Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Arzu Akbulut, Filiz Taşcan We found trivial conservation laws by conservation theorem and exact solutions modified extended tanh-function method of (1+1)-dimensional nonlinear coupled Klein–Gordon–Zakharov equation. The travelling wave solutions are expressed by the hyperbolic, trigonometric and rational functions. It is shown that the suggested method provides a powerful mathematical instrument for solving nonlinear equations in the science of mathematics, physics and engineering.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Zhibo Zhang, Enyuan Wang, Nan Li A rock mechanics test system and an acoustic emission (AE) test system have been employed to conduct AE event location experiments on sandstone samples under uniaxial compression. According to the single-link cluster (SLC) method, an SLC structure is set up. Each single link is defined as a spatial vector, which can be calculated based on the spatial coordinates of the nearest neighbor of a certain AE event minus the spatial coordinates of this event. Based on the basic theory of fractal geometry, a spherical surface-covering method is proposed herein to analyze the fractal property of spatial vector direction in an SLC structure. To study the change in fractal dimension over time, a sliding event window is used. The results show that the window length and slide step length have little effect on the change in trend of fractal dimension. Before buckling failure of a rock sample, the fractal dimension curve shows a clear downward trend. The reason for the change in fractal dimension is related to the generation mechanism of AE events. Therefore, a clear decrease in fractal dimension can be taken as precursor to rock buckling failure under uniaxial compression. This research provides a new method for studying the evolution process of rock buckling failure and has important significance regarding the monitoring of rock stability and early warning of rock burst.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Jun Luo, Yi Yang Given an unshielded continuum K ⊂ C , the core decomposition D K L C of K with respect to local connectedness is known to exist [2]. Such a decomposition D K L C is monotone, locally connected under quotient topology, and refines every other monotone decomposition D ′ which is locally connected under quotient topology. Let ∼ be the closed equivalence whose classes form the decomposition D K L C . Then ∼ contains a symmetric closed relation RK which requires (x, y) ∈ RK if and only if x and y lie in a prime end impression of C ∖ K . We also say that the relation RK respects prime end impressions. From Akin’s viewpoint of dynamical systems, the equivalence ∼ may be obtained as one of the limit relations of RK , through transfinite process. We will propose a direct approach to realize this limit relation in a concrete construction. More or less, such an approach builds the elements of D K L C “from below ” . If K is an unshielded disconnected compactum, the core decomposition D K F S of K with respect to the finitely suslinian property has been obtained in [3]. In this case, we consider a symmetric closed relation that “respects limit continua” and show that the same approach also works, in constructing D K F S from below.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): P.R.L. Alves, L.G.S. Duarte, L.A.C.P. da Mota In the reconstruction scheme, the global fitting is a basis for the approach to the time evolution of dynamic systems directly from time series. A new theory of dynamic characterization is present in the aim of this work. The least squares method determines the predictors in the Algebraic Computation environment. The program for diagnosing of time series run in a Maple environment. The computational routine determines a new quantifier of chaos. A test for theory and computational tools in periodic, chaotic and random systems is in the scope of this paper. An application of the method in a real-world time series gives a satisfactory result.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Andrey Gavrilov, Evgeny Loskutov, Dmitry Mukhin A method for optimal data simulation using random evolution operator is proposed. We consider a discrete data-driven model of the evolution operator that is a superposition of deterministic function and stochastic forcing, both parameterized with artificial neural networks (particularly, three-layer perceptrons). An important property of the model is its data-adaptive state-dependent (i.e. inhomogeneous over phase space) stochastic part. The Bayesian framework is applied to model construction and explained in detail. Particularly, the Bayesian criterion of model optimality is adopted to determine both the model dimension and the number of parameters (neurons) in the deterministic as well as in the stochastic parts on the base of statistical analysis of the data sample under consideration. On an example of data generated by the stochastic Lorenz-63 system we investigate this criterion and show that it allows to find a stochastic model which adequately reproduces invariant measure and other statistical properties of the system. Also, we demonstrate that the state-dependent stochastic part is optimal only for large enough duration of the time series.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Paride O. Lolika, Steady Mushayabasa, Claver P. Bhunu, Chairat Modnak, Jin Wang We present a mathematical model for the transmission dynamics of brucellosis that incorporates the effects of seasonality. We analyze the basic reproduction number associated with the time-periodic model and establish results on the threshold dynamics. Meanwhile, we perform an optimal control study on the use of animal vaccination and environmental decontamination as disease control measures against brucellosis infection. Our results show that seasonality plays an important role in shaping the long-term dynamics of brucellosis, which subsequently impacts the design of its optimal control strategies.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Debaldev Jana, Aniket Banerjee, G.P. Samanta Every population should exploit a specially variable and diverse environment so as to increase their Darwinian fitness. Dynamics of any local population depends upon attributes of the local habitat. Although, use of refuge habitat by prey population can reduce their risk of predation, refuge use may also involve cost such as increased interspecific and intraspecific competition within the refuge patch. Surveys in the Sunderban mangrove ecosystem show that two detritivorous prey fishes Liza parsia and Liza tade coexist in nature by using refuges with the presence of the predator fish population Lates calcarifer. In view of such observations in mind, a three component model conceiving of two competing prey and one predator is considered in the present study with the inclusion of Holling type-II response function incorporating a fraction of prey refuge. The geographic position of these refuge patches tend to determine the population of prey residing in these patches which ultimately leads to the interspecific competition inclusion between prey. Here, we have differentiated the geographic position of the refuge patch into five different cases, for example, disjoint refuge patch (no competition between refuge prey population), partially overlapping refuge patch (competition between non-refuge and partially refuge prey population), only one prey refuge patch (competition between one prey population entirely and non-refuge prey population of the other) and common refuge patch (competition between both refuge and non-refuge prey in and out of the common patch). Equilibrium abundance of each population and the stability criterion are absolutely motivated by the interspecific competition strategies by both prey due to their patch selection. Mathematical results and numerical results support these hypothesis.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Rahul Saha, Geetha G Cryptanalysis analyses various combinations among plaintexts, ciphertexts and random keys; even using differential methods or analog methods, the attackers can interpret the keys depending upon the operations in the round functions or any subset of the algorithm. The previous research emphasizes on creation of different cryptographic functions, however the randomness of such functions has not been researched significantly so far. In this paper, we have shown a random function generator which can be used for any cryptographic algorithm. This generator outputs the combination of functions in random and cannot be traced back due its randomness. The objective of our research work is not to identify a particular boolean function that is balanced or symmetric based on its input variables, our proposed work provides a random combination of generic boolean functions as used in MD5 or SHA series, block cipher round functions and stream ciphers. Moreover, the random selection of input variables for a particular function also makes it desirable for cryptographic function modules. The results of our experimentation show that the functions generated by the proposed generator provide a good non-linearity, resiliency and balanced effect.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Bashir Ahmad, Rodica Luca We investigate the existence and uniqueness of solutions for a system of nonlinear Caputo type sequential fractional integro-differential equations with coupled Riemann–Stieltjes integral boundary conditions, by using the Leray–Schauder alternative and the Banach contraction principle. Two examples are also presented to illustrate our results.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Prabir Panja, Shyamal Kumar Mondal, Dipak Kumar Jana In this paper, a three species predator-prey interaction model among Phytoplankton, Zooplankton and Fish has been developed in the presence of toxicant. It is assumed that Phytoplankton grows logistically and its growth rate is affected by toxin release in the environment through different natural and human activities. It is considered that Zooplankton consumes only Phytoplankton. Fish consumes both Phytoplankton as well as Zooplankton. Here, it is also assume that Phytoplankton releases some toxin in the environment which makes some death on Zooplankton. Also, it is considered that toxicant in the environment are increased constantly for different natural and human behavior. Then, different equilibrium points have been determined and the stability of the proposed system has been analyzed around these equilibrium points. Hopf bifurcation analysis has been done with respect to the contact rate between Phytoplankton and Environmental toxin (γ), constant rate of increase of Environmental toxin (A) , depletion rate of environmental toxin (d 1), harvesting effort (E) and rate of releasing toxin by Phytoplankton (ρ). From this study, it is seen that these parameters have a big role on the stability of this predator prey system. Finally, some numerical simulation results have been shown to verify our analytical findings.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Kang-Kang Wang, Ya-Jun Wang, Sheng-Hong Li, Jian-Cheng Wu In this paper, we aim to investigating in detail the stability of the system, the mean extinction time and stochastic resonance (SR) phenomena caused by a multiplicative periodic signal for a dual time-delayed metapopulation system subjected to cross-correlated noises. By use of the fast descent method, the small time delay approximation method and the SR theory, we obtain the expressions of the steady state probability distribution function, the mean first-passage time and signal-to-noise ratio (SNR). Numerical results indicate that the multiplicative, additive and association noises together with time delay τ can all accelerate the transition from the stable state of big density to the extinction one and play significant roles in weakening the stability and shortening the mean extinction time of the metapopulation. In particular, the additive noise and time delay τ can result in the crash of the system, while another time delay θ can strengthen the biological system stability and extend the declining time for the population. On the other hand, with respect to the SNR, the figures show that time delay τ plays entirely antipodal roles in motivating stochastic resonance (SR) in a variety of different situations. Conversely, the multiplicative noise intensity Q and time delay θ all along produce negative effect on exciting the SR. Meanwhile, the increase of the weak additive noise intensity M can stimulate the SR phenomenon, but the bigger values of M will suppress the SNR and SR phenomenon. The strength of the noise correlation λ plays an important role in restraining the SR in most cases except that it does in the plot of SNR-Q.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): D. Aguilar-Velázquez, L. Guzmán-Vargas We investigated the synchronization and temporal correlations in a simple signaling network model. The model, which is able to display spatio-temporal avalanches and diverse fractal time correlations, consists of Boolean units located in a small-world network. We evaluated the synchrony between pairs of sets of units by means of the global lability synchronization measure, which is based on the probability of change of the total number of synchronized signals, for a range of evolutions of the system with different correlation dynamics. Here, we show that the global lability distribution exhibits power-law scaling for large-scale dynamics identified with 1/f signals, whereas a breakdown in the scaling behavior emerges when there are deviations toward either short-term correlated or uncorrelated dynamics. Furthermore, we extend our study to interacting multilayer networks, which consist of two small-world networks with different correlation dynamics in each layer. We evaluated the change in the correlation and the synchronization dynamics displayed by the system in terms of the coupling parameter between layers. Our results show that long-range correlated fluctuations naturally emerge or are still present even when coupled layers initially display different correlation dynamics. Moreover, the correlation-synchronization between pairs of global lability events closely follows a power-law scaling when networks are coupled, indicating that there exists a high correlation over long time scales due to information transmission.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Wencheng Guo, Jiandong Yang Aiming at improving the regulation quality of a hydro-turbine governing system with sloping ceiling tailrace tunnel, Hopf bifurcation control using nonlinear state feedback is studied. Firstly, the nonlinear mathematical model of a hydro-turbine governing system with sloping ceiling tailrace tunnel is presented. Then, a novel control strategy using nonlinear state feedback with polynomial functions is proposed, and Hopf bifurcation control of a hydro-turbine governing system using the proposed control strategy is described. Finally, the application and functional mechanism of the proposed control strategy are analyzed by comparison with the PID strategy. The results indicate that: The proposed nonlinear state feedback control strategy is able to make the frequency of a hydro-turbine unit return to the initial value (i.e. rated frequency), and the regulation quality and the response speed are obviously better than those under the PID strategy. The effect of the linear term of nonlinear state feedback control strategy is to modify the system's linear stability, in order to eliminate or delay an existing bifurcation. Altering the nonlinear term can change the stability of bifurcation solutions, for example, converting a Hopf bifurcation from subcritical to supercritical.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Ying Xu, Ya Jia, Jun Ma, Ahmed Alsaedi, Bashir Ahmad Synapse plays an important role in signal exchange and information encoding between neurons. Electric and chemical synapses are often used to investigate the synchronization in electrical activities of neurons. In this paper, memristor is used to connect two neurons and the phase synchronization in electrical activities is discussed. Inter-spike interval (ISI) is calculated from the sampled time series for membrane potential, and the dependence of coupling intensity on phase synchronization of neuron is investigated and the effect of electromagnetic induction is considered. Furthermore, the synchronization stability of network is detected under noise; a statistical synchronization factor is also calculated. It is found synchronization can be enhanced under memristor coupling and appropriate noise is also helpful for synchronization stability.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Emile Franc Doungmo Goufo One of the questions that has recently predominated the literature is the generation and modulation of strange chaotic attractors, namely the ones with multi scrolls. The fractional theory might be useful in addressing the questions. We use the Caputo fractional derivative together with Haar wavelet numerical scheme to investigate a three-dimensional system that generates chaotic four-wing attractors. Some conditions of stability at the origin (the trivial equilibrium point) are provided for the model. The error analysis shows that the method converges and is concluded thanks to Fubini–Tonelli theorem for non-negative functions and the Mean value theorem for definite integrals. Graphical simulations, performed for some different value of the derivative order α show existence, as expected, of chaotic dynamics characterized by orbits with four scrolls, typical to strange attractors. Hence, fractional calculus appears to be useful in generating and modulating chaotic multi-wing attractors.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Giovanni Bella This paper explores the possibility of complex dynamics in a variant of the [29] model of endogenous growth. In particular, we derive the exact parametric configuration that allows for the emergence of a double-pulse homoclinic orbit, and the rise of globally indeterminate solutions, in the same area where local determinacy was found. Our results confirm that irregular patterns and oscillating solutions can be obtained along a subsidiary homoclinic orbit at which the periodic loop starts to double, so that the system might perpetually oscillate around the long run equilibrium, being thus confined in a stationary trapping region outside the neighborhood of the steady state. The economic implications of these results are finally discussed.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Frank T. Werner, Benjamin K. Rhea, R. Chase Harrison, Robert N. Dean Presented is an electronic implementation of a matched filter intended for chaos-based communication systems. While implementing the transmitter side of such systems is relatively trivial, the receiver has proven challenging to develop. Most chaotic systems lack a known fixed basis function, making it difficult to develop a matched filter for them. Instead, their receivers rely on more complicated or less effective techniques to compensate for the presence of noise. However, a previously developed manifold piecewise linear chaotic system has been shown to have an exact analytic solution. This solution has enabled the development of a matched filter for use in any communication system based on this chaotic system. The original communication system operated at a fundamental frequency of 84 Hz, much too low for any practical applications. Therefore, newer communication systems have been designed to operate at higher frequencies. In this work, the performance of this matched filter has been evaluated with a 18.4 kHz chaotic oscillator.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Sami H. Altoum, Hakeem A. Othman, Hafedh Rguigui The quantum white noise (QWN) Gaussian kernel operators (with operators parameters) acting on nuclear algebra of white noise operators is introduced by means of QWN- symbol calculus. The quantum-classical correspondence is studied. An integral representation in terms of the QWN- derivatives and their adjoints is obtained. Under some conditions on the operators parameters, we show that the composition of the QWN-Gaussian kernel operators is a QWN-Gaussian kernel operator with other parameters. Finally, we characterize the QWN-Gaussian kernel operators using important connection with the QWN- derivatives and their adjoints.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): L. De Micco, M. Antonelli, H.A. Larrondo This paper deals with a family of interesting 2D-quadratic maps proposed by Sprott, in his seminal paper [1], related to “chaotic art”. Our main interest about these maps is their great potential for using them in digital electronic applications because they present multiple chaotic attractors depending on the selected point in the parameter’s space. Only results for the analytical representation of these maps have been published in the open literature. Consequently, the objective of this paper is to extend the analysis to the digital version, to make possible the hardware implementation in a digital medium, like field programmable gate arrays (FPGA) in fixed-point arithmetic. Our main contributions are: (a) the study of the domains of attraction in fixed-point arithmetic, in terms of period lengths and statistical properties; (b) the determination of the threshold of the bus width that preserves the integrity of the domain of attraction and (c) the comparison between two quantifiers based on respective probability distribution functions (PDFs) and the well known maximum Lyapunov exponent (MLE) to detect the above mentioned threshold.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Asma Rouatbi, Khaled Omrani Two conservative differences schemes for the nonlinear dispersive Benjamin–Bona–Mahony–KdV (BBM-KdV) equation are proposed. The first scheme is two-level and nonlinear-implicit. The second scheme is three-level and linear implicit. Existence of its difference solutions has been shown. It is proved by the discrete energy method that the two schemes are uniquely solvable, unconditionally stable and the convergence is of second-order in the maximum norm. An iterative algorithm is proposed for solving the nonlinear scheme. The particular case known as the RLW equation is also discussed numerically in detail. Furthermore, three invariants of motion are evaluated to determine the conservation properties of the problem. Interaction of solitary waves with different amplitudes are shown. The three invariants of the motion are evaluated to determine the conservation proprieties of the system. The temporal evaluation of a Maxwellian initial pulse is then studied. Some numerical examples are given in order to validate the theoretical results.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Yuyang Chen, Kaiming Bi, Songnian Zhao, David Ben-Arieh, Chih-Hang John Wu This paper proposes a new, information-transmission-based behavior-switch that applies the individual fear factor (IFF) to describe how information regarding current disease epidemics can cause human behavior change in a disease-dynamic system. This research is a first attempt to mathematically model how an individual's emotions influence behavior. The approach can be used to study the relationship of information dissemination (e.g., broadcasting, public health education, news media, etc.) and human behaviors during disease outbreaks. The expression of IFF and a mathematical IFF model that combines human behaviors with a classic SIR model is presented, and an optimal strategy that reduces the number of infected individuals and financial loss due to switch behaviors is proposed. In particular, model stability is analyzed and corresponding necessary conditions are determined. This novel modeling approach shows that information transmission influence individual fear, resulting in a variety of human behaviors and leading to numerous disease consequences.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Ahmad Zareie, Amir Sheikhahmadi, Adel Fatemi Measurement of the spreading capability of nodes has been one of the most attractive challenges in the field of social networks. Because of the huge number of nodes in a network, it has appealed to many researchers to find an accurate measure which can potentially detect the spreading capability and rankings of nodes. Most of the available methods determine the spreading capability of nodes based on their topological locations. In this paper, however, we have proposed a new measure based on the basic notions in information theory to detect the spreading capability of nodes in networks on the basis of their topological information. The simulation and experimental results of investigating real-world and artificial networks show that the proposed measure is more accurate and efficient than the similar ones.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Yao Liu, Yashun Wang, Xun Chen, Chunhua Zhang, Yuanyuan Tan The determination of fractal dimension of rough surface profile curve is important for characterizing the fractal features of rough surface microscopic topography. There are many methods to calculate the fractal dimension, such as the power spectrum method (PSM), the structure function method (SFM), the variation method, the R/S analysis method, the wavelet transform method and etc., among which the PSM and SFM are widely used methods. This study aims to improve the computational accuracy of the fractal dimension of the profile curve. For this purpose, the two-stage method based on PSM and SFM are proposed. Firstly, we analyze the principle of calculating the fractal dimension of profile curve using PSM and SFM. Then, based on PSM and SFM, we propose a two-stage method for determining the fractal dimension of profile curve. Simulation results show that the two-stage method for fractal dimension of profile curve can greatly reduce the error compared with the original PSM and SFM. Finally, the fractal dimensions of the profile curve of the cuboid specimen are calculated by the original PSM and SFM and the two-stage method respectively. The experimental results show that the proposed method provides more precise results for determining the fractal dimension.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Peiyuan Sun, Xuesong Liu, Enze Wang, Mingfeng He, Qiuhui Pan We study the evolution of cooperation by modeling interactional individuals with compensation mechanism on a two-dimensional square lattice. In this model, the payoff to cooperators is the same no matter what types their neighbors are, while the payoff to defectors depends on whether there exists cooperative neighbor. In addition, cooperators will obtain some compensation from the payoffs of defectors. We find that a larger compensation coefficient in the model leads to the higher cooperation, which means the compensation mechanism partly promotes cooperation. In addition, the simulation results suggest that decreasing either the payoff of defectors without cooperative neighbors or the payoff of defectors with cooperative neighbors will promote cooperation.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Yahui Sun, Ling Hong, Jun Jiang The stochastic sensitivity function (SSF) method is extended to estimate the stationary probability distribution around periodic attractors of nonautonomous nonlinear dynamical systems subjected to Poisson white noise in this paper. After deriving the stochastic sensitivity functions of period-N cycle of mapping systems based on the characteristic of Poisson process, non-autonomous dynamical systems around periodic attractors are converted to mapping systems by constructing a stroboscopic map, and then the stochastic sensitivity functions of periodic attractors of nonautonomous nonlinear systems can be obtained by adopting the results of mapping systems. It is found that the stochastic sensitivity functions depend on the product of noise intensity and the arrival rate of Poisson processes. To illustrate the validity of the proposed method, a Henon map driven by Poisson processes and a Mathieu–Duffing oscillator under Poisson white noise are studied.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): B. Nana, S.B. Yamgoué, R. Tchitnga, P. Woafo The paper is devoted to theoretical and experimental investigations of an electromechanical system consisting of DC motor, a physical pendulum with the repulsive magnets. The work consists of modeling, simulation and experimental measurements to validate the analytical predictions and the numerical simulation of the earlier introduced mathematical model. The parameters of the model are estimated using the experimental data. The analyzed system shows several types of non-linear effects, including hysteresis, jump phenomena, chaos and periodic dynamics. Good agreement between real and simulated behavior of the system is obtained.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Chen Chu, Jinzhuo Liu, Chen Shen, Jiahua Jin, Yunxuan Tang, Lei Shi Evolutionary prisoner's dilemma game in structured populations on a weighted square lattice, on which the edge weight represents the relationship between agents and adaptively changes in time, has been proved to be an efficient way that can promote cooperation. In fact, such an adaptive link weight introduces a new time scale τa , not necessarily equal to the time scale of game strategy τ ɛ. Inspired from aforementioned above, we investigate the effect of w = τ ɛ τ a on the evolution of cooperative behavior. Through numerical simulation, we find cooperation can be promoted effectively with a larger value of w, which is related to the increase of average link weight in the structured population.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Dawei Ding, Jie Yan, Nian Wang, Dong Liang This paper investigates the synchronization of fractional order complex-variable dynamical networks with time-varying coupling. Based on information of the complex network's configuration, an effective adaptive pinning control strategy to adjust simultaneously coupling strength and feedback gain is designed. Besides, we also consider the synchronization in complex networks with time-varying coupling weight. By constructing suitable Lyapunov function and using the presented lemma, some sufficient criteria are derived to achieve the synchronization of fractional order complex-variable dynamical networks under the corresponding update law. The update law is only dependent on the states of the complex dynamical networks, which do not need any other information such as the characteristic of the uncoupled nodes of the networks. Further, the result extends the synchronization condition of the real-variable dynamical networks to the complex-valued field. Finally, the correctness and feasibility of the proposed theoretical results are verified by two examples of fractional complex-variable dynamic networks.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Qiu Li, MingChu Li, Lin Lv, Cheng Guo, Kun Lu Infectious diseases have proven to be remarkably resilient foes of human health and so the prevention and control of infectious diseases have been attracting the attention of all countries over the world. Vaccination is an effective way to prevent the spread of infectious diseases. However, vaccination is a long-standing social dilemmas due to the vaccine’s risk by itself and the spread of infectious diseases in the population depends on not only the pathogen itself, but also the impact of social network structures. In this paper, we propose a new prediction model of infectious diseases with new vaccination strategies based on network structures and dynamic replicator. In our model, we consider not only the subsidies of vaccine failure but also the incentive strategy for medical treatment to promote individuals to take the initiative to vaccinate. At the same time, in decision-making phase, we use weighted average benefits of all participants to update their strategies due to individual difference. Simulation experiments show that the our proposed model is much effective and better than other existing models. We also use Jacobian matrix to prove the stability of dynamic equilibrium for our proposed model.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Enrique de Amo, Manuel Díaz Carrillo, Juan Fernández-Sánchez We introduce and study representation systems for the numbers in the unit interval [0, 1]. We call them ϕm -systems (where ϕm is a pseudo-golden ratio). With the aid of these representation systems, we define a family hm of strong negations and an increasing function gm which is the inverse of the generator of hm . The functions hm and gm are singular, and we study several properties; among which we calculate the Hausdorff dimensions of certain sets that are related to them. Finally, we prove that gm is an infinite convolution, and the sequence of coefficients in the Fourier series of its associated Stieltjes measure does not converge to zero.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): A. Ahmadian, S. Salahshour, M. Ali-Akbari, F. Ismail, D. Baleanu This paper focuses on providing a new scheme to find the fuzzy approximate solution of fractional differential equations (FDEs) under uncertainty. The Caputo-type derivative base on the generalized Hukuhara differentiability is approximated by a linearization formula to reduce the corresponding uncertain FDE to an ODE under fuzzy concept. This new approach may positively affect on the computational cost and easily apply for the other types of uncertain fractional-order differential equation. The performed numerical simulations verify the proficiency of the presented scheme.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Xingpeng Zhang, Dong Li, Xiaohong Zhang Randomness is a common phenomenon in nonlinear systems. And conditions to reach synchronization are more complex and difficult when chaotic systems have random parameters. So in this paper, an adaptive scheme for synchronization of chaotic system with random parameters by using the fuzzy impulsive method and combining the properties of Wiener process and Ito differential is investigated. The main concepts of this paper are applying fuzzy method to approximate the nonlinear part of system, then using Ito differential to study the Wiener process of random parameters of chaotic system, and realizing synchronization under fuzzy impulsive method. The stability is analyzed by Lyapunov stability theorem. At the end of the paper, numerical simulation is presented to illustrate the effectiveness of the results obtained in this paper.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Ali Durdu, Yılmaz Uyaroğlu In this study, a novel three dimensional autonomous chaotic attractor was found and secure communication masking application was performed with optimal fractional order, which offers more precise and faster results than first order chaotic equations, via Pecaro Carroll synchronization algorithm. The shortest synchronization time was investigated with optimal fractional order value. In the novel secure communication synchronization application with fractional order chaotic system, there is an angle of 45° between the signals sent and received, which clearly shows that the system can be employed in secure communication.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): R. Sahadevan, P. Prakash Lie symmetry analysis and invariant subspace methods of differential equations play an important role separately in the study of fractional partial differential equations. The former method helps to derive point symmetries, symmetry algebra and admissible exact solution, while the later one determines admissible invariant subspace as well as to derive exact solution of fractional partial differential equations. In this article, a comparison between Lie symmetry analysis and invariant subspace methods is presented towards deriving exact solution of the following coupled time fractional partial differential equations: (i) system of fractional diffusion equation, (ii) system of fractional KdV type equation, (iii) system of fractional Whitham-Broer-Kaup’s type equation, (iv) system of fractional Boussinesq-Burgers equation and (v) system of fractional generalized Hirota-Satsuma KdV equation.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Gabriel Gajardo, Werner Kristjanpoller We apply MF-ADCCA to analyze the presence and asymmetry of the cross-correlations between Latin-American and US stock market indices and crude oil market. We find that multifractality exists in this cross-correlations, and that there is asymmetry on its behavior. The asymmetry degree changes accordingly to the series considered for the trend behavior. We find that fluctuation sizes greatly influence the asymmetry in the cross-correlation exponent, increasing for large fluctuations when we consider the trend of the crude oil price. We also find no clear differences in the exponents with different scales under different trends of the WTI, contrary to other studies in asymmetric scaling behavior. When we examine the time varying features of the asymmetry degree we find that the US indices show a consistent behavior in time for both trends, where the cross-correlation exponents tend to be larger for downward trends. On the other hand, given the more heterogeneous individual properties of Latin-American indices, the asymmetry behavior varies more depending on the trend considered.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Chris G. Antonopoulos, Charalampos Skokos, Tassos Bountis, Sergej Flach In the study of subdiffusive wave-packet spreading in disordered Klein–Gordon (KG) nonlinear lattices, a central open question is whether the motion continues to be chaotic despite decreasing densities, or tends to become quasi-periodic as nonlinear terms become negligible. In a recent study of such KG particle chains with quartic (4th order) anharmonicity in the on-site potential it was shown that q − Gaussian probability distribution functions of sums of position observables with q > 1 always approach pure Gaussians ( q = 1 ) in the long time limit and hence the motion of the full system is ultimately “strongly chaotic”. In the present paper, we show that these results continue to hold even when a sextic (6th order) term is gradually added to the potential and ultimately prevails over the 4th order anharmonicity, despite expectations that the dynamics is more “regular”, at least in the regime of small oscillations. Analyzing this system in the subdiffusive energy domain using q-statistics, we demonstrate that groups of oscillators centered around the initially excited one (as well as the full chain) possess strongly chaotic dynamics and are thus far from any quasi-periodic torus, for times as long as t = 10 9 .

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Viktor Avrutin, Jose D. Morcillo, Zhanybai T. Zhusubaliyev, Fabiola Angulo Power electronic DC/AC converters (inverters) play an important role in modern power engineering for a broad variety of applications including solar and wind energy systems as well as electric and hybrid cars drives. It is well known that the waveform of the output voltage (or current) of an inverter may be significantly distorted by phase restricted high frequency oscillations, frequently referred to as bubbling. However, the reasons leading to the appearance of this undesired effect are still not completely understood. Considering as an example a 2D model of a PWM H-bridge single-phase inverter, the present paper reports the appearance of two different kinds of bubbling. In the first case, the appearance of bubbling occurs suddenly and is related to the change of periodicity. We show that high-periodic, quasiperiodic and chaotic oscillations may exhibit bubbling, and also that solutions with and without bubbling may coexist. In the second case, the appearance of bubbling occurs gradually in the parameter domain where the investigated system undergoes border collisions of so-called persistence type. As a result, the appearance of the bubbling of the second kind does not change the periodicity of the motion but nevertheless disturbs the waveform. We discuss some differences in the properties of the second kind of bubbling from the first one, and present numerical techniques for its detection.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): I.A. Shepelev, D.V. Shamshin, G.I. Strelkova, T.E. Vadivasova We study the boundaries of existence of traveling waves and stationary spatial structures in an active medium model by varying the control parameters. The medium is represented by a ring of diffusively coupled FitzHugh–Nagumo neurons, which, when uncoupled, can demonstrate excitable, self-sustained oscillatory or bistable dynamics depending on control parameter values. The dynamical regimes realized in the medium are compared with those ones observed in an individual FitzHugh–Nagumo neuron. Possible bifurcations of traveling waves are analyzed when the dynamics of the medium elements changes. We also explore the influence of the relaxation level of FitzHugh–Nagumo neurons on the medium dynamics.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Can Li, Zun-Guang Guo, Zhi-Yu Zhang Brucellosis is a major problem worldwide in public health and existing work mainly focused on severity estimation based on the real data. However, global analysis on brucellosis transmission model is not well understood. In this paper, we presented a dynamical model of brucellosis transmission coupled with sheep and human populations and global analysis is shown based on Lyapunov functions. We found that the global dynamics of brucellosis model is determined by basic reproduction number R 0: if R 0 < 1, then the disease-free equilibrium is globally asymptotically stable; otherwise, the endemic equilibrium is globally asymptotically stable. We hope that our study may provide theoretical basis for the further work on brucellosis control.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Debajyoti Saha, Pankaj Kumar Shaw, Sabuj Ghosh, M.S. Janaki, A.N.S. Iyengar The evidence of finite nonlinear interaction in a DC glow discharge plasma has been demonstrated by estimating phase coherence index for different types external forcing techniques likewise noise, sinusoidal, square etc. The existence of finite phase coherence index i.e finite correlation prompts us to carry out nonlinearity analysis using delay vector variance (DVV). Finite nonlinear interaction obtained from phase coherence index values is observed to be predominant at a particular amplitude of square forcing which corroborates our nonlinearity analysis using DVV. Existence of phase coherence index has been demonstrated introducing continuous wavelet transform (CWT). Characterization of the difference in the phase distribution by the difference in the waveform in real space instead of dealing in Fourier space has been facilitated by introducing structure function or path length for different orders to study and identify the dynamical system. The expression of path length eventually enables us to evaluate the phase coherence index. The transition in the dynamics is observed through recurrence plot techniques which is an efficient method to observe the critical regime transitions in dynamics.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Peng Gang Sun, Xiaoke Ma, Juan Chi By identifying important nodes (driver nodes), the minimum dominating set (MDS) provides an effective model to dominate complex networks. However, in many networks, the skeleton of driver nodes selected using the MDS is usually connected, which motivates us to explore a new framework and try to dominate a network by identifying its minimum skeleton. We define the minimum skeleton of a graph as a subgraph induced from the nodes within the minimum connected dominating set (MCDS), and the problem can be solved by a maximum spanning tree-based algorithm. For the domination of complex networks, in general, the MCDS needs more driver nodes, and is more robust than the MDS against link attack. Interestingly, for the MDS, it is harder to control the networks with weaker communities, while for the MCDS, this finding tends to be observed on the networks with homogeneous community sizes. In addition, for the MDS, the curves for the percentage of driver nodes on the networks with variable community strengths shift downward as the average degree of the networks increases, while for the MCDS, the curves, like power functions rotate clockwise. For the both, it tends to be harder to control the networks with stronger overlapping, and the number of driver nodes is dependent on the networks’ degree distribution.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Fengji Peng, Wen Wang, Shengyou Wen We prove that, given 0 ≤ β ≤ α and α ≤ λ ≤ β + α , there exist compact subsets X, Y of the Euclidean space R ⌈ α ⌉ such that dim A X = α , dim A Y = β and dim A ( X × Y ) = λ , where ⌈α⌉ is the smallest integer ≥ α and dim A denotes Assouad dimension. In the proof an Assouad dimension formula for uniform Cantor sets is established.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Kálmán Klapcsik, Ferenc Hegedűs In this study, a nonlinear investigation of a periodically driven gas bubble in glycerine is presented. The bifurcation structure of the bubble oscillator (Keller–Miksis equation) is explored in the pressure amplitude-frequency parameter plane of the excitation by means of initial (high resolution bi-parametric plots) and boundary value problem solvers at various ambient temperatures. The range of the applied temperature covers two orders of magnitude difference in the liquid viscosity which is the main damping factor of the system. Therefore, the evolution of the harmonic and ultraharmonic resonances are presented starting with an overdamped behaviour (there are no resonances in the parameter space) and ending up with a fully developed bifurcation superstructure. The results reveal a complex period bubbling mechanism organized in a Farey-tree; inside each bubble a fine substructure of alternating chaotic and periodic bands exist. The description of the bifurcation structure presented throughout the paper can help to understand the mechanism of dissipation on the behaviour of nonlinear systems in more detail.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Jiaquan Zhang, Dan Lu, Shunkun Yang Epidemic spreading has been intensively studied in SIS epidemic model. Although the mean-field theory of SIS model has been widely used in the research, there is a lack of comparative results between different theoretical calculations, and the differences between them should be systematically explained. In this paper, we have compared different theoretical solutions for mean-field theory and explained the underlying reason. We first describe the differences between different equations for mean-field theory in different networks. The results show that the difference between mean-field reaction equations is due to the different probability consideration for the infection process. This finding will help us to design better theoretical solutions for epidemic models.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): D. Youmbi Fouego, E.D. Dongmo, P. Woafo This work deals with the analysis of the voltage amplitude generated in a linear load by an array of Van der Pol (VDP) and Hindmarsh–Rose (HR) oscillators. For the array of Van der Pol oscillators, it is found that after a threshold number of oscillators under which the power is equal to zero, the power increases with the size of the array. A high order nonlinearity in the damping of the Van der Pol oscillator increases the power. In the case of the array of HR oscillators, it is shown that varying the coupling coefficient leads to the appearance of chaotic dynamics in the system. Contrary to the case of the VDP oscillators, it is found that the voltage amplitudes decrease when the size of the array of the HR oscillators increases. These results can be linked to the mechanism of biological oscillators powering biological organs.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Vladimir A. Maximenko, Alexander E. Hramov, Alexey A. Koronovskii, Vladimir V. Makarov, Dmitry E. Postnov, Alexander G. Balanov The spatially discrete-continuous dynamical systems, that are composed of a spatially extended medium coupled with a set of lumped elements, are frequently met in different fields, ranging from electronics to multicellular structures in living systems. Due to the natural heterogeneity of such systems, the calculation of Lyapunov exponents for them appears to be a challenging task, since the conventional techniques in this case often become unreliable and inaccurate. The paper suggests an effective approach to calculate Lyapunov exponents for discrete-continuous dynamical systems, which we test in stability analysis of two representative models from different fields. Namely, we consider a mathematical model of a 1D transferred electron device coupled with a lumped resonant circuit, and a phenomenological neuronal model of spreading depolarization, which involves 2D diffusive medium. We demonstrate that the method proposed is able reliably recognize regular, chaotic and hyperchaotic dynamics in the systems under study.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Marcel Ausloos, Roy Cerqueti, Tariq A. Mir This paper explores a real-world fundamental theme under a data science perspective. It specifically discusses whether fraud or manipulation can be observed in and from municipality income tax size distributions, through their aggregation from citizen fiscal reports. The study case pertains to official data obtained from the Italian Ministry of Economics and Finance over the period 2007–2011. All Italian (20) regions are considered. The considered data science approach concretizes in the adoption of the Benford first digit law as quantitative tool. Marked disparities are found, - for several regions, leading to unexpected “conclusions”. The most eye browsing regions are not the expected ones according to classical imagination about Italy financial shadow matters.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Xue-feng Zhang, Feng-bao Yang, Xu-zhu Wang In 1982, Dubois and Prade investigated the relationship between belief function, plausibility function and basic probability assignment when the involved universe is finite. In this paper, the similar results on their relationships are obtained with a continuous universe. As an important facility to connect possibility distribution in continuous universes and discrete probability values, basic probability histogram is defined by means of measurement amplitude, which is a notion with both probability and possibility features. A theorem about how to calculate a suitable sample size for estimation is proposed based on the researches on basic probability histograms. Through the theorem, we can directly calculate the appropriate samples size for any population distribution. Even with small samples, a reasonable estimation can be obtained with a non-normal distribution.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): Zhenghong Deng, Chunmiao Ma, Xudong Mao, Shenglan Wang, Zhenxi Niu, Li Gao Understanding the evolution of cooperation among selfish individuals remains a large challenge. Network reciprocity has been proved to be an efficient way that can promote cooperation and has spawned many studies focused on network. Traditional evolutionary games on graph assumes players updating their strategies based on their current payoff, however, historical payoff may also play an indispensable role in agent's decision making processes. Another unavoidable fact in real word is that not all players can know exactly their historical payoff. Based on these considerations, in this paper, we introduce historical payoff and use a tunable parameter u to control the agent's fitness between her current payoff and historical payoff. When u equals to zero, it goes back to the traditional version; while positive u incorporates historical payoff. Besides, considering the limited knowledge of individuals, the structured population is divided into two types. Players of type A, whose proportion is v, calculate their fitness using historical and current payoff. And for players of type B, whose proportion is 1 − v , their fitness is merely determined by their current payoff due to the limited knowledge. Besides, the proportion of these types keeps unchanged during the simulations. Through numerous simulations, we find that historical payoff can promote cooperation. When the contribution of historical payoff to the fitness is larger, the facilitating effect becomes more striking. Moreover, the larger the proportion of players of type A, the more obvious this promoting effect seems.

Abstract: Publication date: November 2017 Source:Chaos, Solitons & Fractals, Volume 104 Author(s): S. Behnia, M. Yahyavi, R. Habibpourbisafar Hierarchy of one-dimensional ergodic chaotic maps with Tsallis type of q-deformation are studied. We find that in the chaotic region, these maps with q-deformation are ergodic as the Birkhoff ergodic theorem predicts. q-deformed maps are defined as ratios of polynomials of degree N. Hence, by using the Stieltjes transform approach (STA), invariant measure is proposed. In addition, considering Sinai-Ruelle-Bowen (SRB) measure, Kolmogorov-Sinai (KS) entropy for q-deformed maps is calculated analytically. The new q-deformed scheme have ability to keep previous significant properties such as ergodicity, sensitivity to initial condition. By adding q-parameter to the hierarchy in order increase the randomness and one-way computation, we present a new scheme for watermarking. The introduced algorithm tries to improve the problem of failure of encryption such as small key space, encryption speed and level of security. To illustrate the effectiveness of the proposed scheme, some security analyses are presented. By considering the obtained results, it can be concluded that, this scheme have a high potential to be adopted for watermarking. It can be concluded that, the proposed novel watermarking scheme for image authentication can be applied for practical applications.