N.N. Verichev

This paper studies autonomous and non-autonomous dynamical systems with cylindrical phase space and one and a half degrees of freedom. The diversity of applications varying from mechanical to quantum ones as well as a wide variety of dynamic properties from regular to chaotic ones allows considering such systems to be basic in models of modern nonlinear dynamics. Qualitative study has been carried out, which allowed establishing structures and bifurcations of trajectories in the phase space of autonomous system. The same has been done for the structures of trajectories in a space of Poincare mapping for the non-autonomous case. For non-autonomous system, the parameters - domain corresponding to the main resonance (i.e. to the simple synchronization) has been investigated. Using the method of the averaging, affinity of the bifurcation diagrams as a decomposition of the parameter space of autonomous system and of the effective parameters of non-autonomous one has been established. This establishes the correspondence of qualitative structures and bifurcations of trajectories of autonomous system to the structures and bifurcations of trajectories of a Poincare mapping for the non-autonomous one, which, in particular, is exemplified by the correspondence of the period doubling bifurcations of the limit cycle in the autonomous case to the period-doubling bifurcations of a tore in non-autonomous case. Results of the qualitative study are interpreted in terms of the rotation characteristic as a dependence on the constant component of its speed of rotation on the system parameters and initial conditions. Qualitative forms of the rotation characteristics for the various parameters indicating its non-trivial patterns corresponding to the chaotic attractors are present.