223 research outputs found
On passage through resonances in volume-preserving systems
Resonance processes are common phenomena in multiscale (slow-fast) systems.
In the present paper we consider capture into resonance and scattering on
resonance in 3-D volume-preserving slow-fast systems. We propose a general
theory of those processes and apply it to a class of viscous Taylor-Couette
flows between two counter-rotating cylinders. We describe the phenomena during
a single passage through resonance and show that multiple passages lead to the
chaotic advection and mixing. We calculate the width of the mixing domain and
estimate a characteristic time of mixing. We show that the resulting mixing can
be described using a diffusion equation with a diffusion coefficient depending
on the averaged effect of the passages through resonances.Comment: 23 pages and 9 Figure
On the accuracy of conservation of adiabatic invariants in slow-fast systems
Let the adiabatic invariant of action variable in slow-fast Hamiltonian
system with two degrees of freedom have two limiting values along the
trajectories as time tends to infinity. The difference of two limits is
exponentially small in analytic systems. An iso-energetic reduction and
canonical transformations are applied to transform the slow-fast systems to
form of systems depending on slowly varying parameters in a complexified phase
space. On the basis of this method an estimate for the accuracy of conservation
of adiabatic invariant is given for such systems.Comment: 27 pages, 14 figure
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