Lyapunov exponents of orthogonal-plus-normal cocycles
Sam Bednarski, Anthony Quas

TL;DR
This paper derives exact formulas and asymptotic behavior for Lyapunov exponents of matrix products combining orthogonal matrices with small Gaussian perturbations, revealing their dependence on the perturbation size.
Contribution
It provides a novel exact expression for Lyapunov exponents of orthogonal-plus-normal cocycles and analyzes their asymptotic behavior as the perturbation parameter approaches zero.
Findings
Exact formula for the jth Lyapunov exponent in terms of Gram-Schmidt orthogonalization.
Asymptotic expansion of Lyapunov exponents showing quadratic dependence on epsilon.
Demonstration of the exponents' behavior as epsilon tends to zero.
Abstract
We consider products of matrices of the form where is a sequence of orthogonal matrices and has independent standard normal entries and the are mutually independent. We study the Lyapunov exponents of the cocycle as a function of , giving an exact expression for the th Lyapunov exponent in terms of the Gram-Schmidt orthogonalization of . Further, we study the asymptotics of these exponents, showing that .
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Taxonomy
TopicsAdvanced Differential Equations and Dynamical Systems · Matrix Theory and Algorithms · Quantum chaos and dynamical systems
