arXiv subjects

M. Edelman

Publications and source records attributed to M. Edelman.

7 recordsLinked to original sources

arXiv2008

Superdiffusion in the Dissipative Standard Map

We consider transport properties of the chaotic (strange) attractor along unfolded trajectories of the dissipative standard map. It is shown that the diffusion process is normal except of the cases when a control parameter is close to some special values that correspond to the ballistic mode dynamics. Diffusion near the related crisises is anomalous and non-uniform in time: there are large time intervals during which the transport is normal or ballistic, or even superballistic. The anomalous superdiffusion seems to be caused by stickiness of trajectories to a non-chaotic and nowhere dense invariant Cantor set that plays a similar role as cantori in Hamiltonian chaos. We provide a numerical example of such a sticky set. Distribution function on the sticky set almost coincides with the distribution function (SRB measure) of the chaotic attractor.

nlin.CD↗

arXiv2007

Dynamics of the Chain of Oscillators with Long-Range Interaction: From Synchronization to Chaos

We consider a chain of nonlinear oscillators with long-range interaction of the type 1/l^{1+alpha}, where l is a distance between oscillators and 0< alpha <2. In the continues limit the system's dynamics is described by the Ginzburg-Landau equation with complex coefficients. Such a system has a new parameter alpha that is responsible for the complexity of the medium and that strongly influences possible regimes of the dynamics. We study different spatial-temporal patterns of the dynamics depending on alpha and show transitions from synchronization of the motion to broad-spectrum oscillations and to chaos.

nlin.CD↗

arXiv2005

Stickiness of Trajectories in a Perturbed Anosov System

We consider a perturbation of the Anosov-type system, which leads to the appearance of a hierarchical set of islands-around-islands. We demonstrate by simulation that the boundaries of the islands are sticky to trajectories. This phenomenon leads to the distribution of Poincare recurrences with power-like tails in contrast to the exponential distribution in the Anosov-type systems.

nlin.CD↗

arXiv2005

Multiscale behavior and fractional kinetics from the data of solar wind - magnetosphere coupling

Multiscale phenomena are ubiquitous in nature as well in laboratories. A broad range of interacting space and time scales determines the dynamics of many systems which are inherently multiscale. In most research disciplines multiscale phenomena are not only prominent, but also they have often played the dominant role. In the solar wind - magnetosphere interaction, multiscale features coexist along with the global or coherent features. Underlying these phenomena are the mathematical and theoretical approaches such as phase transitions, turbulence, self-organization, fractional kinetics, percolation, etc. The fractional kinetic equations provide a suitable mathematical framework for multiscale behavior. In the fractional kinetic equations the multiscale nature is described through fractional derivatives and the solutions of these equations yield non-convergent moments, showing strong multiscale behavior. Using a Lévy-flights approach, we analyze the data of the magnetosphere and the solar wind. Based on this analysis we propose a model of the multiscale features and compare it with the solutions of diffusion type equations. The equation with fractional spatial derivative shows strong multiscale behavior with divergent moments. On the other hand the equation with space dependent diffusion coefficients yield convergent moments, indicating Gaussian type solutions and absence of long tails typically associated with multiscale behavior.

physics.space-ph↗

arXiv2005

Chaotic and pseudochaotic attractors of perturbed fractional oscillator

We consider a nonlinear oscillator with fractional derivative of the order alpha. Perturbed by a periodic force, the system exhibits chaotic motion called fractional chaotic attractor (FCA). The FCA is compared to the ``regular'' chaotic attractor. The properties of the FCA are discussed and the ``pseudochaotic'' case is demonstrated.

nlin.CD↗

arXiv2002

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

We study dynamics of the atom-photon interaction in cavity quantum electrodynamics (QED), 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. 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 $δ$ and atomic recoil frequency $α$. It is demonstrated how the atom-photon dynamics with a given value of $α$ depends on the values of $δ$ and initial conditions. Two types of Lévy flights, one corresponding to the ballistic motion of the atom and another one 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 $ \simτ^{1.13}$ at $δ=1.2$ to a superdiffusion with $ \sim τ^{2.2}$ at $δ=1.92$ 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 $δ$.

nlin.CD↗

arXiv2002

Pseudochaos

A family of the billiard-type systems with zero Lyapunov exponent is considered as an example of dynamics which is between the regular one and chaotic mixing. This type of dynamics is called ``pseudochaos''. We demonstrate how the fractional kinetic equation can be introduced for the pseudochaos and how the main critical exponents of the fractional kinetics can be evaluated from the dynamics. Problems related to pseudochaos are discussed: Poincare recurrences, continued fractions, log-periodicity, rhombic billiards, and others. Pseudochaotic dynamics and fractional kinetics can be applied to streamlines or magnetic field lines behavior.

nlin.CD↗