Symmetric Jacobian for local Lyapunov exponents and Lyapunov stability analysis revisited

The stability analysis introduced by Lyapunov and extended by Oseledec provides an excellent tool to describe the character of nonlinear n-dimensional flows by n global exponents if these flows are stationary in time. However, here we discuss two shortcomings: (a) the local exponents fail to indicate the origin of instability where trajectories start to diverge. Instead, their time evolution contains a much stronger chaos than the trajectories, which is only eliminated by integrating over a long time. Therefore, shorter time intervals cannot be characterized correctly, which would be essential to analyse changes of chaotic character as in transients. (b) Although Oseledec uses an n dimensional sphere around a point x to be transformed into an n dimensional ellipse in first order, this local ellipse has not yet been evaluated. The aim of this contribution is to eliminate these two shortcomings. Problem (a) disappears if the Oseledec method is replaced by a frame with a ‘constraint’ as performed by Rateitschak and Klages (RK) [Rateitschak K, Klages R, Lyapunov instability for a periodic Lorentz gas thermostated by deterministic scattering. Phys Rev E 2002;65:036209/1–11]. The reasons why this method is better will be illustrated by comparing different systems. In order to analyze shorter time intervals, integrals between consecutive Poincaré points will be evaluated. The local problem (b) will be solved analytically by introducing the ‘symmetric Jacobian for local Lyapunov exponents’ and its orthogonal submatrix, which enable to search in the full phase space for extreme local separation exponents. These are close to the RK exponents but need no time integration of the RK frame. Finally, four sets of local exponents are compared: Oseledec frame, RK frame, symmetric Jacobian for local Lyapunov exponents and its orthogonal submatrix.