ESTIMATION OF CHAOTIC BEHAVIOR BOUNDARY NEAR SINGULAR VORTEX WHEN NONSTATIONARY PERTURBATION IS NOT SMALL

The chaotic advection of a passive particle in a vortices generating by a topographic background current is analyzed [1-2]. The background currents concept [1-3] represents a wide class of vortical geophysical flows models admitting chaotic solutions. We will consider such a two-layer model, which give us an opportunity to take into consideration stratification of a fluid [3]. Thus, the dimensionless current function we used looks like [2]: ( ) () ()0lnW t y r aK rψ = − − −. It is known that the regular advection region near an elliptic point in models with singularities always exists for any perturbation, even if it is very big. The estimation of minimal regular advection boundaries may clear a passive particle transport problem in the models [3]. On the other hand, this estimation is useful to the oceanology in part of interpretation of point vortices as a finite-core vortices [1]. The question about why all trajectories are regular in the vicinity of the singular point, though they all are unstable from linear approach also is opened. A positive Lyapunov exponent is known results in occurrence of chaotic properties of trajectories [3]. It is suggested to estimate the boundaries of the regular behavior of trajectories by using the Lyapunov vector projection to the normal of a trajectory. Let us consider the eigenvector projection of the positive eigenvalue on an unperturbed trajectory normal: ( ) 2 2 ( ) /x yпр u uυ λ υ υ υ = − − +nh. The local stability is determined by eigenvalues of the matrix x y x y u u L υ υ which look like 0 0 /y x x yu u V V rλ υ υ ′ = ± − = ± − , when 0 1(1 / ) ()V r aK r = +.The divergence of trajectories which will be calculated as the sum of the absolute value of the Lyapunov vector projections with respect to its sign toward the normal is supposed to give the boundary of regular motion. The equation for this divergence looks like: