Numerical Techniques for Approximating Lyapunov Exponents and Their Implementation

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 integration for the Lyapunov exponents is carried out. 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 This work was supported by National Science Foundation Focused Research Grant Numbers DMS0139895, DMS-0139874, and DMS-0139824, and Grant Numbers DMS-0511533, DMS-0513438, and DMS0812800. e-mail: [email protected] e-mail: [email protected] Corresponding author. e-mail: [email protected] 1 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.