Fractional Dynamics Research Articles

Simulation of nonlinear fractional dynamics arising in the modeling of cognitive decision making using a new fractional neural network

By the rapid growth of available data, providing data‐driven solutions for nonlinear (fractional) dynamical systems becomes more important than before. In this paper, a new fractional neural network model that uses fractional order of Jacobi functions as its activation functions for one of the hidden layers is proposed to approximate the solution of fractional differential equations and fractional partial differential equations arising from mathematical modeling of cognitive‐decision‐making processes and several other scientific subjects. This neural network uses roots of Jacobi polynomials as the training dataset, and the Levenberg‐Marquardt algorithm is chosen as the optimizer. The linear and nonlinear fractional dynamics are considered as test examples showing the effectiveness and applicability of the proposed neural network. The numerical results are compared with the obtained results of some other networks and numerical approaches such as meshless methods. Numerical experiments are presented confirming that the proposed model is accurate, fast, and feasible.

Fractional dynamics on circulant multiplex networks: optimal coupling and long-range navigation for continuous-time random walks

This work analyzes fractional continuous-time random walks on two-layer multiplexes. A node-centric dynamics is used, in which it is assumed a Poisson distribution of a walker to become active, while a jump to one of its neighbors depends on the connection weight. Synthetic multiplexes with well known topology are used to illustrate dynamical features obtained by numerical simulations, while exact analytical expressions are presented for multiplexes assembled by circulant layers with finite number of nodes. Special attention is given to the effect of inter- Dx and intra-layer Di coefficients on the system’s behavior. In opposition to usual discrete time dynamics, the relaxation time has a well defined minimum at an optimal value of . It is found that, even for the enhanced diffusion condition, the walkers mean square displacement increases linearly with time.

Health assessment of LFP automotive batteries using a fractional-order neural network

A recurrent neural network with fractional order dynamics is used for assessing the health of LFP rechargeable automotive batteries through incremental capacity analysis. The proposed algorithm learns a dynamical model of the battery voltage from samples of current and voltage taken on a vehicle. The output of this dynamical model is the sum of two terms: the over-potential and the open circuit voltage. The over-potential model includes an element with fractional order dynamics. The open circuit voltage model is designed to depend on the same parameters as the incremental capacity curve, so the expression of the learnt model can be directly related to the health of the battery. The performance of the new method is assessed in three different batteries with varying states of health. A usable estimation of the incremental capacity curve was attained for low to moderate discharge currents. The state of health estimation produced by the fractional order network was consistently better than statistical and fuzzy models, LSTM and Echo State Networks for all the batteries under study.

Solutions of fractional gas dynamics equation by a new technique

In this paper, a novel technique is formed to obtain the solution of a fractional gas dynamics equation. Some reproducing kernel Hilbert spaces are defined. Reproducing kernel functions of these spaces have been found. Some numerical examples are shown to confirm the efficiency of the reproducing kernel Hilbert space method. The accurate pulchritude of the paper is arisen in its strong implementation of Caputo fractional order time derivative on the classical equations with the success of the highly accurate solutions by the series solutions. Reproducing kernel Hilbert space method is actually capable of reducing the size of the numerical work. Numerical results for different particular cases of the equations are given in the numerical section.

Exogenous phosphorus compounds interact with nitrogen availability to regulate dynamics of soil inorganic phosphorus fractions in a meadow steppe

Abstract. Here we investigated the effects of P compounds (KH2PO4 and Ca(H2PO4)2) with different addition rates of 0, 20, 40, 60, 80, and 100 kg P ha−1 yr−1 and NH4NO3 addition (0 and 100 kg N ha−1 yr−1) on soil labile inorganic phosphorus (IP) (dicalcium phosphate, Ca2-P), moderate-cycling IP, and recalcitrant IP fractions in a calcareous grassland of northeastern China. Soil moderate-cycling IP fractions, not readily available to plants but transforming into soil-available P quickly, include variscite (Al-P), strengite (Fe-P) and octacalcium phosphate (Ca8-P); recalcitrant IP fractions include hydroxylapatite (Ca10-P) and occluded P (O-P). Soil labile and moderate-cycling IP fractions and total P significantly increased with increasing P addition rates, with higher concentrations detected for KH2PO4 than for Ca(H2PO4)2 addition. Combined N and P treatments showed lower soil labile IP and moderate-cycling IP fractions compared to ambient N conditions, due to enhanced plant productivity. Moderate-cycling IP was mainly regulated by P addition and plant P uptake to further enhance labile IP and total P concentrations with KH2PO4 and Ca(H2PO4)2 addition. Soil labile IP was also directly and negatively affected by soil pH and plant P uptake with Ca(H2PO4)2 addition. Ca(H2PO4)2 addition significantly increased the soil recalcitrant IP (Ca10-P) fraction, while KH2PO4 addition showed no impact on it. A significant positive correlation was detected between soil labile IP, moderate-cycling IP fractions and soil Olsen-P which illustrated that labile IP and moderate-cycling IP fractions were important sources for soil-available P. Our results suggest that moderate-cycling IP fractions are essential for grassland P biogeochemical cycling and the chemical form of P fertilizer should be considered during fertilization management for maintaining soil-available P.

Soil carbon and nitrogen fraction dynamics affected by tillage erosion

Understanding the impact of tillage erosion on soil organic carbon (SOC) and nitrogen (N) fractions is essential for targeted soil conservation in mountainous and hilly areas. However, little is known about this issue. In this study, we selected a tillage erosion-dominated hillslope from the Sichuan Basin, China, to determine the effect of tillage erosion on particulate OC (POC), dissolved OC (DOC), light fraction OC (LFOC), ammonium N (NH4+-N), nitrate N (NO3−-N) and alkali-hydrolysable N (AN). Additionally, we investigated the microbial activities in relation to soil C and N dynamics, including soil microbial biomass, β-glucosidase and urease activities. Tillage erosion induced serious soil loss in upper slope positions and soil deposition in lower slope positions. The observations of the various labile OC fraction distributions across the hillslope suggest that tillage erosion exerts less impact on DOC and LFOC dynamics but a notable effect on POC. The distribution pattern in total organic carbon under tillage erosion mainly depends on POC redistribution. The POC redistribution is a major factor affecting microbial activities. The AN is more prone to the tillage erosion impact than NH4+-N and NO3−-N. Effective soil conservation measures should be taken to weaken the adverse impacts of tillage erosion on POC and AN redistribution in sloping farmlands.

Lamperti transformation of scaled Brownian motion and related Langevin equations

Recently, scaled Brownian motion has attracted considerable attention in the context of single particle tracking experiments displaying anomalous fractional dynamics. Its probability density function coincides with the one for fractional Brownian motion. On the other hand, scaled Brownian motion displays weak ergodicity breaking. In this paper we show that by applying the so-called Lamperti transformation we are able to transform the scaled Brownian motion into ergodic process. This allows us to estimate the distribution and moments of the studied process having only one, appropriately long trajectory. We apply the same method to the scaled fractional Brownian motion and stable Lévy motion. It appears that the method works also in the case of long-range correlations and heavy-tailed distributions. We confirm our theoretical results using numerical simulations.

An Efficient Approach for Solving Fractional Dynamics of a Predator-Prey System

In this work, we execute a generally new analytical technique, the modified generalized Mittag-Leffler function method (MGMLFM) for solving nonlinear partial differential equations containing fractional derivative emerging in predator-prey biological population dynamics system. This dynamics system are given by a set of fractional differential equations in the Caputo sense. A new solution is constructed in a power series. The stability of equilibrium points is studied. Moreover, numerical solutions for different cases are given and the methodology is displayed. We conducted a comparing between the results obtained by our method with the results obtained by other methods to illustrate the reliability and effectiveness of our main results.

Dynamics of phosphorus fractions in surface soils of different flooding wetlands before and after flow-sediment regulation in the Yellow River Estuary, China

To investigate the dynamics of phosphorus fractions and their influencing factors in the surface soils of estuarine wetlands experiencing different hydrological conditions before and after flow-sediment regulation, soil samples were collected in wetlands (tidal flooding wetlands (TFW), freshwater restoration wetlands (FRW) and freshwater flooding wetlands (FFW) of the Yellow River Estuary in each month from April to October of 2012. Our results showed that the average contents of organic phosphorus (OP) occurred in the following order: FRW soils (60.05 mg/kg) > TFW soils (38.72 mg/kg) > FFW soils (27.56 mg/kg), and accounted for less than 12% of total phosphorus (TP). In contrast to the pattern for OP, FRW soils contained lower inorganic phosphorus (IP) levels than TFW and FFW soils from April to August (p < 0.05). After the flow-sediment regulation, the TP, OP, moderately labile OP (ML-OP) ferrous/aluminum-bound IP (Fe/Al-P), and occluded IP (Oc-P) in the three wetlands decreased. The soluble and loosely bound IP (S/L-P) contents in TFW decreased, while the S/L-P contents in FRW soils increased. The levels of phosphorus fractions were affected by water and salt conditions, soil texture, exchangeable mineral element contents, and nutrient status. The Fe/Al-P and Oc-P in the three types of wetland soils were released, and the increase in S/L-P in FRW soils after the flow-sediment regulation might increase the risk of eutrophication in the coastal waters. The findings of this study could contribute to providing basic data regarding phosphorus fractions in different flooding estuarine wetlands of the Yellow River Estuary and guiding flow-sediment regulations and freshwater restoration to enhance the ecological functions of estuarine wetlands.

Fractional thermal transport and twisted light induced by an optical two-wave mixing in single-wall carbon nanotubes

Photothermally-activated energy transfer effects were analyzed by a vectorial two-wave mixing technique. Single-wall carbon nanotubes in thin film form were experimentally evaluated by nanosecond pulses at 532 nm wavelength. Significant changes in the heat conduction derived by equivalent incident energy from single-beam and two-wave optical interactions in the sample were promoted by different participation of quantum and nonlinear optical phenomena. Fractional analysis was carried out for studying the thermal behavior exhibited by the sample in a nonlinear regime. The fractional dynamics in the thermal transport was employed for describing the interferometric effects that correspond to optical phase-changes induced in nanoscale systems. This report can be a base for future research related to twisted light and photothermally-controlled functions.

Pneumatic conveying of cohesive dairy powder: Experiments and CFD-DEM simulations

We performed an experimental and numerical investigation of pneumatic conveying of cohesive dairy powder. The experiments with fat-filled milk powder (FFMP) fines with an average particle size of 94 μm were carried out in a 2-inch diameter stainless steel pipe consisting of two 2.5 m horizontal sections connected to a 0.65 m vertical section by two bends of 0.4 m radius each. In addition to measurements of pressure drop and powder deposition, an optical technique was used to measure the dynamics (probability densities) of local particle volume fractions as a function of operating conditions. Numerical simulations were performed with a commercial discrete element modelling (DEM) software, EDEM®, coupled with the computational fluid dynamics (CFD) software, FLUENT®. The simulation results in terms of pressure drops and particle volume fractions were compared with the experimental data. A very satisfactory agreement was found. At low gas velocities, cohesive dairy powders easily re-agglomerate after the second 90° bend and then deposit at the bottom of the horizontal pipe. At higher gas velocities, results show intermittent dispersion of particles and less particle deposition is observed even at higher loading ratio.

A New Methodology of Soil Salinization Degree Classification by Probability Neural Network Model Based on Centroid of Fractional Lorenz Chaos Self-Synchronization Error Dynamics

In this article, a new methodology for the centroid variation of chaos self-synchronization error dynamics was used to determine soil salinization degree based on probability neural network (PNN). This was done to overcome the difficulties involved in the handling of a large amount of spectroscopy data, as well as many spectral bands and a low determination rate caused by bands redundancy. The spectral reflectance of saline soils in Xinjiang Uygur Autonomous Region was used as data sources. The results showed that salinization in the area was severe. The proportion of saline, moderately saline, and severely saline soil accounted for 67.3% of the total sample. Fractional-order master/slave chaotic analysis was carried out on the characteristics of soil spectroscopy data with different salinization degrees. The differences between integer order and fractional order of chaotic dynamic error were compared. Simulation results showed that changes in the 0.6 order chaos dynamic error were the most significant and so these were used as PNN input vectors. The PNN model was used to identify the nonlinear hyperspectral signal of soil salinization degrees after chaotic system conversion. The input vector was normalized after insertion into the PNN model input layer and was added to the hidden layer for Gaussian operations. Finally, the hidden layer results were used in the summation layer to calculate the correlation. The verification set classification result was 93.5%. The studies showed that the method proposed in this article could serve as a new way for classifying soil salinization, which has a classification accuracy of 93.5%, and the soil salinization degree can be rapidly determined.

Dinámica de nitrógeno del suelo en agroecosistemas bajo el efecto de abonos verdes

Los abonos verdes (AV) son frecuentemente utilizados en agroecosistemas para mejorar y/o restaurar la fertilidad del suelo. El objetivo de este estudio fue evaluar la dinámica de nitrógeno (N) y carbono (C) del suelo en sistemas de maíz y soya bajo el efecto de AV residuales. En un Vertisol Typic Haplustert ubicado en el Valle del Cauca-Colombia se estableció la asociación Mucuna pruriens var. utilis – maíz (Zea mays), la cual 90 días después de la siembra fue incorporada al suelo como AV o se dispuso sobre la superficie del suelo como acolchado orgánico (AO). Posteriormente fueron sembrados los cultivos maíz-soya en forma intercalada para formar 11 tratamientos bajo el diseño de bloques completos al azar con arreglo factorial 32 +2 y tres repeticiones. Las parcelas principales fueron AV, AO y barbecho (B). Las subparcelas correspondieron a la fertilización con compost, fertilizante de síntesis química industrial y cero fertilizaciones (testigo). En forma paralela fueron sembrados como referentes los monocultivos de maíz y soya manejados de forma tradicional. En las etapas de floración y llenado de grano de los cultivos se midieron como variables en suelo: N total, amonio, nitrato, N-inorgánico total, carbono orgánico; y en tejido vegetal de maíz y soya: carbono y N. Los resultados mostraron que, la aplicación de materiales orgánicos de alta calidad, AV/AO, estimularon la mineralización del C y la dinámica de las diferentes fracciones de N en el suelo, sin cambios significativos en el contenido de estos elementos en el tejido vegetal del cultivo de maíz y soya.

Measurement and control of emergent phenomena emulated by resistive-capacitive networks, using fractional-order internal model control and external adaptive control

A fractional-order internal model control technique is applied to a three-dimensional resistive-capacitive network to enforce desired closed-loop dynamics of first order. In order to handle model mismatch issues resulting from the random allocation of the components within the network, the control law is augmented with a model-reference adaptive strategy in an external loop. By imposing a control law on the system to obey first order dynamics, a calibrated transient response is ensured. The methodology enables feedback control of complex systems with emergent responses and is robust in the presence of measurement noise or under conditions of poor model identification. Furthermore, it is also applicable to systems that exhibit higher order fractional dynamics. Examples of feedback-controlled transduction include cantilever positioning in atomic force microscopy or the control of complex de-excitation lifetimes encountered in many types of spectroscopies, e.g., nuclear magnetic, electron-spin, microwave, multiphoton fluorescence, Förster resonance, etc. The proposed solution should also find important applications in more complex electronic, microwave, and photonic lock-in problems. Finally, there are further applications across the broader measurement science and instrumentation community when designing complex feedback systems at the system level, e.g., ensuring the adaptive control of distributed physiological processes through the use of biomedical implants.

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

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.

Stabilization of Arbitrary Switched Nonlinear Fractional Order Dynamical Systems: Application to Francis Hydro-Turbine Governing System

This paper is a theoretical and practical study on the stabilization of fractional order Lipschitz nonlinear systems under arbitrary switching. The investigated system is a generalization of both switched and fractional order dynamical systems. Firstly, a switched frequency distributed model is introduced as an equivalent for the system. Subsequently, a sufficient condition is obtained for the stabilizability of the system based on the Lyapunov approach. Finally, the results are extended to synthesis mode-dependent state feedback controller for the system. All the results are expressed in terms of coupled linear matrix inequalities, which are solvable by optimization tools and directly reducible to the conditions of the integer order nonlinear switching systems as well as the conventional non-switched nonlinear fractional order systems. The proposed method has various practical implications. As an example, it is utilized to control Francis hydro-turbine governing system. This system is represented as a switching structure and supposed to supply a load suffering abrupt changes driven by an arbitrary switching mechanism. The simulation results support the usefulness of the method.

Fractional dynamics of an erbium-doped fiber laser model

In this paper we investigate the model of the time-fractional dynamics of an erbium-doped fiber laser model (TFDEFL) with Liouville–Caputo (LC), Caputo–Fabrizio–Caputo (CFC) and Atangana–Baleanu–Caputo (ABC) time-fractional derivatives. We employ the homotopy analysis transform method (HATM) to calculate approximate solutions for the TFDEFL model. This method gives the solution in the form of a rapidly convergent series that can ensure the convergence in solving the resultant series. We study the convergence analysis of HATM by computing the interval of convergence through the h-curves, the residual error function and the average residual error, respectively. We also show the effectiveness and accuracy of this method by comparing the approximate solutions based upon the LC, CFC and ABC time-fractional derivatives.

Complexity evolution of chaotic financial systems based on fractional calculus

Abstract Economics and finance are extremely complex nonlinear systems involving human subjects with many subjective factors. There are numerous attribute properties that cannot be described by the theory of integer-order calculus; so it is necessary to theoretically study the internal complexity of the economic and financial system using the method of bifurcation and chaos of fractional nonlinear dynamics. Fractional calculus can more accurately describe the existence characteristics of complex physical, financial or medical systems, and can truly reflect the actual state properties of these systems; therefore the application of fractional order in chaotic systems has great significance to study the mathematical analysis of nonlinear dynamic systems, and the use of fractional calculus theory to model the complexity evolution of fractional chaotic financial systems has attracted more and more scholars’ attention. On the basis of summarizing and analyzing previous studies, this paper qualitatively analyzes the stability of equilibrium solution of fractional-order chaotic financial system, and explores the complexity evolution law of the financial system near the equilibrium point and the occurring conditions of asymptotic chaotic state near this equilibrium point, and simulate the complexity evolution of chaotic financial systems using the Admas-Bashforth-Moulton finite difference method for mapping, phase diagram and time series graph. The research results of this paper provide a reference for government to formulate relevant economic policies, decision-making or further research on the complexity evolution of fractional-order chaotic financial systems.

Dynamic changes and characterization of the protein and carbohydrate fractions of native grass grown in Inner Mongolia during ensiling and the aerobic stage.

ObjectiveTo improve the utility of native grass resources as feed in China, we investigated the dynamics of protein and carbohydrate fractions among Inner Mongolian native grasses, during ensiling and the aerobic stage, using the Cornell Net Carbohydrate and Protein System.MethodsSilages were prepared without or with lactic acid bacteria (LAB) inoculant. We analyzed the protein and carbohydrate fractions and fermentation quality of silages at 0, 5, 15, 20, 30, and 60 d of ensiling, and the stability at 0.5, 2, 5, and 10 d during the aerobic stage.ResultsInner Mongolian native grass contained 10.8% crude protein (CP) and 3.6% water-soluble carbohydrates (WSC) on a dry matter basis. During ensiling, pH and CP and WSC content decreased (p<0.05), whereas lactic acid and ammonia nitrogen (N) content increased (p<0.05). Non-protein N (PA) content increased significantly, whereas rapidly degraded true protein (PB1), intermediately degraded true protein (PB2), total carbohydrate (CHO), sugars (CA), starch (CB1), and degradable cell wall carbohydrate (CB2) content decreased during ensiling (p<0.05). At 30 d of ensiling, control and LAB-treated silages were well preserved and had lower pH (<4.2) and ammonia-N content (<0.4 g/kg of fresh matter [FM]) and higher lactic acid content (>1.0% of FM). During the aerobic stage, CP, extract ether, WSC, lactic acid, acetic acid, PB1, PB2, true protein degraded slowly (PB3), CHO, CA, CB1, and CB2 content decreased significantly in all silages, whereas pH, ammonia-N, PA, and bound true protein (PC) content increased significantly.ConclusionControl and LAB-treated silages produced similar results in terms of fermentation quality, aerobic stability, and protein and carbohydrate fractions. Inner Mongolian native grass produced good silage, nutrients were preserved during ensiling and protein and carbohydrate losses largely occurred during the aerobic stage.

Controllability of Nonlinear Stochastic Fractional Higher Order Dynamical Systems

Abstract This paper deals with the study of controllability of stochastic fractional dynamical systems with 1 &lt; α ≤ 2. Necessary and sufficient condition for controllability of linear stochastic fractional system is obtained. Sufficient conditions for controllability of stochastic fractional semilinear systems, integrodifferential systems, systems with neutral term, systems with delays in control and systems with Lévy noise is formulated and established. The solution is obtained in terms of Mittag-Leffler operator functions by considering bounded operators. The Banach fixed point theorem is used to obtain the desired results from an equivalent nonlinear integral equation of the given system.