Analysis of Numerical Methods Suitable for Computing Lyapunov Exponents

Two standard methods for numerically estimating Lyapunov exponents are reviewed and it is noted that a numerical integration scheme that preserves orthonormality is required. A procedure is introduced for modifying arbitrary rth order numerical schemes to preserve orthonormality. Convergence is shown for the particular case when explicit Euler's method is taken as the arbitrary method. This motivates looking at more general systems of ordinary diierential equations which conserve orthonormality. An arbitrary convergent numerical method is modiied to preserve orthonormality and convergence discussed in this general case. In each case numerical results are presented for the estimation of Lyapunov exponents for the Lorenz equations and results compared with those of other authors.