Mark Edelman

Weak mixing and anomalous kinetics along filamented surfaces

We consider chaotic properties of a particle in a square billiard with a horizontal bar in the middle. Such a system can model field-line windings of the merged surfaces. The system has weak-mixing properties with zero Lyapunov exponent and entropy, and it can be also interesting as an example of a system with intermediate chaotic properties, between the integrability and strong mixing. We show that the transport is anomalous and that its properties can be linked to the ergodic properties of continued fractions. The distribution of Poincare recurrences, distribution of the displacements, and the moments of the truncated distribution of the displacements are obtained. Connections between different exponents are found. It is shown that the distribution function of displacements and its truncated moments as a function of time exhibit log-periodic oscillations (modulations) with a universal period T(log)=pi(2)/12 ln 2. We note that similar results are valid for a family of billiard, particularly for billiards with square-in-square geometry. (c) 2001 American Institute of Physics.

Fractional dissipative standard map

Using kicked differential equations of motion with derivatives of noninteger orders, we obtain generalizations of the dissipative standard map. The main property of these generalized maps, which are called fractional maps, is long-term memory. The memory effect in the fractional maps means that their present state of evolution depends on all past states with special forms of weights. Already a small deviation of the order of derivative from the integer value corresponding to the regular dissipative standard map (small memory effects) leads to the qualitatively new behavior of the corresponding attractors. The fractional dissipative standard maps are used to demonstrate a new type of fractional attractors in the wide range of the fractional orders of derivatives.

Fractional standard map

Properties of the phase space of the standard map with memory are investigated. This map was obtained from a kicked fractional differential equation. Depending on the value of the map parameter and the fractional order of the derivative in the original differential equation, this nonlinear dynamical system demonstrates attractors (fixed points, stable periodic trajectories, slow converging and slow diverging trajectories, ballistic trajectories, and fractal-like structures) and/or chaotic trajectories. At least one type of fractal-like sticky attractors in the chaotic sea was observed.

Dynamics of the chain of forced oscillators with long-range interaction : From synchronization to chaos

We consider a chain of nonlinear oscillators with long-range interaction of the type 1l(1+alpha), where l is a distance between oscillators and 0<alpha<2. In the continuous limit, the systems dynamics is described by a fractional generalization of 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, especially near alpha=2 and alpha=1. We study different spatiotemporal patterns of the dynamics depending on alpha and show transitions from synchronization of the motion to broad-spectrum oscillations and to chaos.

Fractional Maps as Maps with Power-Law Memory

The study of systems with memory requires methods which are different from the methods used in regular dynamics. Systems with power-law memory in many cases can be described by fractional differential equations, which are integro-differential equations. To study the general properties of nonlinear fractional dynamical systems we use fractional maps, which are discrete nonlinear systems with power-law memory derived from fractional differential equations. To study fractional maps we use the notion of α-families of maps depending on a single parameter α > 0 which is the order of the fractional derivative in a nonlinear fractional differential equation describing a system experiencing periodic kicks. α-families of maps represent a very general form of multi-dimensional nonlinear maps with power-law memory, in which the weight of the previous state at time t i in defining the present state at time t is proportional to \({(t - t_{i})}^{\alpha -1}\). They may be applicable to studying some systems with memory such as viscoelastic materials, electromagnetic fields in dielectric media, Hamiltonian systems, adaptation in biological systems, human memory, etc. Using the fractional logistic and standard α-families of maps as examples we demonstrate that the phase space of nonlinear fractional dynamical systems may contain periodic sinks, attracting slow diverging trajectories, attracting accelerator mode trajectories, chaotic attractors, and cascade of bifurcations type trajectories whose properties are different from properties of attractors in regular dynamical systems.

Universal fractional map and cascade of bifurcations type attractors.

We modified the way in which the Universal Map is obtained in the regular dynamics to derive the Universal α-Family of Maps depending on a single parameter α>0, which is the order of the fractional derivative in the nonlinear fractional differential equation describing a system experiencing periodic kicks. We consider two particular α-families corresponding to the Standard and Logistic Maps. For fractional α<2 in the area of parameter values of the transition through the period doubling cascade of bifurcations from regular to chaotic motion in regular dynamics corresponding fractional systems demonstrate a new type of attractors--cascade of bifurcations type trajectories.

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 for the cases when a control parameter is close to some special values that correspond to the ballistic mode dynamics. Diffusion near the related crises is anomalous and nonuniform 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 nonchaotic 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.

New Types of Solutions of Non-linear Fractional Differential Equations

Using the Riemann-Liouville and Caputo Fractional Standard Maps (FSM) and the Fractional Dissipative Standard Map (FDSM) as examples, we investigate types of solutions of non-linear fractional differential equations. They include periodic sinks, attracting slow diverging trajectories (ASDT), attracting accelerator mode trajectories (AMT), chaotic attractors, and cascade of bifurcations type trajectories (CBTT). New features discovered include attractors which overlap, trajectories which intersect, and CBTTs.

On the fractional Eulerian numbers and equivalence of maps with long term power-law memory (integral Volterra equations of the second kind) to Grünvald-Letnikov fractional difference (differential) equations

In this paper, we consider a simple general form of a deterministic system with power-law memory whose state can be described by one variable and evolution by a generating function. A new value of the systems variable is a total (a convolution) of the generating functions of all previous values of the variable with weights, which are powers of the time passed. In discrete cases, these systems can be described by difference equations in which a fractional difference on the left hand side is equal to a total (also a convolution) of the generating functions of all previous values of the systems variable with the fractional Eulerian number weights on the right hand side. In the continuous limit, the considered systems can be described by the Grünvald-Letnikov fractional differential equations, which are equivalent to the Volterra integral equations of the second kind. New properties of the fractional Eulerian numbers and possible applications of the results are discussed.

Caputo standard α-family of maps: Fractional difference vs. fractional

In this paper, the author compares behaviors of systems which can be described by fractional differential and fractional difference equations using the fractional and fractional difference Caputo standard α-Families of maps as examples. The author shows that properties of fractional difference maps (systems with falling factorial-law memory) are similar to the properties of fractional maps (systems with power-law memory). The similarities (types of attractors, power-law convergence of trajectories, existence of cascade of bifurcations and intermittent cascade of bifurcations type trajectories, and dependence of properties on the memory parameter α) and differences in properties of falling factorial- and power-law memory maps are investigated.