2,855 research outputs found
Review of Some Promising Fractional Physical Models
Fractional dynamics is a field of study in physics and mechanics
investigating the behavior of objects and systems that are characterized by
power-law non-locality, power-law long-term memory or fractal properties by
using integrations and differentiation of non-integer orders, i.e., by methods
of the fractional calculus. This paper is a review of physical models that look
very promising for future development of fractional dynamics. We suggest a
short introduction to fractional calculus as a theory of integration and
differentiation of non-integer order. Some applications of
integro-differentiations of fractional orders in physics are discussed. Models
of discrete systems with memory, lattice with long-range inter-particle
interaction, dynamics of fractal media are presented. Quantum analogs of
fractional derivatives and model of open nano-system systems with memory are
also discussed.Comment: 38 pages, LaTe
Fractional dynamics of systems with long-range interaction
We consider one-dimensional chain of coupled linear and nonlinear oscillators
with long-range power wise interaction defined by a term proportional to
1/|n-m|^{\alpha+1}. Continuous medium equation for this system can be obtained
in the so-called infrared limit when the wave number tends to zero. We
construct a transform operator that maps the system of large number of ordinary
differential equations of motion of the particles into a partial differential
equation with the Riesz fractional derivative of order \alpha, when 0<\alpha<2.
Few models of coupled oscillators are considered and their synchronized states
and localized structures are discussed in details. Particularly, we discuss
some solutions of time-dependent fractional Ginzburg-Landau (or nonlinear
Schrodinger) equation.Comment: arXiv admin note: substantial overlap with arXiv:nlin/051201
Optimal random search, fractional dynamics and fractional calculus
What is the most efficient search strategy for the random located target
sites subject to the physical and biological constraints? Previous results
suggested the L\'evy flight is the best option to characterize this optimal
problem, however, which ignores the understanding and learning abilities of the
searcher agents. In the paper we propose the Continuous Time Random Walk (CTRW)
optimal search framework and find the optimum for both of search length's and
waiting time's distributions. Based on fractional calculus technique, we
further derive its master equation to show the mechanism of such complex
fractional dynamics. Numerous simulations are provided to illustrate the
non-destructive and destructive cases.Comment: 12 pages, 7 figure
Hybrid Systems and Control With Fractional Dynamics (II): Control
No mixed research of hybrid and fractional-order systems into a cohesive and
multifaceted whole can be found in the literature. This paper focuses on such a
synergistic approach of the theories of both branches, which is believed to
give additional flexibility and help the system designer. It is part II of two
companion papers and focuses on fractional-order hybrid control. Specifically,
two types of such techniques are reviewed, including robust control of
switching systems and different strategies of reset control. Simulations and
experimental results are given to show the effectiveness of the proposed
strategies. Part I will introduce the fundamentals of fractional-order hybrid
systems, in particular, modelling and stability of two kinds of such systems,
i.e., fractional-order switching and reset control systems.Comment: 2014 International Conference on Fractional Differentiation and its
Application, Ital
Fractional dynamics in the L\'evy quantum kicked rotor
We investigate the quantum kicked rotor in resonance subjected to momentum
measurements with a L\'evy waiting time distribution. We find that the system
has a sub-ballistic behavior. We obtain an analytical expression for the
exponent of the power law of the variance as a function of the characteristic
parameter of the L\'evy distribution and connect this anomalous diffusion with
a fractional dynamics
Fractional Dynamics and Multi-Slide Model of Human Memory
We propose a single chunk model of long-term memory that combines the basic
features of the ACT-R theory and the multiple trace memory architecture. The
pivot point of the developed theory is a mathematical description of the
creation of new memory traces caused by learning a certain fragment of
information pattern and affected by the fragments of this pattern already
retained by the current moment of time. Using the available psychological and
physiological data these constructions are justified. The final equation
governing the learning and forgetting processes is constructed in the form of
the differential equation with the Caputo type fractional time derivative.
Several characteristic situations of the learning (continuous and
discontinuous) and forgetting processes are studied numerically. In particular,
it is demonstrated that, first, the "learning" and "forgetting" exponents of
the corresponding power laws of the memory fractional dynamics should be
regarded as independent system parameters. Second, as far as the spacing
effects are concerned, the longer the discontinuous learning process, the
longer the time interval within which a subject remembers the information
without its considerable lost. Besides, the latter relationship is a linear
proportionality.Comment: Submitted to 36th Annual Conference of the Cognitive Science Society,
Quebec City, Canada, July 23-26 201
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