The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
Roger A Pielke - One of the best experts on this subject based on the ideXlab platform.
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estimating the Lyapunov Exponent spectrum from short time series of low precision
Physical Review Letters, 1991Co-Authors: Xubin Zeng, R Eykholt, Roger A PielkeAbstract:We propose a new method to compute Lyapunov Exponents from limited experimental data. The method is tested on a variety of known model systems, and it is found that the algorithm can be used to obtain a reasonable Lyapunov-Exponent spectrum from only 5000 data points with a precision of ${10}^{\mathrm{\ensuremath{-}}1}$ or ${10}^{\mathrm{\ensuremath{-}}2}$ in three- or four-dimensional phase space, or 10 000 data points in five-dimensional phase space. We also apply our algorithm to the daily-averaged data of surface temperature observed at two locations in the United States to quantitatively evaluate atmospheric predictability.
Artur Avila - One of the best experts on this subject based on the ideXlab platform.
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on the spectrum and Lyapunov Exponent of limit periodic schrodinger operators
Communications in Mathematical Physics, 2009Co-Authors: Artur AvilaAbstract:We exhibit a dense set of limit periodic potentials for which the corresponding one-dimensional Schrodinger operator has a positive Lyapunov Exponent for all energies and a spectrum of zero Lebesgue measure. No example with those properties was previously known, even in the larger class of ergodic potentials. We also conclude that the generic limit periodic potential has a spectrum of zero Lebesgue measure.
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global theory of one frequency schrodinger operators i stratified analyticity of the Lyapunov Exponent and the boundary of nonuniform hyperbolicity
arXiv: Dynamical Systems, 2009Co-Authors: Artur AvilaAbstract:We study Schrodinger operators with a one-frequency analytic potential, focusing on the transition between the two distinct local regimes characteristic respectively of large and small potentials. From the dynamical point of view, the transition signals the emergence of nonuniform hyperbolicity, so the dependence of the Lyapunov Exponent with respect to parameters plays a central role in the analysis. Though often ill-behaved by conventional measures, we show that the Lyapunov Exponent is in fact remarkably regular in a ``stratified sense'' which we define: the irregularity comes from the matching of nice (analytic or smooth) functions along sets with complicated geometry. This result allows us to stablish that the ``critical set'' for the transition has at most codimension one, so for a typical potential the set of critical energies is at most countable, hence typically not seen by spectral measures. Key to our approach are two results about the dependence of the Lyapunov Exponent of one-frequency $\SL(2,\C)$ cocycles with respect to perturbations in the imaginary direction: on one hand there is a severe ``quantization'' restriction, and on the other hand ``regularity'' of the dependence characterizes uniform hyperbolicity when the Lyapunov Exponent is positive. Our method is independent of arithmetic conditions on the frequency.
Victor Galitski - One of the best experts on this subject based on the ideXlab platform.
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Lyapunov Exponent and out of time ordered correlator s growth rate in a chaotic system
Physical Review Letters, 2017Co-Authors: Efim Rozenbaum, Sriram Ganeshan, Victor GalitskiAbstract:: It was proposed recently that the out-of-time-ordered four-point correlator (OTOC) may serve as a useful characteristic of quantum-chaotic behavior, because, in the semiclassical limit ℏ→0, its rate of Exponential growth resembles the classical Lyapunov Exponent. Here, we calculate the four-point correlator C(t) for the classical and quantum kicked rotor-a textbook driven chaotic system-and compare its growth rate at initial times with the standard definition of the classical Lyapunov Exponent. Using both quantum and classical arguments, we show that the OTOC's growth rate and the Lyapunov Exponent are, in general, distinct quantities, corresponding to the logarithm of the phase-space averaged divergence rate of classical trajectories and to the phase-space average of the logarithm, respectively. The difference appears to be more pronounced in the regime of low kicking strength K, where no classical chaos exists globally. In this case, the Lyapunov Exponent quickly decreases as K→0, while the OTOC's growth rate may decrease much slower, showing a higher sensitivity to small chaotic islands in the phase space. We also show that the quantum correlator as a function of time exhibits a clear singularity at the Ehrenfest time t_{E}: transitioning from a time-independent value of t^{-1}lnC(t) at t t_{E}. We note that the underlying physics here is the same as in the theory of weak (dynamical) localization [Aleiner and Larkin, Phys. Rev. B 54, 14423 (1996)PRBMDO0163-182910.1103/PhysRevB.54.14423; Tian, Kamenev, and Larkin, Phys. Rev. Lett. 93, 124101 (2004)PRLTAO0031-900710.1103/PhysRevLett.93.124101] and is due to a delay in the onset of quantum interference effects, which occur sharply at a time of the order of the Ehrenfest time.
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Lyapunov Exponent and out of time ordered correlator s growth rate in a chaotic system
Physical Review Letters, 2017Co-Authors: Efim Rozenbaum, Sriram Ganeshan, Victor GalitskiAbstract:It was proposed recently that the out-of-time-ordered four-point correlator (OTOC) may serve as a useful characteristic of quantum-chaotic behavior, because, in the semiclassical limit $\ensuremath{\hbar}\ensuremath{\rightarrow}0$, its rate of Exponential growth resembles the classical Lyapunov Exponent. Here, we calculate the four-point correlator $C(t)$ for the classical and quantum kicked rotor---a textbook driven chaotic system---and compare its growth rate at initial times with the standard definition of the classical Lyapunov Exponent. Using both quantum and classical arguments, we show that the OTOC's growth rate and the Lyapunov Exponent are, in general, distinct quantities, corresponding to the logarithm of the phase-space averaged divergence rate of classical trajectories and to the phase-space average of the logarithm, respectively. The difference appears to be more pronounced in the regime of low kicking strength $K$, where no classical chaos exists globally. In this case, the Lyapunov Exponent quickly decreases as $K\ensuremath{\rightarrow}0$, while the OTOC's growth rate may decrease much slower, showing a higher sensitivity to small chaotic islands in the phase space. We also show that the quantum correlator as a function of time exhibits a clear singularity at the Ehrenfest time ${t}_{E}$: transitioning from a time-independent value of ${t}^{\ensuremath{-}1}\mathrm{ln}C(t)$ at $tl{t}_{E}$ to its monotonic decrease with time at $tg{t}_{E}$. We note that the underlying physics here is the same as in the theory of weak (dynamical) localization [Aleiner and Larkin, Phys. Rev. B 54, 14423 (1996); Tian, Kamenev, and Larkin, Phys. Rev. Lett. 93, 124101 (2004)] and is due to a delay in the onset of quantum interference effects, which occur sharply at a time of the order of the Ehrenfest time.
Xubin Zeng - One of the best experts on this subject based on the ideXlab platform.
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estimating the Lyapunov Exponent spectrum from short time series of low precision
Physical Review Letters, 1991Co-Authors: Xubin Zeng, R Eykholt, Roger A PielkeAbstract:We propose a new method to compute Lyapunov Exponents from limited experimental data. The method is tested on a variety of known model systems, and it is found that the algorithm can be used to obtain a reasonable Lyapunov-Exponent spectrum from only 5000 data points with a precision of ${10}^{\mathrm{\ensuremath{-}}1}$ or ${10}^{\mathrm{\ensuremath{-}}2}$ in three- or four-dimensional phase space, or 10 000 data points in five-dimensional phase space. We also apply our algorithm to the daily-averaged data of surface temperature observed at two locations in the United States to quantitatively evaluate atmospheric predictability.
R Eykholt - One of the best experts on this subject based on the ideXlab platform.
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estimating the Lyapunov Exponent spectrum from short time series of low precision
Physical Review Letters, 1991Co-Authors: Xubin Zeng, R Eykholt, Roger A PielkeAbstract:We propose a new method to compute Lyapunov Exponents from limited experimental data. The method is tested on a variety of known model systems, and it is found that the algorithm can be used to obtain a reasonable Lyapunov-Exponent spectrum from only 5000 data points with a precision of ${10}^{\mathrm{\ensuremath{-}}1}$ or ${10}^{\mathrm{\ensuremath{-}}2}$ in three- or four-dimensional phase space, or 10 000 data points in five-dimensional phase space. We also apply our algorithm to the daily-averaged data of surface temperature observed at two locations in the United States to quantitatively evaluate atmospheric predictability.
