Lyapunov exponent and susceptibility

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2017-08-23

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Abstract

Lyapunov exponents characterize the chaotic nature of dynamical systems by quantifying the growth rate of uncertainty associated with imperfect measurement of the initial conditions. Finite-time estimates of the exponent, however, experience fluctuations due to both the initial condition and the stochastic nature of the dynamical path. The scale of these fluctuations is governed by the Lyapunov susceptibility, the finiteness of which typically provides a sufficient condition for the law of large numbers to apply. Here, we obtain a formally exact expression for this susceptibility in terms of the Ruelle dynamical zeta function. We further show that, for systems governed by sequences of random matrices, the cycle expansion of the zeta function enables systematic computations of the Lyapunov susceptibility and of its higher-moment generalizations. The method is here applied to a class of dynamical models that maps to static disordered spin chains with interactions stretching over a varying distance, and is tested against Monte Carlo simulations.

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Scholars@Duke

Charbonneau

Patrick Charbonneau

Professor of Chemistry

Professor Charbonneau studies soft matter. His work combines theory and simulation to understand the glass problem, protein crystallization, microphase formation, and colloidal assembly in external fields.

Pfister

Henry Pfister

Jeffrey N. Vinik Associate Professor of Electrical and Computer Engineering

Henry D. Pfister received his Ph.D. in Electrical Engineering in 2003 from the University of California, San Diego and is currently a professor in the Electrical and Computer Engineering Department of Duke University with a secondary appointment in Mathematics.  Prior to that, he was an associate professor at Texas A&M University (2006-2014), a post-doctoral fellow at the École Polytechnique Fédérale de Lausanne (2005-2006), and a senior engineer at Qualcomm Corporate R&D in San Diego (2003-2004).  His current research interests include information theory, error-correcting codes, quantum computing, and machine learning.

He received the NSF Career Award in 2008 and a Texas A&M ECE Department Outstanding Professor Award in 2010.  He is a coauthor of the 2007 IEEE COMSOC best paper in Signal Processing and Coding for Data Storage and a coauthor of a 2016 Symposium on the Theory of Computing (STOC) best paper.  He has served the IEEE Information Theory Society as a member of the Board of Governors (2019-2022), an Associate Editor for the IEEE Transactions on Information Theory (2013-2016), and a Distinguished Lecturer (2015-2016).  He was also the General Chair of the 2016 North American School of Information Theory.


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