Showing papers on "Fractional dynamics published in 2020"

Analysis for fractional dynamics of Ebola virus model.

Abstract: Ebola virus is very challenging problem of the world The main purpose of this work is to study fractional Ebola virus model An efficient computational method based on iterative scheme is proposed to solve fractional Ebola model numerically Stability of proposed method is also discussed Efficiency of proposed method is shown by listing CPU time Proposed computational method will work for long time domain Numerical results are presented graphically The main reason for using this technique is low computational cost and high accuracy It is also shown how the approximate solution varies for fractional and integer order Ebola virus model

Mathematical Economics: Application of Fractional Calculus

Abstract: Mathematical economics is a theoretical and applied science in which economic objects, processes, and phenomena are described by using mathematically formalized language [...]

Applicability of time fractional derivative models for simulating the dynamics and mitigation scenarios of COVID-19

Abstract: Fractional calculus provides a promising tool for modeling fractional dynamics in computational biology, and this study tests the applicability of fractional-derivative equations (FDEs) for modeling the dynamics and mitigation scenarios of the novel coronavirus for the first time. The coronavirus disease 2019 (COVID-19) pandemic radically impacts our lives, while the evolution dynamics of COVID-19 remain obscure. A time-dependent Susceptible, Exposed, Infectious, and Recovered (SEIR) model was proposed and applied to fit and then predict the time series of COVID-19 evolution observed over the last three months (up to 3/22/2020) in China. The model results revealed that 1) the transmission, infection and recovery dynamics follow the integral-order SEIR model with significant spatiotemporal variations in the recovery rate, likely due to the continuous improvement of screening techniques and public hospital systems, as well as full city lockdowns in China, and 2) the evolution of number of deaths follows the time FDE, likely due to the time memory in the death toll. The validated SEIR model was then applied to predict COVID-19 evolution in the United States, Italy, Japan, and South Korea. In addition, a time FDE model based on the random walk particle tracking scheme, analogous to a mixing-limited bimolecular reaction model, was developed to evaluate non-pharmaceutical strategies to mitigate COVID-19 spread. Preliminary tests using the FDE model showed that self-quarantine may not be as efficient as strict social distancing in slowing COVID-19 spread. Therefore, caution is needed when applying FDEs to model the coronavirus outbreak, since specific COVID-19 kinetics may not exhibit nonlocal behavior. Particularly, the spread of COVID-19 may be affected by the rapid improvement of health care systems which may remove the memory impact in COVID-19 dynamics (resulting in a short-tailed recovery curve), while the death toll and mitigation of COVID-19 can be captured by the time FDEs due to the nonlocal, memory impact in fatality and human activities.

Fractal and fractional dynamics for a 3D autonomous and two-wing smooth chaotic system

Abstract: Some existing chaotic systems cannot display dynamics with attractors showing a fractal representation. This is due, not only to the nature of the phenomenon under description, but also to the type of derivative operator used to express the whole model. Now, the question to be asked now is can we use a derivative operator that triggers the appearance of a fractal structure in the dynamics of the system! In this work, we use the fractal-fractional derivative with a fractional order to analyze a multi-dimensional autonomous system that happens to be chaotic with multi-wing attractors. The fractal-fractional operator, which is a combination of fractal process and fractional differentiation, is a relatively new concept whose properties and features are still under investigation. After recalling the basic concepts behind fractal-fractional operator, we analyze the model both in the integer standard case and the generalized case. The integer case reveals that, under certain conditions on the parameters involved, the model is characterized by a two-wing attractor instead of four wings. Due to the impact of such a fractal-fractional operator, the system is able to maintain the two-wing attractor. Additionally, such attractor that can self-replicate in a fractal process and the observe self replication can multiply as the fractal-fractional derivative order changes. This results reveal a great feature of the fractal-fractional derivative with a fractional order, that was still unknown.

Non-local network dynamics via fractional graph Laplacians

Abstract: We introduce non-local dynamics on directed networks through the construction of a fractional version of a non-symmetric Laplacian for weighted directed graphs. Furthermore, we provide an analytic treatment of fractional dynamics for both directed and undirected graphs, showing the possibility of exploring the network employing random walks with jumps of arbitrary length. We also provide some examples of the applicability of the proposed dynamics, including consensus over multi-agent systems described by directed networks.

Nonlocal biased random walks and fractional transport on directed networks.

Abstract: In this paper, we study nonlocal random walk strategies generated with the fractional Laplacian matrix of directed networks. We present a general approach to analyzing these strategies by defining the dynamics as a discrete-time Markovian process with transition probabilities between nodes expressed in terms of powers of the Laplacian matrix. We analyze the elements of the transition matrices and their respective eigenvalues and eigenvectors, the mean first passage times and global times to characterize the random walk strategies. We apply this approach to the study of particular local and nonlocal ergodic random walks on different directed networks; we explore circulant networks, the biased transport on rings and the dynamics on random networks. We study the efficiency of a fractional random walker with bias on these structures. Effects of ergodicity loss which occur when a directed network is not any more strongly connected are also discussed.

Design of a high-gain observer for the synchronization of chimera states in neurons coupled with fractional dynamics

Abstract: In this paper, we propose a high-gain observer to synchronize chimera states in coupled neurons with fractional dynamics. The observer allows the synchronization with a master–slave topology. The master describes a dynamical system in state-space representation, whereas the slave is described by a high-gain state observer. The fractional differential equations are described by the Riemann–Liouville fractional derivative, also for non-local conformable derivatives and Atangana–Baleanu operators both in Caputo sense. We present numerical simulations involving the synchronization of Hindmarsh–Rose and Hodgking–Huxley models. The numerical simulations showed that the chimera states can be synchronized using fractional derivatives. We believe that the application of fractional operators to synchronization of Chimera states open a new direction of research in the near future.

Fractional econophysics: Market price dynamics with memory effects

Abstract: In recent years, a new branch of the econophysics has appeared and began to actively develop, which can be called fractional econophysics. We can define fractional econophysics as a new direction of research applying methods developed in physical sciences, to describe processes in economics and finance, basically those including power-law memory and spatial nonlocality. The mathematical tool of this branch of econophysics is the fractional calculus. The birth of the fractional econophysics can be dated 2000 and it can be primarily associated with the works of a group, which includes E. Scalas, F. Mainardi, R. Gorenflo, M. Raberto, in the field of the continuous-time finance. The fractional econophysics was born on the border of the centuries: the first paper was submitted to Physica A on 10 December 1999. Then a lot of work was done on the adaptation and application of fractional dynamics methods, physical model and equations, previously used in the physical sciences, to the description of processes in economics and finance. In fact, at the end of 2019 and at the beginning of 2020 there will be a twenty-year anniversary of fractional econophysics. In this paper, using the fractional econophysics approach, we consider dynamics of market prices, in which power-law memory effects are taken into account. We propose new economic model of the price dynamics in the market for a single product. In this model we assume that economic entities (merchants, buyers, suppliers) can remember how stocks of goods and their prices have changed over time.

Exact Solutions of Bernoulli and Logistic Fractional Differential Equations with Power Law Coefficients

Abstract: In this paper, we consider a nonlinear fractional differential equation. This equation takes the form of the Bernoulli differential equation, where we use the Caputo fractional derivative of non-integer order instead of the first-order derivative. The paper proposes an exact solution for this equation, in which coefficients are power law functions. We also give conditions for the existence of the exact solution for this non-linear fractional differential equation. The exact solution of the fractional logistic differential equation with power law coefficients is also proposed as a special case of the proposed solution for the Bernoulli fractional differential equation. Some applications of the Bernoulli fractional differential equation to describe dynamic processes with power law memory in physics and economics are suggested.

Solution to a Zero-Sum Differential Game with Fractional Dynamics via Approximations

Abstract: The paper deals with a zero-sum differential game in which the dynamical system is described by a fractional differential equation with the Caputo derivative of an order $$\alpha \in (0, 1).$$ The goal of the first (second) player is to minimize (maximize) a given quality index. The main contribution of the paper is the proof of the fact that this differential game has the value, i.e., the lower and upper game values coincide. The proof is based on the appropriate approximation of the game by a zero-sum differential game in which the dynamical system is described by a first-order functional differential equation of a retarded type. It is shown that the values of the approximating differential games have a limit, and this limit is the value of the original game. Moreover, the optimal players’ feedback control procedures are proposed that use the optimally controlled approximating system as a guide. An example is considered, and the results of computer simulations are presented.

Green Measures for Time Changed Markov Processes

Abstract: In this paper we study Green measures for certain classes of random time change Markov processes where the random time change are inverse subordinators. We show the existence of the Green measure for these processes under the condition of the existence of the Green measure of the original Markov processes and they coincide. Applications to fractional dynamics in given.

Fractional Dynamics Identification via Intelligent Unpacking of the Sample Autocovariance Function by Neural Networks.

Abstract: Many single-particle tracking data related to the motion in crowded environments exhibit anomalous diffusion behavior. This phenomenon can be described by different theoretical models. In this paper, fractional Brownian motion (FBM) was examined as the exemplary Gaussian process with fractional dynamics. The autocovariance function (ACVF) is a function that determines completely the Gaussian process. In the case of experimental data with anomalous dynamics, the main problem is first to recognize the type of anomaly and then to reconstruct properly the physical rules governing such a phenomenon. The challenge is to identify the process from short trajectory inputs. Various approaches to address this problem can be found in the literature, e.g., theoretical properties of the sample ACVF for a given process. This method is effective; however, it does not utilize all of the information contained in the sample ACVF for a given trajectory, i.e., only values of statistics for selected lags are used for identification. An evolution of this approach is proposed in this paper, where the process is determined based on the knowledge extracted from the ACVF. The designed method is intuitive and it uses information directly available in a new fashion. Moreover, the knowledge retrieval from the sample ACVF vector is enhanced with a learning-based scheme operating on the most informative subset of available lags, which is proven to be an effective encoder of the properties inherited in complex data. Finally, the robustness of the proposed algorithm for FBM is demonstrated with the use of Monte Carlo simulations.

Fractional dynamics and synchronization of Kuramoto oscillators with nonlocal, nonsingular and strong memory

Abstract: This work deals with the synchronization of fractional order Kuramoto oscillator. The main idea is to prove that, using fractional derivatives we can get synchronization in a Kuramoto system when the critical gain value is under the computed value. The Atangana-Baleanu-Caputo derivative is used to prove the synchronization in a system with 5, 10 and 45 coupled oscillators. The analysis developed in the present work proves that even when we use a lower value of the critical gain, we can get synchronization in some oscillator. We present the numerical solution using the Adams method.

Solvability of fractional dynamic systems utilizing measure of noncompactness

Abstract: Fractional dynamics is a scope of study in science considering the action of systems These systems are designated by utilizing derivatives of arbitrary orders In this effort, we discuss the sufficient conditions for the existence of the mild solution (m-solution) of a class of fractional dynamic systems (FDS) We deal with a new family of fractional m-solution in Rn for fractional dynamic systems To accomplish it, we introduce first the concept of (F, ψ)-contraction based on the measure of noncompactness in some Banach spaces Consequently, we establish requisite fixed point theorems (FPTs), which extend existing results following the Krasnoselskii FPT and coupled fixed point results as a outcomes of derived one Finally, we give a numerical example to verify the considered FDS, and we solve it by iterative algorithm constructed by semianalytic method with high accuracy The solution can be considered as bacterial growth system when the time interval is large

Active fault-tolerant synchronisation of fractional-order chaotic gyroscope system

Abstract: In this paper, a fractional-order adaptive sliding mode controller is designed to synchronise a fractional-order chaotic gyroscope with fractional dynamics. To this end, the error dynamic is define...

Generalized Space–Time Fractional Dynamics in Networks and Lattices

Abstract: We analyze generalized space–time fractional motions on undirected networks and lattices. The continuous-time random walk (CTRW) approach of Montroll and Weiss is employed to subordinate a space fractional walk to a generalization of the time fractional Poisson renewal process. This process introduces a non-Markovian walk with long-time memory effects and fat-tailed characteristics in the waiting time density. We analyze ‘generalized space–time fractional diffusion’ in the infinite d-dimensional integer lattice \( \mathbb {Z}^d\). We obtain in the diffusion limit a ‘macroscopic’ space–time fractional diffusion equation. Classical CTRW models such as with Laskin’s fractional Poisson process and standard Poisson process which occur as special cases are also analyzed. The developed generalized space–time fractional CTRW model contains a four-dimensional parameter space and offers therefore a great flexibility to describe real-world situations in complex systems.

Rough Homogenisation with Fractional Dynamics

Abstract: We review recent developments of slow/fast stochastic differential equations, and also present a new result on Diffusion Homogenisation Theory with fractional and non-strong-mixing noise and providing new examples. The emphasise of the review will be on the recently developed effective dynamic theory for two scale random systems with fractional noise: Stochastic Averaging and `Rough Diffusion Homogenisation Theory'. We also study the geometric models with perturbations to symmetries.

Fractional Order Internal Model Control Strategies for a Submerged Nanorobot

Abstract: Fractional calculus is a well-known tool that improves the closed loop performance of processes when compared to integer order approaches. The study presents a generalization of the popular Internal Model Control (IMC) strategy for fractional order processes of any complexity. The controller is based on the inverse fractional order transfer function to which a fractional order filter is added. The velocity of a scalable nanorobot operating in submersed environments is the chosen process to exemplify the proposed control strategy. The Fractional Order Internal Model Control (FOIMC) is developed for the fractional dynamics of the robot. The validity of method is proved through simulations regarding reference tracking, disturbance rejection and robustness to gain variations.

Dispersive Transport Described by the Generalized Fick Law with Different Fractional Operators

Abstract: The approach based on fractional advection–diffusion equations provides an effective and meaningful tool to describe the dispersive transport of charge carriers in disordered semiconductors. A fractional generalization of Fick’s law containing the Riemann–Liouville fractional derivative is related to the well-known fractional Fokker–Planck equation, and it is consistent with the universal characteristics of dispersive transport observed in the time-of-flight experiment (ToF). In the present paper, we consider the generalized Fick laws containing other forms of fractional time operators with singular and non-singular kernels and find out features of ToF transient currents that can indicate the presence of such fractional dynamics. Solutions of the corresponding fractional Fokker–Planck equations are expressed through solutions of integer-order equation in terms of an integral with the subordinating function. This representation is used to calculate the ToF transient current curves. The physical reasons leading to the considered fractional generalizations are elucidated and discussed.

Non-minimum phase viscoelastic properties of soft biological tissues.

Abstract: Understanding the viscoelastic properties of biological tissues is important because they can reveal tissue structure. This study analyzes the viscoelastic properties of soft biological tissues using a fractional dynamics model. We conducted a dynamic viscoelastic test on several porcine samples, i.e., liver, breast, and skeletal muscle tissues, using a plate-plate rheometer. We found that some soft biological tissues have non-minimum phase properties, i.e., the relationship between compliance and phase delay is not uniquely related to the non-integer derivative order in the fractional dynamics model. The experimental results show that the actual phase delay is larger than that estimated from compliance. We propose an empirical model to represent these non-minimum phase properties; a fractional Maxwell model with the fractional Hilbert transform term is proposed. The model and experimental results were highly correlated in terms of compliance and phase diagrams, and complex mechanical impedance. We also show that the amount of additional phase delay, defined as the increase in actual phase delay compared to that estimated from compliance, differs with tissue type.

Fractional Dynamics in Soccer Leagues

Abstract: This paper addresses the dynamics of four European soccer teams over the season 2018–2019. The modeling perspective adopts the concepts of fractional calculus and power law. The proposed model embeds implicitly details such as the behavior of players and coaches, strategical and tactical maneuvers during the matches, errors of referees and a multitude of other effects. The scale of observation focuses the teams’ behavior at each round. Two approaches are considered, namely the evaluation of the team progress along the league by a variety of heuristic models fitting real-world data, and the analysis of statistical information by means of entropy. The best models are also adopted for predicting the future results and their performance compared with the real outcome. The computational and mathematical modeling lead to results that are analyzed and interpreted in the light of fractional dynamics. The emergence of patterns both with the heuristic modeling and the entropy analysis highlight similarities in different national leagues and point towards some underlying complex dynamics.