The paper gives a systematic analysis of singularities of transition
processes in dynamical systems. General dynamical systems with dependence on
parameter are studied. A system of relaxation times is constructed. Each
relaxation time depends on three variables: initial conditions, parameters $k$
of the system and accuracy $\epsilon$ of the relaxation. The singularities of
relaxation times as functions of $(x_0,k)$ under fixed $\epsilon$ are studied.
The classification of different bifurcations (explosions) of limit sets is
performed. The relations between the singularities of relaxation times and
bifurcations of limit sets are studied. The peculiarities of dynamics which
entail singularities of transition processes without bifurcations are described
as well. The analogue of the Smale order for general dynamical systems under
perturbations is constructed. It is shown that the perturbations simplify the
situation: the interrelations between the singularities of relaxation times and
other peculiarities of dynamics for general dynamical system under small
perturbations are the same as for the Morse-Smale systems.