After presenting a classification of dynamical systems, tools of differential geometry are discussed with respect to the analysis of trajectories of dynamical systems. The description of terms defining flow characteristics of dynamical systems (e.g., divergence, rotation) is followed by discussing linearization techniques for dynamical systems.
Together with an investigation of flow behavior close to a
critical point and cycles a concept for the local analysis of a
dynamical system close to an arbitrary trajectory is presented.
This approach basically investigates perturbations orthogonal to
the chosen trajectory by determining eigenvalues and eigenvectors
of a matrix which is closely related to the Jacobian matrix of the
dynamical system but with lower dimension.