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This web site provides access to information on the computation of Lyapunov Characteristic Exponents (LCEs) based on the work done by the Dynamics and Controls Group at the University of Southern California. Actual MATLAB code to compute LCEs is available for downloading/viewing in PDF, MS-Word and HTML format. A list of publications related to the computation of LCEs published by the group is also available.

LCEs play a key role in the study of nonlinear dynamical systems. LCEs provide a way to characterize the asymptotic behavior of nonlinear dynamical systems by measuring the mean exponential growth (or shrinking) of perturbations with respect to a nominal trajectory. LCEs are a measure of the sensitivity of the solutions of a given dynamical system to small changes in the initial conditions. One feature of chaos is the sensitive dependence on initial conditions; for a chaotic dynamical system at least one LCE must be positive. Since for non-chaotic systems all LCEs are non-positive, the presence of a positive LCE has often been used to help determine if a system is chaotic or not.

Currently you can download (view) MATLAB code to compute the LCEs for discrete dynamical systems that use the accurate, efficient and numerically stable schemes developed by the group. Programs to compute the LCEs for continuous dynamical systems are currently being developed and will be made available here when completed.

The readme file contains detailed information on the programs and how to use them. Basically three different programs are provided for the computation of the LCEs of discrete dynamical systems. The program "lyapunov.m", computes the trajectory and all the LCEs. The other two programs “thqrbp1.m” and “thqrbp2.m” are optimized to compute only the largest p LCEs. The program “thqrbp1.m” is more efficient when p is close to n (the dimension of the system), and “thqrbp2.m” is more efficient when p is close to one. Each of the programs uses two user-defined function files as input, samples of these system dependent input files are provided.

The programs provided here are “free”. All the authors request from the users is to acknowledge the use of our programs in their work by either referencing the papers in the publications list or stating the use of the programs and providing the URL : www.usc.edu\go\DynCon.

The University of Southern California does not screen or control the content on this website and thus does not guarantee the accuracy, integrity, or quality of such content. All content on this website is provided by and is the sole responsibility of the person from which such content originated, and such content does not necessarily reflect the opinions of the University administration or the Board of Trustees