Abstract: Publication date: May 2019Source: Chaos, Solitons & Fractals, Volume 122Author(s): Necdet BİLDİK, Sinan DENİZ In this paper, we use Atangana–Baleanu derivative which is defined with the Mittag–Leffler function and has all the properties of a classical fractional derivative for solving the system of fractional differential equations. The classical model of polluted lakes system is modified by using the concept of fractional differentiation with nonsingular and nonlocal fading memory. The new numerical scheme recommended by Toufik and Atangana is used to analyze the modified model of polluted lakes system. Some numerical illustrations are presented to show the effect of the new fractional differentiation.

Abstract: Publication date: May 2019Source: Chaos, Solitons & Fractals, Volume 122Author(s): P. Gatabazi, J.C. Mba, E. Pindza, C. Labuschagne The study uses grey Lotka–Volterra model (GLVM) of two and three dimensions for assessing the interaction between cryptocurrencies. The 2-dimensional study is on Bitcoin and Litecoin while the 3-dimensional study is on Bitcoin, Litecoin and Ripple. Records from 28-April-2013 to 10-February-2018 provide forecasting values for Bitcoin and Litecoin through 2-dimensional GLVM study, while records from 7-August-2013 to 10-February-2018 provide forecasting values of Bitcoin, Litecoin and Ripple through 3-dimensional GLVM study. The behaviour of Bitcoin and Litecoin or both Bitcoin, Litecoin and Ripple in future is proposed by looking at the 100 last forecasting values of n-dimensional GLVM study, n={2,3}. Lyapunov exponents of the 2 and 3-dimensional Lotka–Volterra models reveals that it is a chaotic dynamical system. Plots of 2 and 3-dimensional Lotka–Volterra models for filtered datasets suggest also a chaos. Using the mean absolute percentage error criterion, it was found that the accuracy of the GLVM is better than that of the grey model (GM(1,1)). By analysing the 2-dimensional GLVM, Bitcoin and Litecoin are found in the competition known as mutualism or equivalently a win-win situation where Bitcoin transaction is constant while Litecoin transaction has the increasing trend. The 3-dimensional GLVM analysis evokes however, an increasing trend in transacting both Bitcoin, Litecoin and Ripple where Bitcoin keep relatively higher transaction counts.

Abstract: Publication date: May 2019Source: Chaos, Solitons & Fractals, Volume 122Author(s): Fatmawati, Muhammad Altaf Khan, Muftiyatul Azizah, Windarto, Saif Ullah In the present paper, we propose a mathematical model that describes the dynamics of competition between commercial and rural banks in Indonesia through two different fractional operators Atangana-Baleanu and Caputo. We present a parameter estimation of the Lotka–Volterra competition model by using the genetic algorithm method. Parameter estimation is done based on annual profit data of commercial and rural banks in Indonesia. The estimation results capable to predict the profit of commercial and rural banks every year which is not much different from the real data. Next, the competition model between commercial and rural banks in Indonesia is explored in the fractional sense of Atangana–Baleanu and Caputo derivative. The fractional model is examined through the Atangana–Baleanu and Caputo fractional derivative and present the results. A recent numerical procedure is used to obtain the graphical results using various values of the fractional order parameter for the dynamics of the model. A comparison of both the operators for various values of the fractional order parameters are given. We discussed briefly the results and then summarized briefly in section conclusion.

Abstract: Publication date: May 2019Source: Chaos, Solitons & Fractals, Volume 122Author(s): Zhenya Yan The nonlinear Schrödinger (NLS) equation with fully nonlinear dispersion (called NLS(m, n) equation) has been introduced by Yan [Phys. Lett. A 355 (2006) 212] and shown to possess the single-hump compactons for m=n>1. In this paper, we further investigated the focusing NLS(n, n) equation and find that it possesses the multi-hump compactons for n > 1, whose properties are analyzed in detail. Particularly, we surprisedly find that the maximal intensities of the multi-hump compactons approach to the natural base e as n→1+. Moreover, we numerically study the stabilities and interactions of the single-hump and double-hump compactons such that some stable multi-hump compactons and elastic interactions are found for some small values of the parameter n. These multi-hump compactons will be useful for understanding the soliton-like solutions and applying them in the related fields of nonlinear science.

Abstract: Publication date: May 2019Source: Chaos, Solitons & Fractals, Volume 122Author(s): André Gusso, Ricardo L. Viana, Amanda C. Mathias, Iberê L. Caldas We investigate theoretically the nonlinear dynamics and the emergence of chaos in suspended beam micro/nanoelectromechanical (MEMS/NEMS) resonators actuated by two-sided electrodes. Through the analysis of phase diagrams we have found that the system presents a rich and complex nonlinear behavior. Multistability is observed in a significant region of the relevant parameter space, involving periodic and chaotic attractors. Complex and varied routes to chaos were also found. Basins of attraction with strongly intermingled attractors provide further evidence of multistability. The basins are analyzed in greater detail. Their fractal dimensions and uncertainty exponent are calculated using the well known box counting and uncertainty methods. The results for the uncertainty exponent are compared with those obtained with yet another approach, based on the recently proposed basin entropy method. The comparison provides a test for the new approach, which we conclude that is a reliable alternative method of calculation. Very low uncertainty exponents have been obtained, indicating that some basins have extremely intermingled attractors, what may have significant influence in the experimental investigation and practical applications of the resonators. We also conclude that the observation of chaos in this system is favored by lower frequencies of excitation and comparatively small quality factors (larger dissipation).

Abstract: Publication date: May 2019Source: Chaos, Solitons & Fractals, Volume 122Author(s): Jun Tanimoto, Xie An A new Cellular Automata traffic model based on Revised S-NFS model was established, which considers traffic density ahead of a car in next 50 [m] and also accounts for a decision making process of whether a lane change should be tried or not so as to diminish the frequency of meaningless lane-changes. It intends to be applied as one of the protocols to improve traffic efficiency in premise with Intelligence Traffic System (ITS) that is able to provide information on traffic density next hundred meters in front of a focal vehicle. A series of systematic simulations reveals that the presented lane changing protocol enhances traffic flux vis-à-vis the conventional lane change rule based on the traditional incentive criterion and safe criterion. Social dilemma analysis suggests our new protocol mitigates a strong social dilemma encouraged by a competition between a cooperator; not intending any lane-changes and a defector; trying to lane-changes to minimize his own travel time.

Abstract: Publication date: May 2019Source: Chaos, Solitons & Fractals, Volume 122Author(s): Hayder Natiq, Santo Banerjee, A.P. Misra, M.R.M. Said In this work, the dynamical behaviors of a low-dimensional model, which governs the interplay between a driver associated with pressure gradient and relaxation of instability due to magnetic field perturbations, are investigated. Besides that, two nonlinear controllers are constructed precisely to shift the equilibria of the plasma model apart from each other. Simulation results show that shifting the equilibria can change the spacing of chaotic attractors, and subsequently break the butterfly wings into one or two symmetric pair of coexisting chaotic attractors. Furthermore, stretching the equilibria of the system apart enough from each other gives rise to degenerate the butterfly wings into several periodic orbits. In addition, with appropriate initial conditions, the complex multistability behaviors including the coexistence of butterfly chaotic attractor with two point attractors, the coexistence of transient transition chaos with completely quasi-periodic behavior, and the coexistence of symmetric Hopf bifurcations are also observed.

Abstract: Publication date: April 2019Source: Chaos, Solitons & Fractals, Volume 121Author(s): Abouzar Kaboudian, Elizabeth M. Cherry, Flavio H. Fenton The study of complex systems has emerged as an important field with many discoveries still to be made. Computer simulation and visualization provide important tools for studying complex dynamics including chaos, solitons, and fractals, but available computing power has been a limiting factor. In this work, we describe a novel and highly efficient computing and visualization paradigm using a Web Graphics Library (WebGL 2.0) methodology along with our newly developed library (Abubu.js). Our approach harnesses the power of widely available and highly parallel graphics cards while maintaining ease of use by simplifying programming through hiding implementation details, running in a web browser without the need for compilation, and avoiding the use of plugins. At the same time, it allows for interactivity, such as changing parameter values on the fly, and its computing is so fast that zooming in on a region of a fractal like the Mandelbrot set can incur no delay despite having to recalculate values for the entire plane. We demonstrate our approach using a wide range of complex systems that display dynamics from fractals to standing and propagating waves in 1, 2 and 3 dimensions. We also include some models with instabilities that can lead to chaotic dynamics. For all the examples shown here we provide links to the codes for anyone to use, modify and further develop with other models. Overall, the enhanced visualization and computation capabilities provided by WebGL together with Abubu.js have great potential to facilitate new discoveries about complex systems.

Abstract: Publication date: April 2019Source: Chaos, Solitons & Fractals, Volume 121Author(s): E. Haihong, Hu Yingxi, Peng Haipeng, Zhao Wen, Xiao Siqi, Niu Peiqing An epidemic is a typical public health emergency that refers to the occurrence and rapid spread of disease. A good epidemic transmission model plays a crucial role in preventing an epidemic. The epidemic transmission model is largely similar to the model of sentiment analysis and transmission on social media. Therefore, this paper intend to use the method of deep learning to explore the key issues of theme and sentiment analysis from the perspective of public opinion analysis.In order to fully extract the features automatically, we combine the following methods: multi-channel inputs, multi-granularity convolution kernels, direct connection with high-speed channels, and this paper proposes the multi-channel and multi-kernel (MCMK) model. Furthermore, we leverage generative adversarial nets to combine several single tasks, called Joint-MCMK model, which achieves information sharing and improves the training speed and model accuracy.To verify the validity of our proposed models, this paper experimented with the short text topic classification dataset TREC [1] and the sentiment analysis dataset IMDB [2]. Results achieved 98.6% and 92.6% respectively, which are superior to the highest existing industry benchmark (96.1% and 92.58%). In addition, this paper compared the training spread differences between the joint-MCMK model and MCMK model, which shows that the joint-MCMK model has better performance at training speed. Finally, the control variable method was used to analyze the multiple effects of the different factors. The optimal value of some relevant parameters in our models were verified by several experiments.

Abstract: Publication date: April 2019Source: Chaos, Solitons & Fractals, Volume 121Author(s): Massimo Materassi In order to construct an analytical model of evolution for the plasma structures in the ionospheric Equatorial Spread F, an attempt is presented to include the plasma perturbations “seeding” the instabilities that cause the Equatorial Spread F in a theoretically consistent way. This is done assuming that the local discrepancies from quasi neutrality and from zero ionization-loss-and-production to be noise terms in the ionospheric density balance equation: doing so, the balance equation becomes a stochastic equation. The program is then to obain the probability that a certain evolution of the ionospheric density takes place from the interplay between plasma dynamics and noise statistics, i.e. to produce a stochastic field theory for the ionospheric density.The expression of this realization probability is obtained in the mathematically tractable cases of noises with Gaussian or exponential distribution. Then, practical limits and possible developments of these achievements are discussed.

Abstract: Publication date: April 2019Source: Chaos, Solitons & Fractals, Volume 121Author(s): H.G. Wu, Y. Ye, B.C. Bao, M. Chen, Q. Xu By substituting two resistive couplings with two memristive couplings, a two-memristor-based hyperchaotic system is proposed. This hyperchaotic system has a plane equilibrium with two zero eigenvalues and three nonzero ones, and the equilibrium plane is divided into three regions with different stabilities, including unstable saddle, stable node, and stable node-focus. Through using the local attraction basin, bifurcation diagrams and Lyapunov exponent spectra, dynamical behaviors are analyzed in the unstable saddle region, from which the bi-stability phenomenon is observed in such a hyperchaotic system. More particularly, the memristor initial boosting behaviors are found, which indicates that the attractor offset boosting is controlled by the memristor initial values. The memristor initial boosting is multi-dimension, nonlinearity and non-monotonic, completely different from the variable offset boosting. Subsequently, PSIM circuit simulations are performed to verify the numerical results.

Abstract: Publication date: April 2019Source: Chaos, Solitons & Fractals, Volume 121Author(s): Yaming Zhang, Yanyuan Su, Li Weigang, Haiou Liu Rumor propagation and behavior spreading are usually closely coupled with each other and the interaction will have great influence on the spreading dynamics. In this paper, we propose a novel interacting model of rumor propagation and behavior spreading in multiplex networks. Specifically, coupled reinforcements are introduced in the derived mean-field equations to describe the interplay between these two dynamical processes. Then the basic reproduction number and the final sizes of rumor and behavior spreading are estimated. Monte Carlo simulations results show that the interacting model are much more consistent with the real data than the classical model. Besides, the attractiveness of rumor and behavior and couple reinforcements are crucial factors affecting the interactive spreading processes. Especially, influenced by the coupled reinforcements, any attractiveness increasing can enhance maximum influences and final spreading sizes of rumor and behavior simultaneously. What’s more, the coupled reinforcements will extend the duration of rumor and behavior spreading. Moreover, the reinforcement of spreaders or infecteds to promote spreading is far stronger than that of stiflers and recovered to suppress spreading. Interestingly, we also find that the final size of rumor propagation is larger than that of behavior spreading, while the range of the final size variation resulting from the coupled reinforcements is opposite. This work may shed some lights on understanding the interaction between rumor and behavior and suggest a promising way to control rumors and irrational behaviors.

Abstract: Publication date: April 2019Source: Chaos, Solitons & Fractals, Volume 121Author(s): Zhaoxing Li, Li Chen Complex networks are prevalent in our lives. A complex network usually is composed of many components. Because the components of a network may suffer from random failures or intentional attacks, it is therefore important to study the robustness of networks in face of perturbations. Because real-world complex networks are practically interdependent, therefore many efforts have been made to investigate the robustness of interdependent or multilayer networks. Existing studies indicate that the robustness of multilayer networks displays first order phase transition, while the robustness of single layer networks only displays second order phase transition. Note that a simple form of a multilayer network is a multipartite network. Intuitively, the robustness of multipartite networks will also possess first order phase transition. In this paper we study the robustness of multipartite networks in face of random node failures. Extensive experiments have been carried out to test the robustness of multipartite networks whose degree distributions follow Poisson distribution. Interestingly, we have found that the robustness of multipartite networks displays second-order-like phase transition which is against the intuitive conclusion.

Abstract: Publication date: April 2019Source: Chaos, Solitons & Fractals, Volume 121Author(s): Xuefan Dong, Yijung Liu, Chao Wu, Ying Lian Calculations were conducted concerning the degree distribution functions of four models with an S-shaped growth characteristic and the preferential attachment rule, based on the mean field theory. Of these models, Model 1 displays the simplest sigmoid function, Model 2 and Model 3 are two extended models, and Model 4 depicts the law of population function. The results show that the graphs of the four degree distribution functions with their different gained control parameters are relatively similar to one another, thus, displaying a power-law form. In addition, a separated method was defined to calculate the degree distribution of single-peak, real-time networks with a symmetric or asymmetric characteristic. Such findings, to a large extent, will enrich the complex theory and could be conducive to understanding the evolutionary dynamics of S-shaped functions as well as other exponential functions.

Abstract: Publication date: April 2019Source: Chaos, Solitons & Fractals, Volume 121Author(s): Xiao-Li Gong, Xi-Hua Liu, Xiong Xiong, Xin-Tian Zhuang In order to analyze the stock market bubble phenomenon, the vector autoregressive moving average (VARMA) model with non-Gaussian innovations and stochastic volatility components (VARMA-t-SV) is constructed for financial modeling. Considering the estimation complexity of VARMA-t-SV model, the Kronecker structure of likelihood function is employed to speed up computation. Then we develop the corresponding Markov chain Monte Carlo (MCMC) sampling method to test the covariance structure specifications. Model comparisons illustrate that the VARMA model with flexible covariance structures perform better performances. The model parameter estimation results show that the fat tail and the heteroscedasticity features are useful in raising the performances compared to the standard form. Finally, using Chinese financial markets data, the effects of monetary policy on stock market bubbles are analyzed based on the VARMA-t-SV model. The empirical studies provide evidence to support the rational asset price bubble theory, namely, the tightening monetary policy may not succeed in shrinking the asset price bubble, which provides suggestions for regulators and investors.

Abstract: Publication date: April 2019Source: Chaos, Solitons & Fractals, Volume 121Author(s): LiuWei Zhao, Jianwei Chang, Jianguo DU In this paper, a two-period game model involving an original manufacturer, a remanufacturer and a retailer was built in context of government subsidies. The system can be regarded as a coupling dynamic of the forward supply chain of Stackelberg game model. Based on this game model, the impact of three government subsidy scenarios (the government subsidizes no one; the government subsidizes remanufacturers; and the government subsidizes the consumers through retailers) on the unit wholesale price, retail price and profit of the two kinds of products is analyzed. Based on the analysis, some dynamic phenomena such as bifurcation and chaos are found. Numerical simulations and the largest Lyapunov exponent are used to provide experimental evidence for the complicated behaviors of the system evolution. The results show that the government subsidy for remanufacturers can reduce the retail price of remanufactured products, increase the retailer's profit, and improve the market competitiveness of remanufactured products, but it makes a small difference to the remanufacturers’ profit; and the government subsidy for consumers has a significant impact on the market competitiveness of remanufactured products.

Abstract: Publication date: April 2019Source: Chaos, Solitons & Fractals, Volume 121Author(s): Yao Xu, Yanzhen Li, Wenxue Li This paper deals with synchronization problem of fractional-order coupled systems (FOCSs) with time-varying delays via periodically intermittent control. Here, nonlinear coupling, time-varying internal delay and time-varying coupling delay are considered when modeling, which makes our model more general in comparison with the most existing fractional-order models. It is the first time that periodically intermittent control is applied to synchronizing FOCSs with time-varying delays. Combining Lyapunov method with graph-theoretic approach, some synchronization criteria are obtained. Moreover, the synchronization criteria we derive depend on the fractional order α, control gain, control rate and control period. Besides, the synchronization issues of fractional-order coupled chaotic systems with time-varying delays and fractional-order coupled Hindmarsh–Rose neuron systems with time-varying delays are also investigated as applications of our theoretical results, and relevant sufficient conditions are derived. Finally, numerical simulations with two examples are provided in order to demonstrate the effectiveness of the theoretical results and the feasibility of control strategy.

Abstract: Publication date: April 2019Source: Chaos, Solitons & Fractals, Volume 121Author(s): Biao Wang, Zhizhong Mao, Keke Huang Process data has been used in most industrial systems to facilitate process control and process monitoring. Even if outliers have been proved to have negative influence on those data-driven techniques, dedicated detection methods are still rare or at a junior phase. Furthermore, due to the fact that most industrial systems are complex and nonlinear, many outlier detection methods developed in the field of data mining are inefficient or cannot be applied directly. In this paper thereby, we propose an outlier detection method dedicated to complex and nonlinear industrial systems. This method is on the basis of dynamic ensemble learning. It is observed that ensemble learning has made great achievement recently, and dynamic ensemble learning usually outperforms other ensemble techniques. Experimental results prove that our dynamic ensemble outlier detection method has better performance for complex nonlinear industrial systems.

Abstract: Publication date: April 2019Source: Chaos, Solitons & Fractals, Volume 121Author(s): Jingfei Jiang, Juan Luis García Guirao, Huatao Chen, Dengqing Cao This paper is concerned with the boundary control of the fractional wave equation when the boundary is subject to persistent external disturbances. By developing the sliding mode control approach to infinite-dimensional fractional order systems, the fractional order sliding mode boundary control law is designed for the infinite dimensional setting. Moreover, based on the fractional asymptotical stability theorem, the asymptotical stability for the fractional wave equation under the control strategies proposed is addressed. Finally, numerical examples are provided to illustrate the viability of the theoretical results.

Abstract: Publication date: April 2019Source: Chaos, Solitons & Fractals, Volume 121Author(s): Sergei A. Plotnikov, Alexander L. Fradkov Biological systems are often composed of various heterogeneous units. It is an important problem to investigate how this heterogeneity affects the network dynamics, namely synchronization phenomenon. We study the heterogeneous networks of FitzHugh–Nagumo oscillators with diffusive coupling and present sufficient conditions for synchronization in these networks using the Lyapunov functions and Linear Matrix Inequalities (LMIs). Starting consideration with the case of two coupled systems further we extend the results to the networks with greater number of nodes. Numerical examples are presented to illustrate the obtained results.

Abstract: Publication date: April 2019Source: Chaos, Solitons & Fractals, Volume 121Author(s): Jacques Kengne, Ruth Line Tagne Mogue, Theophile Fonzin Fozin, Adelaide Nicole Kengnou Telem A novel self-driven RC chaotic jerk circuit with the singular feature of having a smoothly adjustable nonlinearity and symmetry is proposed and investigated. The novel chaotic circuit is mathematically modeled by a third order system with a single nonlinear term in the form ϕk(x)=0.5(exp(kx)−exp(−x)) where parameter k maps a smoothly adjustable control resistor. Obviously for k=1, the system is point symmetric with respect to the origin of the system coordinates since the nonlinear term reduces to the hyperbolic sine function. The case k ≠ 1 corresponds to a non-symmetric system. The numerical experiment reveals a plethora of events including period doubling route to chaos, hysteresis, periodic windows, asymmetric double scroll chaos, symmetric double scroll chaos, and coexisting bifurcations branches as well. This latter phenomenon induces multiple coexisting attractors consisting of two, three, four, five, or six disconnected symmetric or asymmetric attractors for the same set of parameter values when monitoring solely the initial conditions. Laboratory experimental measurements are carried out to confirm the theoretical predictions.

Abstract: Publication date: April 2019Source: Chaos, Solitons & Fractals, Volume 121Author(s): Tomoko Sakiyama, Ikuo Arizono Since Maynard Smith and Price proposed the earliest version of Hawk–Dove (HD) game, it attracted researchers’ attention as one of models of conflict for two players in game theory. In conflict game, the players’ benefit depends on the strategy of opponent for each other. In the classical spatial HD games, if one player adopts defector strategy, it tends to get high payoffs, and therefore increases population of the same strategy, which resulting in an extinction of cooperators. Several studies tried to solve the problem of an extinction of cooperator in spatial HD game. In this paper, we developed a novel spatial HD model replacing the best takes over update rule with different one, and investigated the effect of modifying update rules on the problem of collaborators extinction in space HD games. We found that our model generated characteristic population patterns and represented the survival of cooperators compared with the classical spatial HD model in which the updated rule was fixed.

Abstract: Publication date: April 2019Source: Chaos, Solitons & Fractals, Volume 121Author(s): Hilmi Demiray, Alireza Abdikian In the present work, employing the nonlinear field equations of a hot dusty plasma in the presence of nonthermal electrons and trapped ions, we studied the amplitude modulation of nonlinear waves in such a plasma medium by use of the reductive perturbation method and obtained the modified nonlinear Schrödinger equation. The modulational instability (MI) was investigated and the effects of the proportion of the fast electrons (α), the trapping parameter (b) and the plasma parameters such as the dust-ion temperature ratio (σd), the partial unperturbed electron to dust density (δ), and the ion-electron temperature ratio (σi) on it was discussed. For the investigation of modulational instability problems three parameters P/Q, Kmax and Γmax play the central role. The variations of these parameters with the wave number k and the other physical parameters are discussed and the possibility of occurence of modulational instability is indicated.

Abstract: Publication date: April 2019Source: Chaos, Solitons & Fractals, Volume 121Author(s): Reza Yaghoobi Karimui, Sassan Azadi, Parviz Keshavarzi Attention-deficit/hyperactivity disorder (ADHD) as a behavioral challenge, which affects the people's learning and experiences, is one of the disorders, which leads to reducing the complexity of brain processes and human behaviors. Nevertheless, recent studies often focused on the effects of this disorder in the frequency content of single and multichannel EEG segments and only a few studies that employed the approximate entropy for estimating this reduction. In this study, we provide a different view of this reduction by focusing on the texture of patterns appeared on the auto-recurrence plots obtained from the phase space trajectories reconstructed from the EEG signals recorded under the open-eyes and closed-eyes resting conditions. The outcomes of this analysis generally indicated a significant difference in the texture of recurrence plots, which its reason was the increase of recurrence, parallel and similar behaviors in the trajectories. Evaluating the features extracted from these recurrence plots in the studied children without and with ADHD using the sequential forward selection (SFS) algorithm also provided a remarkable accuracy (90.95% for the testing sets), which is a confirmation on changing the texture of recurrence plots relevant to the EEG signals of ADHD children. Nevertheless, evaluating these results and the results of previous researches with each other represented that the volume of statistical population is an important factor for reducing the rate of separability in the classifiers developed by an EEG segment. Therefore, these findings generally proved that although the ADHD averagely leads to the complexity reduction of EEG processes, the classifiers developed by just an EEG segment cannot be applicable in clinical conditions.

Abstract: Publication date: April 2019Source: Chaos, Solitons & Fractals, Volume 121Author(s): Wei Li, Dongmei Huang, Meiting Zhang, Natasa Trisovic, Junfeng Zhao Fractional-order PID (FOPID) controller, as the results of recent development of fractional calculus, is becoming wide-used in many deterministic dynamical systems, but not in stochastic dynamical systems. This paper explores stochastic bifurcation of a generalized Van del Pol (VDP) system under the control of FOPID controller. Firstly, introducing the transformation between fast-varying and slow-varying variables of the system response, and utilizing the properties of fractional calculus, we obtain a new expression in the form of slow-varying variables for FOPID controller. Based on this work, the stochastic averaging method is applied to obtain the Fokker–Planck–Kolmogorov (FPK) equation and the stationary probability density function (PDF) of the amplitude response. Then a new numerical algorithm is proposed to testify the analytical results in the case of the coexistence of fractional integral and fractional derivative. After that, stochastic bifurcations induced by the order of the fractional integral, the order of the fractional derivative and the coefficient in FOPID controller are investigated in detail. The agreement between analytical and numerical results verifies the correctness and effectiveness of our proposed methods.

Abstract: Publication date: April 2019Source: Chaos, Solitons & Fractals, Volume 121Author(s): Hermann T. Tchokouansi, E. Tchomgo Felenou, Robert Tamwo Tchidjo, Victor K. Kuetche, Thomas B. Bouetou In this article, we derive new traveling wave solution to a nonlinear evolution equation, describing propagation of short wave in inhomogeneous ferrites. Applying Jacobi elliptic function, we derive as series of new exact solutions to the system of our interest which are either periodic or localized solutions. We point out the influence of the inhomogeneous exchange effects on the dynamics of traveling waves obtained. It appears that the soliton responsible of localization is deformed by the presence of inhomogeneities in particular its structure. We discuss some physical implications of these results.

Abstract: Publication date: Available online 8 November 2018Source: Chaos, Solitons & FractalsAuthor(s): Bandhu Prasad This letter refers to [1] where Gp, m matrix and G1,m−3 matrix are erroneous in Sections 5 and 7.1 respectively. We have corrected these sections.