In order to analyze structure of tangent spaces of a transient orbit, we propose a new algorithm which pulls back vectors in tangent spaces along the orbit by using a calculation method of covariant Lyapunov vectors. As an example, the calculation al
gorithm has been applied to a transient orbit converging to an equilibrium in a three-dimensional ordinary differential equations. We obtain vectors in tangent spaces that converge to eigenvectors of the linearized system at the equilibrium. Further, we demonstrate that an appropriate perturbation calculated by the vectors can lead an orbit going in the direction of an eigenvector of the linearized system at the equilibrium.
In the present paper, we study the finite type invariants of Gauss words. In the Polyak algebra techniques, we reduce the determination of the group structure to transformation of a matrix into its Smith normal form and we give the simplified form of
a universal finite type invariant by means of the isomorphism of this transformation. The advantage of this process is that we can implement it as a computer program. We obtain the universal finite type invariant of degree 4, 5, and 6 explicitly. Moreover, as an application, we give the complete classification of Gauss words of rank 4 and the partial classification of Gauss words of rank 5 where the distinction of only one pair remains.
