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Research Papers

# Numerical Techniques for Approximating Lyapunov Exponents and Their Implementation

[+] Author and Article Information
Luca Dieci

School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332dieci@math.gatech.edu

Michael S. Jolly

Department of Mathematics, Indiana University, Bloomington, IN 47405msjolly@indiana.edu

Erik S. Van Vleck1

Department of Mathematics, University of Kansas, Lawrence, KS 66045evanvleck@math.ku.edu

1

Corresponding author.

J. Comput. Nonlinear Dynam 6(1), 011003 (Sep 27, 2010) (7 pages) doi:10.1115/1.4002088 History: Received May 27, 2009; Revised April 20, 2010; Published September 27, 2010; Online September 27, 2010

## Abstract

The algorithms behind a toolbox for approximating Lyapunov exponents of nonlinear differential systems by $QR$ methods are described. The basic solvers perform integration of the trajectory and approximation of the Lyapunov exponents simultaneously. That is, they integrate for the trajectory at the same time, and with the same underlying schemes, as is carried out for integration of the Lyapunov exponents. Separate computational procedures solve small systems for which the Jacobian matrix can be computed and stored, and for large systems for which the Jacobian cannot be stored, and may not even be explicitly known. If it is known, the user has the option to provide the action of the Jacobian on a vector. An alternative strategy is also presented in which one may want to approximate the trajectory with a specialized solver, linearize around the computed trajectory, and then carry out the approximation of the Lyapunov exponents using techniques for linear problems.

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## Figures

Figure 1

First seven approximate Lyapunov exponents versus time for the model of atmospheric variability 23.

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