Abstract: The flow of a smectic liquid crystal in annular geometry, when an electrical potential is applied between the boundaries, is investigated using a dynamical systems approach. In particular, matrix-free continuation methods are implemented to trace branches of solutions as a parameter of the nonlinear system is changed.  Nonlinear systems of this type are generally solved using a Newton-type solver. The methods are called matrix-free methods if the Jacobian is not calculated explicitly but is approximated by its action on a vector with function evaluation, thus the corresponding linear systems may be solved using an iterative method. In this presentation, we discuss the matrix-free methods, and show results of their implementation on a model problem, specifically, the one dimensional K-S equation with periodic boundary.  The stability and the bifurcation points of steady states in the K-S equation are examined for the parameter $\alpha$ between 0 and 40. Further we discuss the implementation of the continuation code to the electroconvection time stepper.