On numerical approaches to the analysis of topology of the phase space for dynamical integrability

In this paper we consider the possibility to use numerical simulations for a computer assisted qualitative analysis of dynamical systems. We formulate a rather general method of recovering the obstructions to dynamical integrability for the systems that after reduction have a small number of degrees of freedom. We generalize this method using the results of KAM theory and stochastic approaches to the families of parameter depending systems. This permits to localize possible integrability regions in the parameter space. We give some examples of application of this approach to dynamical systems having a mechanical origin.