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190
Determining Lyapunov Exponents from a Time Series
 Physica
, 1985
"... We present the first algorithms that allow the estimation of nonnegative Lyapunov exponents from an experimental time series. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related to the exponentially fast divergence or convergence of n ..."
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Cited by 495 (1 self)
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We present the first algorithms that allow the estimation of nonnegative Lyapunov exponents from an experimental time series. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related to the exponentially fast divergence or convergence of nearby orbits in phase space. A system with one or more positive Lyapunov exponents is defined to be chaotic. Our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems: we monitor the longterm growth rate of small volume elements in an attractor. The method is tested on model systems with known Lyapunov spectra, and applied to data for the BelousovZhabotinskii reaction and CouetteTaylor flow. Contents 1.
A practical method for calculating largest Lyapunov exponents from small data sets
 PHYSICA D
, 1993
"... Detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the largest Lyapunov exponent. Lyapunov exponents quantify the exponential divergence of initially close statespace trajectories and estimate the amount of chaos in a system. We present a new m ..."
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Cited by 181 (0 self)
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Detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the largest Lyapunov exponent. Lyapunov exponents quantify the exponential divergence of initially close statespace trajectories and estimate the amount of chaos in a system. We present a new method for calculating the largest Lyapunov exponent from an experimental time series. The method follows directly from the definition of the largest Lyapunov exponent and is accurate because it takes advantage of all the available data. We show that the algorithm is fast, easy to implement, and robust to changes in the following quantities: embedding dimension, size of data set, reconstruction delay, and noise level. Furthermore, one may use the algorithm to calculate simultaneously the correlation dimension. Thus, one sequence of computations will yield an estimate of both the level of chaos and the system complexity.
Spectral Analysis Of Rank One Perturbations And Applications
 LECTURE GIVEN AT THE VANCOUVER SUMMER SCHOOL IN MATHEMATICAL PHYSICS, AUGUST 1014, 1993.
, 1994
"... A review of the general theory of selfadjoint operators of the form A+ ffB where B is rank one is presented. Applications include proofs of localization for Schrodinger operators, results on inverse spectral theory, and examples of operators with singular continuous spectrum. ..."
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Cited by 159 (29 self)
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A review of the general theory of selfadjoint operators of the form A+ ffB where B is rank one is presented. Applications include proofs of localization for Schrodinger operators, results on inverse spectral theory, and examples of operators with singular continuous spectrum.
Sets of matrices all infinite products of which converge. Linear Algebra and its Applications
, 1992
"... An infinite product IIT = lMi of matrices converges (on the right) if limi _ _ M,... Mi exists. A set Z = (Ai: i> l} of n X n matrices is called an RCP set (rightconvergent product set) if all infinite products with each element drawn from Z converge. Such sets of matrices arise in constructing ..."
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Cited by 114 (0 self)
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An infinite product IIT = lMi of matrices converges (on the right) if limi _ _ M,... Mi exists. A set Z = (Ai: i> l} of n X n matrices is called an RCP set (rightconvergent product set) if all infinite products with each element drawn from Z converge. Such sets of matrices arise in constructing selfsimilar objects like von Koch’s snowflake curve, in various interpolation schemes, in constructing wavelets of compact support, and in studying nonhomogeneous Markov chains. This paper gives necessary conditions and also some sufficient conditions for a set X to be an RCP set. These are conditions on the eigenvalues and left eigenspaces of matrices in 2 and finite products of these matrices. Necessary and sufficient conditions are given for a finite set Z to be an RCP set having a limit function M,(d) = rIT = lAd,, where d = (d,,., d,,..>, which is a continuous function on the space of all sequences d with the sequence topology. Finite RCP sets of columnstochastic matrices are completely characterized. Some results are given on the problem of algorithmically
The Lyapunov exponent and joint spectral radius of pairs of matrices are hard  when not impossible  to compute and to approximate
, 1997
"... We analyse the computability and the complexity of various definitions of spectral radii for sets of matrices. We show that the joint and generalized spectral radii of two integer matrices are not approximable in polynomial time, and that two related quantities  the lower spectral radius and th ..."
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Cited by 94 (18 self)
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We analyse the computability and the complexity of various definitions of spectral radii for sets of matrices. We show that the joint and generalized spectral radii of two integer matrices are not approximable in polynomial time, and that two related quantities  the lower spectral radius and the largest Lyapunov exponent  are not algorithmically approximable.
Teichmüller curves, triangle groups, and Lyapunov exponents
, 2005
"... We construct a Teichmüller curve uniformized by the Fuchsian triangle group ∆(m, n, ∞) for every m < n ≤ ∞. Our construction includes the Teichmüller curves constructed by Veech and Ward as special cases. The construction essentially relies on properties of hypergeometric differential operators. ..."
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Cited by 60 (5 self)
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We construct a Teichmüller curve uniformized by the Fuchsian triangle group ∆(m, n, ∞) for every m < n ≤ ∞. Our construction includes the Teichmüller curves constructed by Veech and Ward as special cases. The construction essentially relies on properties of hypergeometric differential operators. We interprete some of the socalled Lyapunov exponents of the Kontsevich–Zorich cocycle as normalized degrees of some natural line bundles on a Teichmüller curves. We determine the Lyapunov exponents for the Teichmüller curves we construct.
On the computation of Lyapunov exponents for continuous dynamical systems
 SIAM J. Numer. Anal
, 1997
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The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations, Part 1: The Stochastic semiflow, Part 2: Existence of stable and unstable manifolds
 98, Memoirs of the American Mathematical Society
, 2002
"... Abstract. The main objective of this paper is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations (see’s) and stochastic partial differential equations (spde’s) near stationary solutions. Such characterization is realized through the longterm behav ..."
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Cited by 35 (13 self)
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Abstract. The main objective of this paper is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations (see’s) and stochastic partial differential equations (spde’s) near stationary solutions. Such characterization is realized through the longterm behavior of the solution field near stationary points. The analysis falls in two parts 1, 2. In Part 1, we prove general existence and compactness theorems for C kcocycles of semilinear see’s and spde’s. Our results cover a large class of semilinear see’s as well as certain semilinear spde’s with Lipschitz and nonLipschitz terms such as stochastic reaction diffusion equations and the stochastic Burgers equation with additive infinitedimensional noise. In Part 2, stationary solutions are viewed as cocycleinvariant random points in the infinitedimensional state space. The pathwise local structure of solutions of semilinear see’s and spde’s near stationary solutions is described in terms of the almost sure longtime behavior of trajectories of the equation in relation to the stationary solution. More specifically, we establish local stable manifold theorems for semilinear see’s and spde’s (Theorems 2.4.12.4.4). These results give smooth stable and unstable manifolds in the neighborhood of a hyperbolic stationary solution of the underlying stochastic equation. The stable and unstable manifolds are stationary, live in a stationary tubular neighborhood of the stationary solution and are asymptotically invariant under the stochastic semiflow of the see/spde. Furthermore, the local stable and unstable manifolds intersect transversally at the stationary point, and the unstable manifolds have fixed finite dimension. The proof uses infinitedimensional multiplicative ergodic theory techniques, interpolation and perfection arguments (Theorem 2.2.1).
Computation Of A Few Lyapunov Exponents For Continuous And Discrete Dynamical Systems
 Appl. Numer. Math
, 1995
"... . In this paper, an error analysis of QR based methods for computing the first few Lyapunov exponents of continuous and discrete dynamical systems is given. Algorithmic developments are discussed. Implementation details, error estimators and testing are also given. 1. INTRODUCTION In this paper, w ..."
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Cited by 33 (9 self)
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. In this paper, an error analysis of QR based methods for computing the first few Lyapunov exponents of continuous and discrete dynamical systems is given. Algorithmic developments are discussed. Implementation details, error estimators and testing are also given. 1. INTRODUCTION In this paper, we consider the computation of a few Lyapunov exponents for continuous and discrete finite dimensional dynamical systems. In [DRV2], we considered approximating all of the exponents of continuous dynamical systems, and gave error analysis and convergence results, as well as algorithmic details, for the class of QR based methods. Here we consider the case in which only the first few exponents are desired. As it turns out, the extension to this case is not automatic, and new theoretical and algorithmic aspects need to be addressed. We still focus on continuous and discrete QR methods, give convergence results, algorithmic details and testing. For continuous dynamical systems, a proper implement...
Lyapunov spectral intervals: theory and computation
 SIAM J. NUMER. ANAL
, 2001
"... Different definitions of spectra have been proposed during the years to characterize the asymptotic behavior of nonautonomous linear systems. Here, we consider the spectrum based on exponential dichotomy of Sacker and Sell and the spectrum defined in terms of upper and lower Lyapunov exponents. A ..."
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Cited by 30 (13 self)
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Different definitions of spectra have been proposed during the years to characterize the asymptotic behavior of nonautonomous linear systems. Here, we consider the spectrum based on exponential dichotomy of Sacker and Sell and the spectrum defined in terms of upper and lower Lyapunov exponents. A main goal of ours is to understand to what extent these spectra are computable. By using an orthogonal change of variables transforming the system to upper triangular form, and the assumption of integral separation for the diagonal of the new triangular system, we justify how popular numerical methods, the socalled continuous QR and SVD approaches, can be used to approximate these spectra. We further discuss how to verify the property of integral separation, and hence to a posteriori infer stability of the attained spectral information. Finally, we discuss the algorithms we have used to approximate the Lyapunov and SackerSell spectra, and present some numerical results.