Dr Alexey Okunev, Loughborough University, Department of Mathematical Sciences
There are two major obstacles to applying the averaging method, resonances and separatrices. We study averaging method for the simplest situation where both these obstacles are present at the same time, timeperiodic perturbations of one-frequency Hamiltonian systems with separatrices. The Hamiltonian depends on a parameter that slowly changes for the perturbed system (so slow-fast Hamiltonian systems with two and a half degrees of freedom are included in our class). Solutions passing through a resonance exhibit a small quasi-random jump, this is called scattering on a resonance. Some solutions passing through a resonance can also be captured into resonance, remaining near the resonance for a long time, but this only happens for small measure of initial data. Far from separatrices there are only finitely many resonances such that capture is possible, however, such resonances can accumulate on separatrices. We estimate how the amplitude of scattering on resonances and the measure of initial data captured into resonances decrease for resonances near separatrices. We also show that the infinite number of resonances near separatrices such that capture is possible can be split into a finite number of series such that dynamics near all resonances in the same series is close to each other and can be written in terms of the Melnikov function. We obtain realistic estimates on the accuracy of averaging method and on the measure of initial data badly described by averaging method (such as initial data captured into resonances) for solutions crossing separatrices. We also prove formulas for probability of capture into different domains after separatrix crossing. Our results can also be applied to perturbations of generic two-frequency integrable systems near separatrices, as they can be reduced to periodic perturbations of one-frequency systems.
Alexey Okunev is currently a postdoc in Mathematics at Loughborough University (UK), before that he completed PhD at Higher school of Economics (Moscow). He is interested in Dynamical systems, in particular averaging method, adiabatic invariants, Milnor attractors, steep and partially hyperbolic skew products with one-dimensional fibre.