Algebraic calculation of Liapunov exponents

Özet

All practical methods of Liapunov exponent calculations involve numerical approximations [1], based either on simulations or on some given time series. A recent algebraic approach has been formualted by Roepstorff [2], which involves the con¬struction of an algebraic basis $S_i$ with n elements for an n dimensional system dynamics, V. The basis is chosen in such a way that $S_i$ and $S_j$ are orthagonal for $i\neq j$ and their Lie brackets with the system dynamics can be represented by a linear combination of the base vectors. The real parts of the eigenvalues of the matrix A formed by the coefficients of this linear relation correspond to the Liapunov exponents of the system. Nevertheless, there is no restriction on how to select a particular basis, nor is there a method to construe one. To construe a particular basis, we thus propose to make the following restriction: let the Lie bracket of any basis vector $S_i$ with the system dynamics vector V be proportional to the basis vector itself. In that case, the matrix A will be diagonal, with the entries corresponding to the Liapunov exponents. In this work, we propose two different methods for constructing the basis vectors. First method we use attempts to make a power series expansion for the basis vectors $S_i$, and to augment the matrix A as the power series involves further terms. This approach increases the number of homogeneous equations, introducing superflous eigenvalues which are not always physically relevant. In the second method, we start with the natural basis and use Lie brackets as an iterative updating formula. This approach, as opposed to the first, restricts the numnber of eigenvalues to the order of the system, i.e., introduces no superflous eigenvalues. However, this method also possesses some problems; namely, converge and the definition of a norm on which further work is in progress.