The Lorenz Attractor is Mixing
TL;DR: In this article, the authors study a class of geometric Lorenz flows, introduced independently by Afraimovic, Bykov & Sil'nikov and by Guckenheimer & Williams, and give a verifiable condition for such flows to be mixing.
Abstract: We study a class of geometric Lorenz flows, introduced independently by Afraimovic, Bykov & Sil'nikov and by Guckenheimer & Williams, and give a verifiable condition for such flows to be mixing. As a consequence, we show that the classical Lorenz attractor is mixing.
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TL;DR: A universal, data-driven decomposition of chaos as an intermittently forced linear system is presented, combining delay embedding and Koopman theory to decompose chaotic dynamics into a linear model in the leading delay coordinates with forcing by low-energy delay coordinates.
Abstract: Understanding the interplay of order and disorder in chaos is a central challenge in modern quantitative science. Approximate linear representations of nonlinear dynamics have long been sought, driving considerable interest in Koopman theory. We present a universal, data-driven decomposition of chaos as an intermittently forced linear system. This work combines delay embedding and Koopman theory to decompose chaotic dynamics into a linear model in the leading delay coordinates with forcing by low-energy delay coordinates; this is called the Hankel alternative view of Koopman (HAVOK) analysis. This analysis is applied to the Lorenz system and real-world examples including Earth’s magnetic field reversal and measles outbreaks. In each case, forcing statistics are non-Gaussian, with long tails corresponding to rare intermittent forcing that precedes switching and bursting phenomena. The forcing activity demarcates coherent phase space regions where the dynamics are approximately linear from those that are strongly nonlinear. The huge amount of data generated in fields like neuroscience or finance calls for effective strategies that mine data to reveal underlying dynamics. Here Brunton et al.develop a data-driven technique to analyze chaotic systems and predict their dynamics in terms of a forced linear model.
383 citations
TL;DR: This work establishes the convergence of a class of numerical algorithms, known as dynamic mode decomposition (DMD), for computation of the eigenvalues and eigenfunctions of the infinite-dimensional Koopman operator, and shows that the singular value decomposition, which is the central part of most DMD algorithms, converges to the proper orthogonal decomposition of observables.
Abstract: We establish the convergence of a class of numerical algorithms, known as Dynamic Mode Decomposition (DMD), for computation of the eigenvalues and eigenfunctions of the infinite-dimensional Koopman operator. The algorithms act on data coming from observables on a state space, arranged in Hankel-type matrices. The proofs utilize the assumption that the underlying dynamical system is ergodic. This includes the classical measure-preserving systems, as well as systems whose attractors support a physical measure. Our approach relies on the observation that vector projections in DMD can be used to approximate the function projections by the virtue of Birkhoff's ergodic theorem. Using this fact, we show that applying DMD to Hankel data matrices in the limit of infinite-time observations yields the true Koopman eigenfunctions and eigenvalues. We also show that the Singular Value Decomposition, which is the central part of most DMD algorithms, converges to the Proper Orthogonal Decomposition of observables. We use this result to obtain a representation of the dynamics of systems with continuous spectrum based on the lifting of the coordinates to the space of observables. The numerical application of these methods is demonstrated using well-known dynamical systems and examples from computational fluid dynamics.
287 citations
Cites background from "The Lorenz Attractor is Mixing"
...This attractor is proven to have the mixing property which implies ergodicity [23]....
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TL;DR: In this paper, a universal, data-driven decomposition of chaos as an intermittently forced linear system is presented, which combines Takens' delay embedding with modern Koopman operator theory and sparse regression to obtain linear representations of strongly nonlinear dynamics.
Abstract: Understanding the interplay of order and disorder in chaotic systems is a central challenge in modern quantitative science. We present a universal, data-driven decomposition of chaos as an intermittently forced linear system. This work combines Takens' delay embedding with modern Koopman operator theory and sparse regression to obtain linear representations of strongly nonlinear dynamics. The result is a decomposition of chaotic dynamics into a linear model in the leading delay coordinates with forcing by low energy delay coordinates; we call this the Hankel alternative view of Koopman (HAVOK) analysis. This analysis is applied to the canonical Lorenz system, as well as to real-world examples such as the Earth's magnetic field reversal, and data from electrocardiogram, electroencephalogram, and measles outbreaks. In each case, the forcing statistics are non-Gaussian, with long tails corresponding to rare events that trigger intermittent switching and bursting phenomena; this forcing is highly predictive, providing a clear signature that precedes these events. Moreover, the activity of the forcing signal demarcates large coherent regions of phase space where the dynamics are approximately linear from those that are strongly nonlinear.
269 citations
TL;DR: In this paper, the convergence of a class of numerical algorithms, known as dynamic mode decomposition (DMD), for computation of the eigenvalues and eigenfunctions of the infinite-dimensional Koopman operator is established.
Abstract: We establish the convergence of a class of numerical algorithms, known as dynamic mode decomposition (DMD), for computation of the eigenvalues and eigenfunctions of the infinite-dimensional Koopman operator. The algorithms act on data coming from observables on a state space, arranged in Hankel-type matrices. The proofs utilize the assumption that the underlying dynamical system is ergodic. This includes the classical measure-preserving systems, as well as systems whose attractors support a physical measure. Our approach relies on the observation that vector projections in DMD can be used to approximate the function projections by the virtue of Birkhoff's ergodic theorem. Using this fact, we show that applying DMD to Hankel data matrices in the limit of infinite-time observations yields the true Koopman eigenfunctions and eigenvalues. We also show that the singular value decomposition, which is the central part of most DMD algorithms, converges to the proper orthogonal decomposition of observables. We use...
215 citations
31 Dec 2010
TL;DR: An overview of homoclinic and heteroclincic bifurcation theory for autonomous vector fields is given in this article, where the main analytic and geometric techniques such as Lin's method, Shil'nikov variables, and center manifolds are discussed.
Abstract: An overview of homoclinic and heteroclinic bifurcation theory for autonomous vector fields is given. Specifically, homoclinic and heteroclinic bifurcations of codimension one and two in generic, equivariant, reversible, and conservative systems are reviewed, and results pertaining to the existence of multi-round homoclinic and periodic orbits and of complicated dynamics such as suspended horseshoes and attractors are stated. Bifurcations of homoclinic orbits from equilibria in local bifurcations are also considered. The main analytic and geometric techniques such as Lin’s method, Shil’nikov variables and homoclinic centre manifolds for analyzing these bifurcations are discussed. Finally, a few related topics, such as topological moduli, numerical algorithms, variational methods, and extensions to singularly perturbed and infinite-dimensional systems, are reviewed briefly.
208 citations
Additional excerpts
...We refer to [6, 12, 15, 262] for further results, addressing primarily ergodic properties, Lorenz-like and other singular hyperbolic attractors....
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References
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TL;DR: In this paper, it was shown that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states, and systems with bounded solutions are shown to possess bounded numerical solutions.
Abstract: Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states. Systems with bounded solutions are shown to possess bounded numerical solutions.
16,554 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
"The Lorenz Attractor is Mixing" refers background in this paper
...…19, 24, 29] and culminating in the work of Tucker [26, 27] gives the following statement (see [25,28] for detailed surveys): The Lorenz equations admit a robust attractor A which supports a “physical” ergodic invariant probability measure ν with a positive Lyapunov exponent Recall that the…...
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...In Section 4, we discuss extensions of our main results and some related future directions....
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TL;DR: In this article, the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts (though not necessarily all) of their phase spaces is discussed.
Abstract: This paper is about the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts (though not necessarily all) of their phase spaces. The main results are for discrete time systems. To put this work into context, recall that for Axiom A attractors the picture has been fairly complete since the 1970’s (see [S1], [B], [R2]). Since then much progress has been made on two fronts: there is a general nonuniform theory that deals with properties common to all diffeomorphisms with nonzero Lyapunov exponents ([O], [P1], [Ka], [LY]), and there are detailed analyses of specific kinds of dynamical systems including, for example, billiards, 1-dimensional and Henon-type maps ([S2], [BSC]; [HK], [J]; [BC2], [BY1]). Statistical properties such as exponential decay of correlations are not enjoyed by all diffeomorphisms with nonzero Lyapunov exponents. The goal of this paper is a systematic understanding of these and other properties for a class of dynamical systems larger than Axiom A. This class will not be defined explicitly, but it includes some of the much studied examples. By looking at regular returns to sets with good hyperbolic properties, one could give systems in this class a simple dynamical representation. Conditions for the existence of natural invariant measures, exponential mixing and central limit theorems are given in terms of the return times. These conditions can be checked in concrete situations, giving a unified way of proving a number of results, some new and some old. Among the new results are the exponential decay of correlations for a class of scattering billiards and for a positive measure set of Henon-type maps.
875 citations
"The Lorenz Attractor is Mixing" refers background in this paper
...However, it is plausible that the Poincaré map P could be modelled by an “unquotiented” Young tower as in [30] to which the ideas in this paper might still be applicable....
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TL;DR: In this paper, the authors considered the setting of a map making "nice" return to a reference set, and defined criteria for the existence of equilibria, speed of convergence to equilibrium, and central limit theorem in terms of the tail of the return time function.
Abstract: The setting of this paper consists of a map making “nice” returns to a reference set. Criteria for the existence of equilibria, speed of convergence to equilibria and for the central limit theorem are given in terms of the tail of the return time function. The abstract setting considered arises naturally in differentiable dynamical systems with some expanding or hyperbolic properties.
794 citations
"The Lorenz Attractor is Mixing" refers methods in this paper
...By conditions (a)–(c), the map f : I → I can be modelled by a Young tower [31] with base Y ....
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Book•
01 Jan 1969
780 citations
"The Lorenz Attractor is Mixing" refers background in this paper
...Anosov [4] showed that geodesic flows for compact manifolds of negative curvature are mixing, and this was generalised [16] to include contact flows....
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