Biomedical subjects

L Kuznetsov

Publications and source records attributed to L Kuznetsov.

3 recordsLinked to original sources

PubMed2002

Scaling invariance of the homoclinic tangle.

The structure of the homoclinic tangle of 11 / 2 degrees of freedom Hamiltonian systems in the neighborhood of the saddle point is invariant under discrete rescaling of the system's parameters. The rescaling constant is derived from the separatrix map and the Melnikov formula. Invariant manifolds for the periodically modulated Duffing oscillator are computed numerically to confirm this property. The scaling is related to the recently found invariance of the separatrix map under a discrete renormalization group. A possibility to extend the scaling invariance to different systems is demonstrated. The equivalency conditions under which two systems have the similarity of their chaotic layer structure near the saddle are derived. A numerical example shows a Duffing oscillator and a pendulum (acted on by different periodic perturbations) with the same structure of the tangle.

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PubMed2001

Chaotic advection near a three-vortex collapse.

Dynamical and statistical properties of tracer advection are studied in a family of flows produced by three point-vortices of different signs. Tracer dynamics is analyzed by numerical construction of Poincaré sections, and is found to be strongly chaotic: advection pattern in the region around the center of vorticity is dominated by a well developed stochastic sea, which grows as the vortex system's initial conditions are set closer to those leading to the collapse of the vortices; at the same time, the islands of regular motion around vortices, known as vortex cores, shrink. An estimation of the core's radii from the minimum distance of vortex approach to each other is obtained. Tracer transport was found to be anomalous: for all of the three numerically investigated cases, the variance of the tracer distribution grows faster than a linear function of time, corresponding to a superdiffusive regime. The transport exponent varies with time decades, implying the presence of multi-fractal transport features. Yet, its value is never too far from 3/2, indicating some kind of universality. Statistics of Poincaré recurrences is non-Poissonian: distributions have long power-law tails. The anomalous properties of tracer statistics are the result of the complex structure of the advection phase space, in particular, of strong stickiness on the boundaries between the regions of chaotic and regular motion. The role of the different phase space structures involved in this phenomenon is analyzed. Based on this analysis, a kinetic description is constructed, which takes into account different time and space scalings by using a fractional equation.

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PubMed2000

Passive particle transport in three-vortex flow

We study transport of tracer particles in a two-dimensional incompressible inviscid flow produced by three point vortices of equal strength. Time dependence of the flow caused by vortex motion gives rise to chaotic tracer trajectories, which fill parts of the flow plane referred to as mixing regions. For general vortex positions, a large connected mixing region (chaotic sea) is formed around vortices. It comprises a number of coherent fluid patches (islands), which do not mix with the rest of the chaotic sea, inside them particle motion is predominantly regular; three near-circular islands surrounding vortices are distinguished by their robust nature. Tracers in the chaotic sea rotate around the center of vorticity in an irregular way. Their trajectories are intermittent, long flights of almost regular motion are caused by trappings in the boundary regions of regular islands. The statistics of tracer rotation exhibits anomalous features, such as faster than linear growth of tracer ensemble variance and asymmetric probability distribution with long power tails. Exponent of the variance growth power law is different for different time ranges. Central part of the tracer distribution and its low (noninteger) moments evolve in a self-similar way, characterized by an exponent, which is different from that of the variance, and contrary to the latter is constant in time. Algebraic tails of the tracer distribution, controlling the behavior of the variance, are responsible for this effect. Long correlations in tracer motion lead to non-Poissonian distribution of Poincare recurrences in the mixing region. Analysis of long recurrences proves, that they are caused by tracer trappings inside boundary layers of islands of regular motion, which always exist inside the mixing region. Statistics of Poincare recurrences and trapping times exhibit power-law decay, indicating absence of a characteristic relaxation time. Values of the decay exponent for recurrences and for escape from the analyzed traps are very close to each other; long correlations are not dominated by a single trap, but are a cumulative effect of all of them, relative importance of a trap is determined by its size, and by its rotation frequency with respect to the background.

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