On Resonances in Near Hamiltonian Systems
with Two Frequencies
Albert D Morozov
Timothy N Dragunov
Department of Mechanics and Mathematics
Nizhny Novgorod State University
Resonances in systems with 3/2 degrees of freedom, which are close to
nonlinear integrable ones, are considered. The report includes three
parts.
- Non-degenerate resonances.
- We discuss the role of limit cycles of the corresponding autonomous system in formation of structure of resonant zones and in the phenomenon of synchronization.
- For systems with nonlinear parametrical perturbation we demonstrate the opportunity of existence of limit cycles in resonant zones. They are two-dimensional invariant tori which are different from Kolmogorov ones.
- Degenerate resonances.
- The problem of existence of degenerate resonances in Hamiltonian systems is discussed.
- The influence of polyharmonical perturbation on the structure of resonant zones is discussed in conservative and non-conservative cases.
- The consideration is illustrated using examples of Duffing-Van der Pole type.
- Non-Area Preserving Maps
-
An example of non-area preserving two-dimensional map is considered which
simulate behavior of solutions in degenerate resonant zone.