Leonid Kon'kov

Dynamics of calcium ions releasing is studied in the framework of a simple electron-conformal model of a cellular Ryanodine channel, a giant protein molecule playing an important role in many biochemical processes. Taking into account only two coupled degrees of freedom (external conformal and internal electron ones), we introduce a Hamiltonian of a cellular Ryanodine channel belonging to the class of spin-boson Hamiltonians. The corresponding equations of motion constitute a nonlinear five-dimensional dynamical system with two isolated integrals of motion. Hamiltonian chaos may arise in that system as a result of a transversal intersection of stable and unstable manifolds of an unperturbed separatrix. The maximal Lyapunov exponent computed is positive in a certain range of values of control parameters. Poincare sections computed demonstrate typical patterns of Hamiltonian chaos with coexisting domains of regular and chaotic motion corresponding to regular and chaotic oscillations of the internal state of the cellular Ryanodine channel. An intermittency of those oscillations is found numerically and explained in terms of a stickiness effect of trajectories to the boundaries of stability islands in the phase space. Thus, even a single cellular Ryanodine channel is able to work in different regimes (regular, chaotic and weakly chaotic) depending on the values of the control parameters.

Problem of sound propagation in a range-dependent underwater sound channel is studied in the scope of the problem of the ray-wave correspondence. Small-scale vertical oscillations of a sound channel inhomogeneity act on near-axial rays in a resonant way. Scattering of rays on resonance leads to forming of a wide chaotic layer with fast mixing in the underlying phase space. The Husimi distribution function is used for examining of dynamics of wavepackets belonging to the chaotic layer. At high frequencies of a signal, a wavepacket diverges rapidly with range. Decreasing of frequency leads to suppressing of resonance induced by vertical oscillations of an inhomogeneity, wavepacket stops diverging and its width in the action space starts to oscillate irregularly. At the frequency of 50 Hz these oscillations are regular, that indicates suppression of chaotic diffusion.