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Lyapunov exponent and susceptibility

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Date
2017-08-23
Authors
Charbonneau, Patrick
Li, Yue Cathy
Pfister, Henry D
Yaida, Sho
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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.
Type
Journal article
Subject
cond-mat.stat-mech
cond-mat.stat-mech
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https://hdl.handle.net/10161/15347
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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

Professor in the Department 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 Corp
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