  Perturbation theory can be reformulated as dynamical theory. Then a sequence
of perturbative approximations is bijective to a trajectory of dynamical system
with discrete time, called the approximation cascade. Here we concentrate our
attention on the stability conditions permitting to control the convergence of
approximation sequences. We show that several types of mapping multipliers and
Lyapunov exponents can be introduced and, respectively, several types of
conditions controlling local stability can be formulated. The ideas are
illustrated by calculating the energy levels of an anharmonic oscillator.
