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Numerical analysis of transient orbits by the pullback method for covariant Lyapunov vector

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 algorithm 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.