Showing papers on "Fractional dynamics published in 2014"

Fractional dynamics on networks: emergence of anomalous diffusion and Lévy flights.

Abstract: We introduce a formalism of fractional diffusion on networks based on a fractional Laplacian matrix that can be constructed directly from the eigenvalues and eigenvectors of the Laplacian matrix. This fractional approach allows random walks with long-range dynamics providing a general framework for anomalous diffusion and navigation, and inducing dynamically the small-world property on any network. We obtained exact results for the stationary probability distribution, the average fractional return probability, and a global time, showing that the efficiency to navigate the network is greater if we use a fractional random walk in comparison to a normal random walk. For the case of a ring, we obtain exact analytical results showing that the fractional transition and return probabilities follow a long-range power-law decay, leading to the emergence of L\'evy flights on networks. Our general fractional diffusion formalism applies to regular, random, and complex networks and can be implemented from the spectral properties of the Laplacian matrix, providing an important tool to analyze anomalous diffusion on networks.

Algorithms for testing of fractional dynamics: a practical guide to ARFIMA modelling

Abstract: In this survey paper we present a systematic methodology which demonstrates how to identify the origins of fractional dynamics. We consider three mechanisms which lead to it, namely fractional Brownian motion, fractional Levy stable motion and an autoregressive fractionally integrated moving average (ARFIMA) process but we concentrate on the ARFIMA modelling. The methodology is based on statistical tools for identification and validation of the fractional dynamics, in particular on an ARFIMA parameter estimator, an ergodicity test, a self-similarity index estimator based on sample p-variation and a memory parameter estimator based on sample mean-squared displacement. A complete list of algorithms needed for this is provided in appendices A–F. Finally, we illustrate the methodology on various empirical data and show that ARFIMA can be considered as a universal model for fractional dynamics. Thus, we provide a practical guide for experimentalists on how to efficiently use ARFIMA modelling for a large class of anomalous diffusion data.

Fractional dynamics and the TeV regime of field theory

Abstract: Fractional dynamics offers a reliable tool for the study of far-from equilibrium processes that display scale-invariant properties, dissipation and long-range correlations. This is particularly attractive when dealing with the complex dynamics generated in the deep ultraviolet regime of quantum field theory. We analyze a simple scalar field Lagrangian using Caputo derivatives and the approximation of low-level fractionality. Results may be extrapolated to more realistic field models and suggest a series of surprising implications regarding phenomena that are expected to emerge beyond the range of the standard model for particle physics.

A numerical investigation of the time distributed-order diffusion model

Abstract: Distributed-order differential models are more powerful tools to describe complex dynamical systems than the classical and fractional-order models because of their nonlocal properties. A time distributed-order diffusion model is investigated. By employing some numerical integration techniques, we approximate the distributed-order fractional model with a multi-term fractional model, which is then solved by an implicit numerical method. The stability and convergence of the numerical method is analyzed. Some numerical results are presented to demonstrate the effectiveness of the method and to exhibit the solution behavior of three different diffusion models. References I. M. Sokolov, A. V. Chechkin and J. Klafter, Distributed-order fractional kinetics, Acta Phys. Pol. B , 35:1323–1341, 2004. http://www.actaphys.uj.edu.pl/_cur/store/vol35/pdf/v35p1323.pdf . F. Liu, P. Zhuang, V. Anh, I. Turner and K. Burrage, Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation, Appl. Math. Comput. , 191:12–20, 2007. doi:10.1016/j.amc.2006.08.162 . R. Metzler, J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. , 339:1–77, 2000. doi:10.1016/S0370-1573(00)00070-3 . A. V. Chechkin, R. Gorenflo and I. M. Sokolov, Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations, Phys. Rev. E , 66:046129, 2002. doi:10.1103/PhysRevE.66.046129 . M. M. Meerschaert, E. Nane and P. Vellaisamy, Distributed-order fractional diffusions on bounded domains, J. Math. Anal. Appl. , 379:216–228, 2011. doi:10.1016/j.jmaa.2010.12.056 . C. F. Lorenz and T. T. Hartley, Variable order and distributed order fractional operators, Nonlinear Dynam. , 29:57–98, 2002. http://link.springer.com/article/10.1023/A:1016586905654 . M. Caputo, Mean fractional-order-derivatives differential equations and filters. Ann. Univ. Ferrara , 41:73–84, 1995. http://link.springer.com/article/10.1007%2FBF02826009 . K. Diethelm and N. J. Ford, Numerical analysis for distributed-order differential equations, J. Comp. Appl. Math. , 225:96–104, 2009. doi:10.1016/j.cam.2008.07.018 . H. Ye, F. Liu, V. Anh and I. Turner, Maximum principle and numerical method for the multi-term time-space Riesz–Caputo fractional differential equations, Appl. Math. Comput. , 227:531–540, 2014. doi:10.1016/j.amc.2013.11.015 . F. Liu, M. M. Meerschaert, R. J. McGough, P. Zhuang and Q. Liu, Numerical methods for solving the multi-term time-fractional wave-equation, Fract. Calc. Appl. Anal. , 16:9–25, 2013. doi:10.2478/s13540-013-0002-2 . Z.-Z. Sun and X. Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math. , 56:193–209, 2006. doi:10.1016/j.apnum.2005.03.003 .

Time-Optimal Control of Linear Fractional Systems

Abstract: Problem of time-optimal control of linear systems with fractional dynamics is treated in the paper from the convex-analytic standpoint. A linear system of fractional differential equations involving Riemann--Liouville derivatives is considered. A method to construct a control function that brings trajectory of the system to the terminal state in the shortest time is proposed in terms of attainability sets and their support functions.

Fractional Dynamics of Computer Virus Propagation

Abstract: We propose a fractional model for computer virus propagation. The model includes the interaction between computers and removable devices. We simulate numerically the model for distinct values of the order of the fractional derivative and for two sets of initial conditions adopted in the literature. We conclude that fractional order systems reveal richer dynamics than the classical integer order counterpart. Therefore, fractional dynamics leads to time responses with super-fast transients and super-slow evolutions towards the steady-state, effects not easily captured by the integer order models.

Velocity autocorrelation of a free particle driven by a Mittag-Leffler noise: fractional dynamics and temporal behaviors.

Abstract: We investigate the dynamical phase diagram of the generalized Langevin equation of the free particle driven by a Mittag-Leffler noise and show critical curves and a critical value of the exponent parameter of the Mittag-Leffler function that mark different dynamical regimes. By considering that the modeling of a Mittag-Leffer memory kernel corresponds to a power-law second-order memory kernel, we show that the generalized Langevin equation of the velocity autocorrelation function (VACF) is transformed in a fractional Langevin equation. In the superdiffusive case our results exhibit oscillations and negative correlations of the VACF that are not provided by the usual power-law noise model.

Model predictive control for fractional-order system a modeling and approximation based analysis

Abstract: A widely recognized advanced control methodology model predictive control is applied to solve a classical servo problem in the context of linear fractional-order (FO) system with the help of an approximation method In model predictive control, a finite horizon optimal control problem is solved at each sampling instant to obtain the current control action The optimization delivers an optimal control sequence and the first control thus obtained is applied to the plant An important constituent of this type of control is the accuracy of the model For a system with fractional dynamics, accurate model can be obtained using fractional calculus One of the methods to implement such a model for control purpose is Oustaloup's recursive approximation This method delivers equivalent integer-order transfer function for a fractional-order system, which is then utilized as an internal model in model predictive control Analytically calculated output equation for FO system has been utilized to represent process model to make simulations look more realistic by considering current and initial states in process model The paper attempts to present the effect of modeling and approximations of fractional-order system on the performance of model predictive control strategy

On universality in fractional dynamics

Abstract: In this paper the author presents the results of the preliminary investigation of fractional dynamical systems based on the results of numerical simulations of fractional maps. Fractional maps are equivalent to fractional differential equations describing systems experiencing periodic kicks. Their properties depend on the value of two parameters: the non-linearity parameter, which arises from the corresponding regular dynamical systems; and the memory parameter which is the order of the fractional derivative in the corresponding non-linear fractional differential equations. The examples of the fractional Standard and Logistic maps demonstrate that phase space of non-linear 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. The author argues that discovered properties should be evident in the natural (biological, psychological, physical, etc.) and engineering systems with power-law memory.

Fractional dynamics of tethered particles in oscillatory Stokes flows

Abstract: A mechanistic model for the low-Reynolds-, high-Strouhal-number behaviour of a system consisting of a spherical particle attached to an inelastic tether under uniform sinusoidal cross-flow is presented. Unsteady history drag and virtual mass effects are considered for both the sphere and the tether. The mechanics of the problem is such that the resulting coupled fractional differential equations are linear and solvable analytically. The stationary solutions obtained in this work show that there are limiting dimensions for the length and thickness of the tether when compared to the radius of the particle that allow for the motion of the particle–tether system to simulate the motion of a free particle. These conditions exist for the range of small oscillation amplitudes that are required for keeping the particle Reynolds number smaller than unity while oscillating the particle–tether system at high frequencies (Strouhal numbers larger than unity). The fractional order model for the particle–tether system is compared against detailed experimental results for tethered particles for a wide range of experimental frequencies, including the low-frequency range where tether effects are measurable.

Rotational invariance of stochastic processes with application to fractional dynamics

Abstract: In this work we study a notion of stochastic processes which are rotationally invariant. This concept allows for simpler modelling of multi-dimensional data which exhibits rotational symmetry and is particularly useful for the treatment of systems with a complex memory structure. For this reason we introduce rotationally invariant extensions of popular fractional dynamics models. Next, we discuss a few related concepts, which may simplify analysis of a system even if rotational symmetry is broken. Finally, we show exemplary real-data application of the introduced ideas for a set of subdiffusive trajectories.

Fractional Dynamics and Multi-Slide Model of Human Memory

Abstract: 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.

Fractional dynamics of a model for HIV and TB coinfection

Abstract: In this paper we study a fractional order model for HIV and TB coinfection. We consider vertical transmission for HIV and treatment for both diseases. We analyze numerical results of the proposed model for different values of the order of the fractional derivative. The results are in agreement with the integer order model and reveal that we can extend the dynamical evolution up to new types of transients.