Lyapunov exponent and susceptibility
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
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https://hdl.handle.net/10161/15347Collections
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Show full item recordScholars@Duke
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.
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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