arXiv subjects

Mark Edelman

Publications and source records attributed to Mark Edelman.

24 records · Page 2Linked to original sources

arXiv2013

Fractional Standard Map: Riemann-Liouville vs. Caputo

Properties of the phase space of the standard maps with memory obtained from the differential equations with the Riemann-Liouville and Caputo derivatives are considered. Properties of the attractors which these fractional dynamical systems demonstrate are different from properties of the regular and chaotic attractors of systems without memory: they exist in the asymptotic sense, different types of trajectories may lead to the same attracting points, trajectories may intersect, and chaotic attractors may overlap. Two maps have significant differences in the types of attractors they demonstrate and convergence of trajectories to the attracting points and trajectories. Still existence of the the most remarkable new type of attractors, "cascade of bifurcation type trajectories", is a common feature of both maps.

nlin.CD↗

arXiv2013

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.

nlin.CD↗

arXiv2012

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.

nlin.CD↗

arXiv2012

Fractional Maps and Fractional Attractors. Part I: $α$-Families of Maps

In this paper we present a uniform way to derive families of maps from the corresponding differential equations describing systems which experience periodic kicks. The families depend on a single parameter - the order of a differential equation $α> 0$. We investigate general properties of such families and how they vary with the increase in $α$ which represents increase in the space dimension and the memory of a system (increase in the weights of the earlier states). To demonstrate general properties of the $α$-families we use examples from physics (Standard $α$-family of maps) and population biology (Logistic $α$-family of maps). We show that with the increase in $α$ systems demonstrate more complex and chaotic behavior.

nlin.CD↗

arXiv2011

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.

nlin.CD↗

arXiv2009

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 parameter of the map and the fractional order of the derivative in the original differential equation this nonlinear dynamical system demonstrates attractors (fixed points, stables 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.

nlin.CD↗