Biomedical subjects

G M Zaslavsky

Publications and source records attributed to G M Zaslavsky.

13 recordsLinked to original sources

PubMed2003

Semiclassical quantization of separatrix maps.

Quantization of energy balance equations, which describe a separatrixlike motion is presented. The method is based on an exact canonical transformation of the energy-time pair to the action-angle canonical pair, (E,t)-->(I,theta). Quantum mechanical dynamics can be studied in the framework of the new Hamiltonian. This transformation also establishes a relation between a wide class of the energy balance equations and dynamical localization of classical diffusion by quantum interference, that was studied in the field of quantum chaos. An exact solution for a simple system is presented as well.

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PubMed2003

Breaking time for the quantum chaotic attractor.

A model of a quantum dissipative system is considered in the regime when the classical limit corresponds to a chaotic attractor, and the breaking time tau(Planck) of the classical-quantum correspondence is obtained. The model describes a periodically kicked harmonic oscillator (or a particle in a constant magnetic field) with a dissipation. Another analog of this problem is the dissipative kicked Harper model. It is shown that in the limit of the so-called dying attractor, the breaking time tau(Planck) can be arbitrarily large.

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PubMed2003

Space-time complexity in Hamiltonian dynamics.

New notions of the complexity function C(epsilon;t,s) and entropy function S(epsilon;t,s) are introduced to describe systems with nonzero or zero Lyapunov exponents or systems that exhibit strong intermittent behavior with "flights," trappings, weak mixing, etc. The important part of the new notions is the first appearance of epsilon-separation of initially close trajectories. The complexity function is similar to the propagator p(t(0),x(0);t,x) with a replacement of x by the natural lengths s of trajectories, and its introduction does not assume of the space-time independence in the process of evolution of the system. A special stress is done on the choice of variables and the replacement t-->eta=ln t, s-->xi=ln s makes it possible to consider time-algebraic and space-algebraic complexity and some mixed cases. It is shown that for typical cases the entropy function S(epsilon;xi,eta) possesses invariants (alpha,beta) that describe the fractal dimensions of the space-time structures of trajectories. The invariants (alpha,beta) can be linked to the transport properties of the system, from one side, and to the Riemann invariants for simple waves, from the other side. This analog provides a new meaning for the transport exponent mu that can be considered as the speed of a Riemann wave in the log-phase space of the log-space-time variables. Some other applications of new notions are considered and numerical examples are presented.

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PubMed2002

Chaos and flights in the atom-photon interaction in cavity QED.

We study dynamics of the atom-photon interaction in cavity quantum electrodynamics, considering a cold two-level atom in a single-mode high-finesse standing-wave cavity as a nonlinear Hamiltonian system with three coupled degrees of freedom: translational, internal atomic, and the field. The system proves to have different types of motion including Lévy flights and chaotic walkings of an atom in a cavity. The corresponding equations of motion for expectation values of the atom and field variables have two characteristic time scales: fast Rabi oscillations of the internal atomic and field quantities and slow translational oscillations of the center of the atom mass. It is shown that the translational motion, related to the atom recoils, is governed by an equation of a parametric nonlinear pendulum with a frequency modulated by the Rabi oscillations. This type of dynamics is chaotic with some width of the stochastic layer that is estimated analytically. The width is fairly small for realistic values of the control parameters, the normalized detuning delta and atomic recoil frequency alpha. We consider the Poincaré sections of the dynamics, compute the Lyapunov exponents, and find a range of the detuning, |delta| less, similar 3, where chaos is prominent. It is demonstrated how the atom-photon dynamics with a given value of alpha depends on the values of delta and initial conditions. Two types of Lévy flights, one corresponding to the ballistic motion of the atom and the other corresponding to small oscillations in a potential well, are found. These flights influence statistical properties of the atom-photon interaction such as distribution of Poincaré recurrences and moments of the atom position x. The simulation shows different regimes of motion, from slightly abnormal diffusion with approximately tau(1.13) at delta=1.2 to a superdiffusion with approximately tau(2.2) at delta=1.92 that corresponds to a superballistic motion of the atom with an acceleration. The obtained results can be used to find new ways to manipulate atoms, to cool and trap them by adjusting the detuning delta.

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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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PubMed2002

Quantum localization for a kicked rotor with accelerator mode islands.

Dynamical localization of classical superdiffusion for the quantum kicked rotor is studied in the semiclassical limit. Both classical and quantum dynamics of the system become more complicated under the conditions of mixed phase space with accelerator mode islands. Recently, long time quantum flights due to the accelerator mode islands have been found. By exploration of their dynamics, it is shown here that the classical-quantum duality of the flights leads to their localization. The classical mechanism of superdiffusion is due to accelerator mode dynamics, while quantum tunneling suppresses the superdiffusion and leads to localization of the wave function. Coupling of the regular type dynamics inside the accelerator mode island structures to dynamics in the chaotic sea proves increasing the localization length. A numerical procedure and an analytical method are developed to obtain an estimate of the localization length which, as it is shown, has exponentially large scaling with the dimensionless Planck's constant (tilde)h<<1 in the semiclassical limit. Conditions for the validity of the developed method are specified.

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PubMed2001

Theory and applications of ray chaos to underwater acoustics.

Chaotic ray dynamics in deep sea propagation models is considered using the approaches developed in the theory of dynamical chaos. It has been demonstrated that the mechanism of emergence of ray chaos due to overlapping of nonlinear ray-medium resonances should play an important role in long range sound propagation. Analytical estimations, supported by numerical simulations, show that for realistic values of spatial periods and sound speed fluctuation amplitudes associated with internal-wave-induced perturbations, the resonance overlapping causes stochastic instability of ray paths. The influence of the form of the smooth unperturbed sound speed profile on ray sensitivity to the perturbation is studied. Stability analysis has been conducted by constructing the Poincaré maps and examining depth differences of ray trajectories with close take-off angles. The properties of ray travel times, including fractal properties of the time front fine structures, under condition of ray chaos have been investigated. It has been shown that the coexistence of chaotic and regular rays, typical for dynamical chaos, leads to the appearance of gaps in ray travel time distributions, which are absent in unperturbed waveguides. This phenomenon has a prototype in theory of dynamical chaos called the stochastic particle acceleration. It has been shown that mesoscale inhomogeneities with greater spatial scales than that of internal waves, create irregular local waveguide channels in the vicinity of the axis (i.e., sound speed minimum) of the unperturbed waveguide. Near-axial rays propagating at small grazing angles, "jump" irregularly between these microchannels. This mechanism determines chaotic behavior of the near-axial rays.

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PubMed2001

Quantum breaking time scaling in superdiffusive dynamics.

We show that the breaking time of quantum-classical correspondence depends on the type of kinetics and the dominant origin of stickiness. For sticky dynamics of quantum kicked rotor, when the hierarchical set of islands corresponds to the accelerator mode, we demonstrate by simulation that the breaking time scales as tau(Planck's over 2pi) approximately (1/Planck's over 2pi)(1/mu) with the transport exponent mu>1 that corresponds to superdiffusive dynamics [B. Sundaram and G. M. Zaslavsky, Phys. Rev. E 59, 7231 (1999)]. We discuss also other possibilities for the breaking time scaling and transition to the logarithmic one tau(Planck's over 2pi) approximately ln(1/Planck's over 2pi) with respect to Planck's over 2pi.

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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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PubMed1999

Anomalous transport and quantum-classical correspondence.

We present evidence that anomalous transport in the classical standard map results in strong enhancement of fluctuations in the localization length of quasienergy states in the corresponding quantum dynamics. This generic effect occurs even far from the semiclassical limit and reflects the interplay of local and global quantum suppression mechanisms of classically chaotic dynamics. Possible experimental scenarios are also discussed.

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PubMed1999

Chaotic advection in compressible helical flow.

Compressible helical flow with div v not equal to 0 drastically increases the area of chaotic dynamics and mixing properties when the helicity parameter is spatially dependent. We show that the density dependence on the z coordinate can be incorporated in new variables in a way that leads to a Hamiltonian formulation of the system. This permits the application of various important results like the Kolmogorov-Arnold-Moser theory and, particularly, an understanding of why and in which sense the compressible helical flow is "more chaotic" than the incompressible one. Simulation demonstrates this property for an analog of the ABC flow. An interesting type of the dynamical system with "dense" island chains is described.

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PubMed1999

Anomalous diffusion in a running sandpile model.

To explore the character of underlying transport in a sandpile, we have followed the motion of tracer particles. Moments of the distribution function of the particle positions, =D(0)t(nnu(n)), are determined as a function of the elapsed time. The numerical results show that the transport mechanism for distances less than the sandpile length is superdiffusive with an exponent nu(n) close to 0.75, for n<1.

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PubMed1999

Immense delocalization from fractional kinetics.

We observe immense delocalization of the order of 10(9) for a kicked Harper model when a control parameter K is taken to be K*=6.349 972. This "magic" value corresponds to special phase space topology in the classical limit, when a hierarchical self-similar set of sticky islands emerges. The origin of the effect is of the general nature and similar immense delocalization can be found in other systems.

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