Volume 1, Issue 2
Qr-Based Methods for Computing Lyapunov Exponents of Stochastic Differential Equations

Felix Carbonell, Rolando Biscay & Juan Carlos Ji

DOI:

Int. J. Numer. Anal. Mod. B, 1 (2010), pp. 147-171

Published online: 2010-01

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  • Abstract

Lyapunov exponents (LEs) play a central role in the study of stability properties and asymptotic behavior of dynamical systems. However, explicit formulas for them can be derived for very few systems, therefore numerical methods are required. Such is the case of random dynamical systems described by stochastic differential equations (SDEs), for which there have been reported just a few numerical methods. The first attempts were restricted to linear equations, which have obvious limitations from the applications point of view. A more successful approach deals with nonlinear equation defined over manifolds but is effective for the computation of only the top LE. In this paper, two numerical methods for the efficient computation of all LEs of nonlinear SDEs are introduced. They are, essentially, a generalization to the stochastic case of the well known QR-based methods developed for ordinary differential equations. Specifically, a discrete and a continuous QR method are derived by combining the basic ideas of the deterministic QR methods with the classical rules of the differential calculus for the Stratanovich representation of SDEs. Additionally, bounds for the approximation errors are given and the performance of the methods is illustrated by means of numerical simulations.

  • Keywords

Lyapunov Exponents Stochastic Differential Equations QR-decomposition numerical methods

  • AMS Subject Headings

34D08 37M25 60H35.

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COPYRIGHT: © Global Science Press

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@Article{IJNAMB-1-147, author = {Felix Carbonell, Rolando Biscay and Juan Carlos Ji}, title = { Qr-Based Methods for Computing Lyapunov Exponents of Stochastic Differential Equations}, journal = {International Journal of Numerical Analysis Modeling Series B}, year = {2010}, volume = {1}, number = {2}, pages = {147--171}, abstract = {Lyapunov exponents (LEs) play a central role in the study of stability properties and asymptotic behavior of dynamical systems. However, explicit formulas for them can be derived for very few systems, therefore numerical methods are required. Such is the case of random dynamical systems described by stochastic differential equations (SDEs), for which there have been reported just a few numerical methods. The first attempts were restricted to linear equations, which have obvious limitations from the applications point of view. A more successful approach deals with nonlinear equation defined over manifolds but is effective for the computation of only the top LE. In this paper, two numerical methods for the efficient computation of all LEs of nonlinear SDEs are introduced. They are, essentially, a generalization to the stochastic case of the well known QR-based methods developed for ordinary differential equations. Specifically, a discrete and a continuous QR method are derived by combining the basic ideas of the deterministic QR methods with the classical rules of the differential calculus for the Stratanovich representation of SDEs. Additionally, bounds for the approximation errors are given and the performance of the methods is illustrated by means of numerical simulations.}, issn = {}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnamb/330.html} }
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