stadium.9.3 The Kicked Quantum Rotator.10 Controlling Chaos.10.1 Stabilization of Unstable Orbits.10.2 The OGY Method.10.3 Time-Delayed Feedback Control.10.4 Parametric Resonance from Unstable Periodic Orbits.11 Synchronization of Chaotic Systems.11.1 Identical Systems with Symmetric Coupling.11.2 Master-Slave Configurations.11.3 Generalized Synchronization.11.4 Phase Synchronization of Chaotic Systems.12 Spatiotemporal Chaos.12.1 Models for Space-Time Chaos.12.2 Characterization of Space-Time Chaos.12.3 Nonlinear Nonequilibrium Space-Time Dynamics.Outlook.Appendix.A Derivation of the Lorenz Model.B Stability Analysis and the Onset of Convection and Turbulence in the Lorenz Model.C The Schwarzian Derivative.D Renormalization of the One-Dimensional Ising Model.E Decimation and Path Integrals for External Noise.F Shannon's Measure of Information.F.1 Information Capacity of a Store.F.2 Information Gain.G Period Doubling for the Conservative H-enon Map.H Unstable Periodic Orbits.Remarks and References.Index.

1,693 citations

Book
01 Dec 1982
TL;DR: The first homoclinic explosion in the Lorenz equation was described in this article, where the authors proposed an approach to the problem of finding the position of the first homocalinic explosion by using the Maxima-in-z method.
Abstract: 1. Introduction and Simple Properties.- 1.1. Introduction.- 1.2. Chaotic Ordinary Differential Equations.- 1.3. Our Approach to the Lorenz Equations.- 1.4. Simple Properties of the Lorenz Equations.- 2. Homoclinic Explosions: The First Homoclinic Explosion.- 2.1. Existence of a Homoclinic Orbit.- 2.2. The Bifurcation Associated with a Homoclinic Orbit.- 2.3. Summary and Some General Definitions.- 3. Preturbulence, Strange Attractors and Geometric Models.- 3.1. Periodic Orbits for the Hopf Bifurcation.- 3.2. Preturbulence and Return Maps.- 3.3. Strange Attractor and Homoclinic Explosions.- 3.4. Geometric Models of the Lorenz Equations.- 3.5. Summary.- 4. Period Doubling and Stable Orbits.- 4.1. Three Bifurcations Involving Periodic Orbits.- 4.2. 99.524 100.795. The x2y Period Doubling Window.- 4.3. 145 166. The x2y2 Period Doubling Window.- 4.4. Intermittent Chaos.- 4.5. 214.364 ?. The Final xy Period Doubling Window.- 4.6. Noisy Periodicity.- 4.7. Summary.- 5. From Strange Attractor to Period Doubling.- 5.1. Hooked Return Maps.- 5.2. Numerical Experiments.- 5.3. Development of Return Maps as r Increases: Homoclinic Explosions and Period Doubling.- 5.4. Numerical Experiments on Periodic Orbits.- 5.5. Period Doubling and One-Dimensional Maps.- 5.6. Global Approach and Some Conjectures.- 5.7. Summary.- 6. Symbolic Description of Orbits: The Stable Manifolds of C1 and C2.- 6.1. The Maxima-in-z Method.- 6.2. Symbolic Descriptions from the Stable Manifolds of C1 and C2.- 6.3. Summary.- 7. Large r.- 7.1. The Averaged Equations.- 7.2. Analysis and Interpretation of the Averaged Equations.- 7.3. Anomalous Periodic Orbits for Small b and Large r.- 7.4. Summary.- 8. Small b.- 8.1. Twisting Around the z-Axis.- 8.2. Homoclinic Explosions with Extra Twists.- 8.3. Periodic Orbits Without Extra Twisting Around the z-Axis.- 8.4. Heteroclinic Orbits Between C1 and C2.- 8.5. Heteroclinic Bifurcations.- 8.6. General Behaviour When b = 0.25.- 8.7. Summary.- 9. Other Approaches, Other Systems, Summary and Afterword.- 9.1. Summary of Predicted Bifurcations for Varying Parameters ?, b and r.- 9.2. Other Approaches.- 9.3. Extensions of the Lorenz System.- 9.4. Afterword - A Personal View.- Appendix A. Definitions.- Appendix B. Derivation of the Lorenz Equations from the Motion of a Laboratory Water Wheel.- Appendix C. Boundedness of the Lorenz Equations.- Appendix D. Homoclinic Explosions.- Appendix E. Numerical Methods for Studying Return Maps and for Locating Periodic Orbits.- Appendix F. Computational Difficulties Involved in Calculating Trajectories which Pass Close to the Origin.- Appendix G. Geometric Models of the Lorenz Equations.- Appendix H. One-Dimensional Maps from Successive Local Maxima in z.- Appendix I. Numerically Computed Values of k(r) for ? = 10 and b = 8/3.- Appendix J. Sequences of Homoclinic Explosions.- Appendix K. Large r the Formulae.

1,463 citations

Journal ArticleDOI
TL;DR: The sparsity of the discretized systems for the computation of limit cycles and their bifurcation points is exploited by using the standard Matlab sparse matrix methods.
Abstract: MATCONT is a graphical MATLAB software package for the interactive numerical study of dynamical systems. It allows one to compute curves of equilibria, limit points, Hopf points, limit cycles, period doubling bifurcation points of limit cycles, and fold bifurcation points of limit cycles. All curves are computed by the same function that implements a prediction-correction continuation algorithm based on the Moore-Penrose matrix pseudo-inverse. The continuation of bifurcation points of equilibria and limit cycles is based on bordering methods and minimally extended systems. Hence no additional unknowns such as singular vectors and eigenvectors are used and no artificial sparsity in the systems is created. The sparsity of the discretized systems for the computation of limit cycles and their bifurcation points is exploited by using the standard Matlab sparse matrix methods. The MATLAB environment makes the standard MATLAB Ordinary Differential Equations (ODE) Suite interactively available and provides computational and visualization tools; it also eliminates the compilation stage and so makes installation straightforward. Compared to other packages such as AUTO and CONTENT, adding a new type of curves is easy in the MATLAB environment. We illustrate this by a detailed description of the limit point curve type.

1,320 citations

Book
01 Jan 1980
TL;DR: Asymptotic solutions of evolution problems bifurcation and stability of steady solution of evolution equations in one dimension imperfection theory and isolated solutions which perturb bifurbcation stability of stable solutions in two dimensions and n dimensions appendices as discussed by the authors.
Abstract: Asymptotic solutions of evolution problems bifurcation and stability of steady solutions of evolution equations in one dimension imperfection theory and isolated solutions which perturb bifurcation stability of steady solutions of evolution equations in two dimensions and n dimensions appendices - bifurcation of steady solution in two dimensions and the stability of the bifurcating solutions appendix - methods of projection for general problems of bifurcation into steady solutions bifurcation of periodic solutions from steady ones (Hopf Bifurcation) in two dimensions bifurcation of periodic solutions in the general case subharmonic bifurcation of forced T-periodic solutions subharmonic bifurcation of forced T-periodic solutions into asymptotically quasi-periodic solutions appendix - secondary subharmonic and symptotically quasi-periodic bifurcation of periodic solutions (of Hopf's type) in the autonomous case stability and bifurcation in conservative systems.

1,180 citations

Book
01 Jan 1983
TL;DR: In this article, the authors present a general framework for the analysis of linear differential equations with respect to the case of a single variable and a single matrix, as well as a case of multiple variables.
Abstract: 1. Introduction.- 1.1 What is Synergetics About?.- 1.2 Physics.- 1.2.1 Fluids: Formation of Dynamic Patterns.- 1.2.2 Lasers: Coherent Oscillations.- 1.2.3 Plasmas: A Wealth of Instabilities.- 1.2.4 Solid-State Physics: Multistability, Pulses, Chaos.- 1.3 Engineering.- 1.3.1 Civil, Mechanical, and Aero-Space Engineering: Post-Buckling Patterns, Flutter, etc.- 1.3.2 Electrical Engineering and Electronics: Nonlinear Oscillations.- 1.4 Chemistry: Macroscopic Patterns.- 1.5 Biology.- 1.5.1 Some General Remarks.- 1.5.2 Morphogenesis.- 1.5.3 Population Dynamics.- 1.5.4 Evolution.- 1.5.5 Immune System.- 1.6 Computer Sciences.- 1.6.1 Self-Organization of Computers, in Particular Parallel Computing.- 1.6.2 Pattern Recognition by Machines.- 1.6.3 Reliable Systems from Unreliable Elements.- 1.7 Economy.- 1.8 Ecology.- 1.9 Sociology.- 1.10 What are the Common Features of the Above Examples?.- 1.11 The Kind of Equations We Want to Study.- 1.11.1 Differential Equations.- 1.11.2 First-Order Differential Equations.- 1.11.3 Nonlinearity.- 1.11.4 Control Parameters.- 1.11.5 Stochasticity.- 1.11.6 Many Components and the Mezoscopic Approach.- 1.12 How to Visualize the Solutions.- 1.13 Qualitative Changes: General Approach.- 1.14 Qualitative Changes: Typical Phenomena.- 1.14.1 Bifurcation from One Node (or Focus) into Two Nodes (or Foci).- 1.14.2 Bifurcation from a Focus into a Limit Cycle (Hopf Bifurcation).- 1.14.3 Bifurcations from a Limit Cycle.- 1.14.4 Bifurcations from a Torus to Other Tori.- 1.14.5 Chaotic Attractors.- 1.14.6 Lyapunov Exponents *.- 1.15 The Impact of Fluctuations (Noise). Nonequilibrium Phase Transitions.- 1.16 Evolution of Spatial Patterns.- 1.17 Discrete Maps. The Poincare Map.- 1.18 Discrete Noisy Maps.- 1.19 Pathways to Self-Organization.- 1.19.1 Self-Organization Through Change of Control Parameters.- 1.19.2 Self-Organization Through Change of Number of Components.- 1.19.3 Self-Organization Through Transients.- 1.20 How We Shall Proceed.- 2. Linear Ordinary Differential Equations.- 2.1 Examples of Linear Differential Equations: The Case of a Single Variable.- 2.1.1 Linear Differential Equation with Constant Coefficient.- 2.1.2 Linear Differential Equation with Periodic Coefficient.- 2.1.3 Linear Differential Equation with Quasiperiodic Coefficient.- 2.1.4 Linear Differential Equation with Real Bounded Coefficient.- 2.2 Groups and Invariance.- 2.3 Driven Systems.- 2.4 General Theorems on Algebraic and Differential Equations.- 2.4.1 The Form of the Equations.- 2.4.2 Jordan's Normal Form.- 2.4.3 Some General Theorems on Linear Differential Equations.- 2.4.4 Generalized Characteristic Exponents and Lyapunov Exponents.- 2.5 Forward and Backward Equations: Dual Solution Spaces.- 2.6 Linear Differential Equations with Constant Coefficients.- 2.7 Linear Differential Equations with Periodic Coefficients.- 2.8 Group Theoretical Interpretation.- 2.9 Perturbation Approach*.- 3. Linear Ordinary Differential Equations with Quasiperiodic Coefficients*.- 3.1 Formulation of the Problem and of Theorem 3.- 3.2 Auxiliary Theorems (Lemmas).- 3.3 Proof of Assertion (a) of Theorem 3.1.1: Construction of a Triangular Matrix: Example of a 2 x 2 Matrix.- 3.4 Proof that the Elements of the Triangular Matrix C are Quasiperiodic in ? (and Periodic in ?j and Ck with Respect to ?: Exampleof a 2 x 2 Matrix.- 3.5 Construction of the Triangular Matrix C and Proof that Its Elements are Quasiperiodic in ? (and Periodic in ?j and Ck with Respect to ?): The Case of an m x m Matrix, all A's Different.- 3.6 Approximation Methods. Smoothing.- 3.6.1 A Variational Method.- 3.6.2 Smoothing.- 3.7 The Triangular Matrix C and Its Reduction.- 3.8 The General Case: Some of the Generalized Characteristic Exponents Coincide.- 3.9 Explicit Solution of (3.1.1) by an Iteration Procedure.- 4. Stochastic Nonlinear Differential Equations.- 4.1 An Example.- 4.2 The Ito Differential Equation and the Ito-Fokker-Planck Equation.- 4.3 The Stratonovich Calculus.- 4.4 Langevin Equations and Fokker-Planck Equation.- 5. The World of Coupled Nonlinear Oscillators.- 5.1 Linear Oscillators Coupled Together.- 5.1.1 Linear Oscillators with Linear Coupling.- 5.1.2 Linear Oscillators with Nonlinear Coupling. An Example. Frequency Shifts.- 5.2 Perturbations of Quasiperiodic Motion for Time-Independent Amplitudes (Quasiperiodic Motion Shall Persist).- 5.3 Some Considerations on the Convergence of the Procedure*.- 6. Nonlinear Coupling of Oscillators: The Case of Persistence of Quasiperiodic Motion.- 6.1 The Problem.- 6.2 Moser's Theorem (Theorem 6.2.1).- 6.3 The Iteration Procedure*.- 7. Nonlinear Equations. The Slaving Principle.- 7.1 An Example.- 7.1.1 The Adiabatic Approximation.- 7.1.2 Exact Elimination Procedure.- 7.2 The General Form of the Slaving Principle. Basic Equations.- 7.3 Formal Relations.- 7.4 The Iteration Procedure.- 7.5 An Estimate of the Rest Term. The Question of Differentiability.- 7.6 Slaving Principle for Discrete Noisy Maps*.- 7.7 Formal Relations*.- 7.8 The Iteration Procedure for the Discrete Case*.- 7.9 Slaving Principle for Stochastic Differential Equations*.- 8. Nonlinear Equations. Qualitative Macroscopic Changes.- 8.1 Bifurcations from a Node or Focus. Basic Transformations.- 8.2 A Simple Real Eigenvalue Becomes Positive.- 8.3 Multiple Real Eigenvalues Become Positive.- 8.4 A Simple Complex Eigenvalue Crosses the Imaginary Axis. Hopf Bifurcation.- 8.5 Hopf Bifurcation, Continued.- 8.6 Frequency Locking Between Two Oscillators.- 8.7 Bifurcation from a Limit Cycle.- 8.8 Bifurcation from a Limit Cycle: Special Cases.- 8.8.1 Bifurcation into Two Limit Cycles.- 8.8.2 Period Doubling.- 8.8.3 Subharmonics.- 8.8.4 Bifurcation to a Torus.- 8.9 Bifurcation from a Torus (Quasiperiodic Motion).- 8.10 Bifurcation from a Torus: Special Cases.- 8.10.1 A Simple Real Eigenvalue Becomes Positive.- 8.10.2 A Complex Nondegenerate Eigenvalue Crosses the Imaginary Axis.- 8.11 Instability Hierarchies, Scenarios, and Routes to Turbulence.- 8.11.1 The Landau-Hopf Picture.- 8.11.2 The Ruelle and Takens Picture.- 8.11.3 Bifurcations of Tori. Quasiperiodic Motions.- 8.11.4 The Period-Doubling Route to Chaos. Feigenbaum Sequence.- 8.11.5 The Route via Intermittency.- 9. Spatial Patterns.- 9.1 The Basic Differential Equations.- 9.2 The General Method of Solution.- 9.3 Bifurcation Analysis for Finite Geometries.- 9.4 Generalized Ginzburg-Landau Equations.- 9.5 A Simplification of Generalized Ginzburg-Landau Equations. Pattern Formation in Benard Convection.- 10. The Inclusion of Noise.- 10.1 The General Approach.- 10.2 A Simple Example.- 10.3 Computer Solution of a Fokker-Planck Equation for a Complex Order Parameter.- 10.4 Some Useful General Theorems on the Solutions of Fokker-Planck Equations.- 10.4.1 Time-Dependent and Time-Independent Solutions of the Fokker-Planck Equation, if the Drift Coefficients are Linear in the Coordinates and the Diffusion Coefficients Constant.- 10.4.2 Exact Stationary Solution of the Fokker-Planck Equation for Systems in Detailed Balance.- 10.4.3 AnExample.- 10.4.4 Useful Special Cases.- 10.5 Nonlinear Stochastic Systems Close to Critical Points: A Summary.- 11. Discrete Noisy Maps.- 11.1 Chapman-Kolmogorov Equation.- 11.2 The Effect of Boundaries. One-Dimensional Example.- 11.3 Joint Probability and Transition Probability. Forward and Backward Equation 305.- 11.4 Connection with Fredholm Integral Equation 306.- 11.5 Path Integral Solution 307.- 11.6 The Mean First Passage Time 308.- 11.7 Linear Dynamics and Gaussian Noise. Exact Time-Dependent Solution of the Chapman-Kolmogorov Equation 310.- 12. Example of an Unsolvable Problem in Dynamics.- 13. Some Comments on the Relation Between Synergetics and Other Sciences.- Appendix A: Moser's Proof of His Theorem.- A.1 Convergence of the Fourier Series.- A.2 The Most General Solution to the Problem of Theorem.- A.3 Convergent Construction.- A.4 Proof of Theorem.- References.

717 citations


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No. of papers in the topic in previous years
YearPapers
202350
2022114
202139
202064
201962
201863