** Authors:**
İnanç BİROL, Avadis HACINLAYAN

** Abstract: **
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 construction 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 construc one.
To construc 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, {\it 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.

** Keywords: **