ABSTRACT
There are basically two alternatives to start with calculating Lyapunov exponents and associated invariant measures. According to Benettin or Talay, the linear equations of the dynamical system are considered. The solutions are simulated in the state space and projected simultaneously on a unit hypersphere in order to apply the multiplicative ergodic theorem of Oseledets. In a reversal of this, following an idea of Khasminskii, the projection can be performed first introducing polar coordinates. This idea was used subsequently by Kozin, Ariaratnam, and by Wedig et al.
The latter method leads to considerable improvements with respect to accuracy and efficiency for both numerical and analytical evaluation schemes. The paper describes this by presenting improved algorithms and techniques. Analytical approximations are made in order to solve higher-dimensional Fokker–Planck equations of spherical angles by means of Galerkin's methods. A further approach is devoted to iterative solutions of parabolic diffusion equations by means of simple differences schemes.
