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119
Interdisciplinary application of nonlinear time series methods
 Phys. Reports
, 1998
"... This paper reports on the application to field measurements of time series methods developed on the basis of the theory of deterministic chaos. The major difficulties are pointed out that arise when the data cannot be assumed to be purely deterministic and the potential that remains in this situatio ..."
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Cited by 88 (4 self)
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This paper reports on the application to field measurements of time series methods developed on the basis of the theory of deterministic chaos. The major difficulties are pointed out that arise when the data cannot be assumed to be purely deterministic and the potential that remains in this situation is discussed. For signals with weakly nonlinear structure, the presence of nonlinearity in a general sense has to be inferred statistically. The paper reviews the relevant methods and discusses the implications for deterministic modeling. Most field measurements yield nonstationary time series, which poses a severe problem for their analysis. Recent progress in the detection and understanding of nonstationarity is reported. If a clear signature of approximate determinism is found, the notions of phase space, attractors, invariant manifolds etc. provide a convenient framework for time series analysis. Although the results have to be interpreted with great care, superior performance can be achieved for typical signal processing tasks. In particular, prediction and filtering of signals are discussed, as well as the classification of system states by means of time series recordings.
Central Limit Theorems and Invariance Principles for Lorenz Attractors
, 2006
"... We prove statistical limit laws for Hölder observations of the Lorenz attractor, and more generally for geometric Lorenz attractors. In particular, we prove the almost sure invariance principle (approximation by Brownian motion). Standard consequences of this result include the central limit theo ..."
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Cited by 40 (17 self)
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We prove statistical limit laws for Hölder observations of the Lorenz attractor, and more generally for geometric Lorenz attractors. In particular, we prove the almost sure invariance principle (approximation by Brownian motion). Standard consequences of this result include the central limit theorem, the law of the iterated logarithm, and the functional versions of these results.
Statistical Limit Theorems for Suspension Flows
, 2004
"... In dynamical systems theory, a standard method for passing from discrete time to continuous time is to construct the suspension flow under a roof function. ..."
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Cited by 39 (17 self)
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In dynamical systems theory, a standard method for passing from discrete time to continuous time is to construct the suspension flow under a roof function.
Exponential error terms for growth functions of negatively curved surfaces
 Amer. J. Math
, 1998
"... Exponential error terms for growth functions on negatively curved surfaces ..."
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Cited by 38 (14 self)
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Exponential error terms for growth functions on negatively curved surfaces
Rapid decay of correlations for nonuniformly hyperbolic flows
 Trans. Amer. Math. Soc
, 2007
"... Abstract. We show that superpolynomial decay of correlations (rapid mixing) is prevalent for a class of nonuniformly hyperbolic flows. These flows are the continuous time analogue of the class of nonuniformly hyperbolic maps for which Young proved exponential decay of correlations. The proof combin ..."
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Cited by 32 (12 self)
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Abstract. We show that superpolynomial decay of correlations (rapid mixing) is prevalent for a class of nonuniformly hyperbolic flows. These flows are the continuous time analogue of the class of nonuniformly hyperbolic maps for which Young proved exponential decay of correlations. The proof combines techniques of Dolgopyat and operator renewal theory. It follows from our results that planar periodic Lorentz flows with finite horizons and flows near homoclinic tangencies are typically rapid mixing. 1.
Spectra of Ruelle transfer operators for Axiom A flows on basic sets
, 2008
"... For Axiom A flows on basic sets satisfying certain additional conditions we prove strong spectral estimates for Ruelle transfer operators similar to these of Dolgopyat [D2] for transitive Anosov flows on compact manifolds with C¹ jointly nonintegrable horocycle foliations. As is now well known, s ..."
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Cited by 25 (10 self)
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For Axiom A flows on basic sets satisfying certain additional conditions we prove strong spectral estimates for Ruelle transfer operators similar to these of Dolgopyat [D2] for transitive Anosov flows on compact manifolds with C¹ jointly nonintegrable horocycle foliations. As is now well known, such results have deep implications in some related areas, e.g. in studying analytic properties of Ruelle zeta functions, closed orbit counting functions, decay of correlations for Hölder continuous potentials. The situation considered here is substantially more difficult than the Anosov case since, even under the additional conditions, in general the geometry of the basic set can be rather complicated.
Stability of mixing and rapid mixing for hyperbolic flows
 Annals of Mathematics
, 2006
"... Abstract. We obtain general results on the stability of mixing and rapid mixing (superpolynomial decay of correlations) for hyperbolic
ows. Amongst Cr Axiom A
ows, r 2, we show that there is a C2open, Crdense set of
ows for which each nontrivial hyperbolic basic set is rapid mixing. This is t ..."
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Cited by 24 (9 self)
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Abstract. We obtain general results on the stability of mixing and rapid mixing (superpolynomial decay of correlations) for hyperbolic
ows. Amongst Cr Axiom A
ows, r 2, we show that there is a C2open, Crdense set of
ows for which each nontrivial hyperbolic basic set is rapid mixing. This is the rst general result on the stability of rapid mixing (or even mixing) for Axiom A
ows that holds in a Cr, as opposed to Holder, topology. 1.
Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function
, 2009
"... Let s0 < 0 be the abscissa of absolute convergence of the dynamical zeta function Z(s) for several disjoint strictly convex compact obstacles Ki ⊂ R N, i = 1,..., κ0, κ0 ≥ 3, and let Rχ(z) = χ(−∆D − z 2) −1 χ, χ ∈ C ∞ 0 (R N), be the cutoff resolvent of the Dirichlet Laplacian −∆D in Ω = RN \ ..."
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Cited by 23 (6 self)
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Let s0 < 0 be the abscissa of absolute convergence of the dynamical zeta function Z(s) for several disjoint strictly convex compact obstacles Ki ⊂ R N, i = 1,..., κ0, κ0 ≥ 3, and let Rχ(z) = χ(−∆D − z 2) −1 χ, χ ∈ C ∞ 0 (R N), be the cutoff resolvent of the Dirichlet Laplacian −∆D in Ω = RN \ ∪ k0 i=1Ki. We prove that there exists σ1 < s0 such that Z(s) is analytic for Re(s) ≥ σ1 and the cutoff resolvent Rχ(z) has an analytic continuation for Im(z) < −σ1, Re(z)  ≥ C> 0.
Spectrum of the Ruelle operator and exponential decay of correlation for open billiard flows
 AMER. J. MATH
, 2001
"... The paper deals with the billiard flow in the exterior of several strictly convex disjoint domains in the plane with smooth boundaries satisfying an additional (visibility) condition. Using a modification of the technique of Dolgopyat, we get spectral estimates for the Ruelle operator related to a ..."
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Cited by 23 (7 self)
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The paper deals with the billiard flow in the exterior of several strictly convex disjoint domains in the plane with smooth boundaries satisfying an additional (visibility) condition. Using a modification of the technique of Dolgopyat, we get spectral estimates for the Ruelle operator related to a Markov family for the nonwandering (trapping) set of the flow similar to those of Dolgopyat in the case of transitive Anosov flows on compact manifolds with smooth jointly nonintegrable horocycle foliations. As a consequence, we get exponential decay of correlation for Hölder continuous potentials on the nonwandering set. Combining the spectral estimate for the Ruelle operator with an argument of Pollicott and Sharp, we also derive the existence of a meromorphic continuation of the dynamical zeta function of the billiard flow to a halfplane Re (s) < hT , where hT is the topological entropy of the billiard flow, and an asymptotic formula with an error term for the number () of closed orbits of least period > 0.