Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Haijun Wang, Xianyi Li Based on the famous Shimizu–Morioka system, this paper proposes a novel five-dimensional Shimizu–Morioka-type hyperchaotic system that has an infinite set of heteroclinic orbits. Of particular interest are the following observed properties of the system: (i) the existence of both ellipse-parabola-type and hyperbola-parabola-type of equilibria; (ii) the strange attractor coexisting either non-isolated equilibria or two pairs of symmetrical equilibria; (iii) the existence of the proposed strange attractors and hyperchaotic attractors bifurcated from the corresponding singularly degenerate heteroclinic cycles; (iv) the existence of an infinite set of both ellipse-parabola-type and hyperbola-parabola-type heteroclinic orbits.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Fu Chaoqi, Wang Ying, Wang Xiaoyang, Gao Yangjun Studying attack strategy of complex networks is the basis of investigating network characteristics such as robustness, invulnerability, and network security. Knowing means of attack can help us take more effective measures to ensure network security. Presently, most research conclusions focus on a single vertex being attacked, and the choice of a set of attack nodes is also limited to a complete understanding of network information. In this paper, considering the effect of cascading failure, we focus on the multi-node attack strategy. Our results showed that the distance between attack targets has a great effect on the attacking effect. Taking both the average avalanche scale and maximum destruction size into account, when the distance between attack targets was 2, the network suffered the most serious damage. If the information about the network was unclear, we presented 3 kinds of conditional attack strategies. Under the condition of different tolerance coefficients and different degrees of known information, each strategy had its own unique advantages. In conclusion, the research in this paper supports the easy and quick selection of attack targets under the condition of incomplete information.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Cristina Flaut, Diana Savin In this paper we present applications of special numbers obtained from a difference equation of degree three. As a particular case of this difference equation of degree three, we obtain the generalized Pell-Fibonacci-Lucas numbers, which were extended to the generalized quaternion algebras. Using properties of these quaternion elements, we can define a set with an interesting algebraic structure, namely, an order on a generalized rational quaternion algebra. Another presented application is in the Coding Theory, since some of these numbers can be used to built cyclic codes with good properties (MDS codes).

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Lingling Huang This paper is concerned with the Diophantine properties of the orbits of real numbers in continued fraction system under the doubling metric. More precisely, let φ be a positive function defined on N . We determine the Lebesgue measure and Hausdorff dimension of the set E ( φ ) = { ( x , y ) ∈ [ 0 , 1 ) × [ 0 , 1 ) : T n x − y < φ ( n ) for i.m. n } , where T is the Gauss map and “i.m.” stands for “infinitely many”.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Andrey V. Andreev, Vladimir V. Makarov, Anastasija E. Runnova, Alexander N. Pisarchik, Alexander E. Hramov We consider a neuronal network model where an external stimulus excites some neurons, which in turn activate other neurons in the network via synapse. We find that the regularity in macroscopic spiking activity of the whole neuronal network maximizes at a certain level of intrinsic noise. A similar resonant behavior, referred to as coherence resonance, is also observed with respect to the stimulus strength, network size, and number of stimulated neurons. The coherence is quantitatively estimated with the signal-to-noise ratio calculated from the average power spectra of the macroscopic signal and with autocorrelation time. Overall synchronization in the neuronal network also exhibits a non-monotonic dependence on the network size.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Steve A. Mendoza, Eliza W. Matt, Diego R. Guimarães-Blandón, Enrique Peacock-López In ecological modeling, seasonality can be represented as an alternation between environmental conditions. This concept of alternation holds common ground between ecologists and chemists, who design time-dependent settings for chemical reactors to influence the yield of a desired product. In this study and for a variety of maps, we consider a switching strategy that alternates between two undesirable dynamics that yields a stable desirable dynamic behavior. By comparing bifurcation diagrams of a map and its alternate version, we can easily find parameter values, which, on their own, yield chaotic orbits. When alternated, however, the parameter values yield a stable periodic orbit. Our analysis of the two-dimensional (2-D) maps is an extension of our previous work with one-dimensional (1-D) maps. In the case of 2-D maps, we consider the Beddington, Free, and Lawton and Udwadia and Raju maps. For these 2-D maps, we not only show that we can find “chaotic” parameters for the so-called “chaos” + “chaos” = “periodic” case, but we find two new “desirable” dynamic situations: “quasiperiodic” + “quasiperiodic” = “periodic” and “chaos” + “chaos” = “periodic coexistence.” In the former case, the alternation of chaotic dynamics yield two different periodic stable orbits implying the coexistence of attractors.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Hui Li, Tianwei Li An Apollonian packing is one of the most beautiful circle packings based on an old theorem of Apollonius of Perga. Ford circles are important objects for studying the geometry of numbers and the hyperbolic geometry. In this paper we pursue a research on the Ford sphere packing, which is not only the three dimensional extension of Ford circle packing, but also a degenerated case of the Apollonian sphere packing. We focus on two interesting sequences in Ford sphere packings. One sequence converges slowly to an infinitesimal sphere touching the origin of the horizontal plane. The other sequence converges at fastest rate to an infinitesimal sphere in a particular position on the plane. All these sequences have their counterparts in Ford circle packings and keep similar features. For example, our finding shows that the x-coordinate of one Ford circle sequence converges to the golden ratio gracefully. We define a Ford sphere group to interpret the Ford sphere packing and its sequences finally.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Mengyao Yu, Kehui Sun, Wenhao Liu, Shaobo He Based on closed-loop modulation coupling pattern and the model of sinusoidal cavity, a high-dimensional sinusoidal cavity hyperchaotic system is proposed. The number of sinusoidal cavities is controlled by the system parameters. By designing a piecewise-linear controller, the grid sinusoidal cavity attractors are obtained. The equilibrium points are theoretically analyzed through mathematical calculation. Taking the two-dimensional grid sinusoidal cavity hyperchaotic map as an example, dynamics of the system are analyzed by phase diagram, equilibrium points, Lyapunov exponents spectrum, bifurcation diagram, complexity and distribution characteristics. The results show that it has rich dynamical behaviors, including complicated phase space trajectory, hyperchaotic behavior, large maximum Lyapunov exponent and typical bifurcations. The proposed hyperchaotic map has advantages in complexity and distribution in the whole parameter space. Therefore, it has good application prospects in secure communication.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Huayong Zhang, Shengnan Ma, Tousheng Huang, Xuebing Cong, Hongju Yang, Feifan Zhang This research investigates pattern self-organization along the route to chaos in a space- and time-discrete predator–prey system, where the prey shows convection movement in space. Through analysis on Turing instability of the system, pattern self-organization conditions are determined. Based on the conditions, simulations are performed under two initial conditions, demonstrating two pattern transitions along the route to chaos. In the first pattern transition, the patterns start from regular stripes, experiencing twisted stripes, then return to regular stripes again. The second pattern transition is much more complex and shows three stages. Especially, an alternation between ordered patterns and disordered chaos is found, contributing greatly to the spatiotemporal complexity of the system. When the system stays at the homogeneous chaotic states, Turing instability driven by convection and diffusion can still force the self-organization of regular striped patterns. The finding in this research provides a new comprehending for pattern self-organization and transition in spatially extended predator–prey systems.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Xiang Li, Suxia Zhang, Xijun Liu, Xiaojing Wang, Anqi Zhou, Peng Liu MAM (mitochondria-associated endoplasmic reticulum membrane complex) is kind of complexes formed between the endoplasmic reticulum (ER) and mitochondria, and it plays an important role in calcium signal transduction of cell. Coherence resonance is cooperative effect of noise and nonlinear system. In the field of life science, coherence resonance reflects the rhythm of life, so studying it by using the theory and method of nonlinear stochastic dynamics has very important biological significance. On the basis of the calcium oscillation model considering the role of MAMs proposed by Szopa et al., and taking two important indexes - the maximal permeability of the Ca2+ channels in the ER membrane (kch ) and the maximal permeability of MCU (mitochondrial calcium uniporter) at MAMs (kMAM ) as the analysis parameters, this paper further studies the bifurcation characteristic of the system, including two-parameter bifurcation and one-parameter bifurcation, and theoretically expounds the reasons for the generation and disappearance of calcium oscillations. Then, on the basis of bifurcation analysis, we focus on the coherence resonance characteristic of calcium signals and its physiological significance when analysis parameters are affected by environmental noise. Graphical abstract

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Torunbalcı Aydın In this paper, bicomplex Fibonacci quaternions are defined. Also, some algebraic properties of bicomplex Fibonacci quaternions which are connected with bicomplex numbers and Fibonacci numbers are investigated. Furthermore, Binet’s formula, Cassini’s identity, Catalan’s identity for these quaternions and real representation of these quaternions are given.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Jinzhuo Liu, Tong Li, Wei Wang, Na Zhao, Feilu Hang Social rewarding is a common but significant mechanism that promotes the evolution of cooperation. However, besides social rewarding, antisocial rewarding is also ordinary. Thus, we study the evolution of cooperation on prison dilemma game with strategy-neutral rewarding, namely a mechanism including social and antisocial rewarding. Two additional strategies, rewarding cooperators (RC) and rewarding defectors (RD), which establish union-like support to aid akin players are introduced. We show that the new mechanism greatly promotes the evolution of cooperation even in the presence of antisocial rewarding. The rewarding cooperators can enjoy both the benefits of their prosocial contributions and the corresponding rewards, thus they can form cooperative clusters to resist the aggression of defectors. On the other hand, due to their inherent greedy, rewarding defectors fail to secure a sustainable future. Our research might provide valuable insights into further exploring the nature of cooperation in the real world.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Yuangan Wang, Honglin Yu In this paper, a fuzzy method is combined with intermittent control method to realize synchronization of chaotic system. Two plant rules of intermittent control are considered to get two theorems. Fuzzy scheme for synchronization is proposed in theorem. Finally, a simulation example is proposed to verify the effectiveness of our results.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Jiahua Jin, Chen Chu, Chen Shen, Hao Guo, Yini Geng, Danyang Jia, Lei Shi Many literatures suggest that incorporating the environment of a focal player (denoted by the average payoff of its all immediate neighbors) into its fitness can promote cooperation in spatial evolutionary games. However, the immediate neighbors influence the focal one to varying degree. Inspired by these, we quantify the focal player's environment with a weighted average payoff of its all immediate neighbors via two interdependent parameters. Numerous simulations show that two moderate parameter pairs favor cooperation, in addition, when the contribution of all immediate neighbors’ payoffs to the environment is negative, the cooperation is promoted remarkably. The generality of this mechanism is verified on different networks and more games. Our work might shed light on the understanding of the evolution of cooperative behaviors in real life.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): B.C. Bao, P.Y. Wu, H. Bao, Q. Xu, M. Chen This paper presents a novel third-order autonomous memristive chaotic oscillator, which is accomplished by parallelly coupling a simple memristive diode bridge emulator into a Sallen–Key low-pass filter (LPF). With the modeling of this oscillator, stability analyses of the equilibrium point and numerical simulations of the phase plane orbit, time-domain sequence, bifurcation diagram, and finite-time Lyapunov exponent spectrum are performed, from which period, quasi-period, chaos, and quasi-period to chaos route are found. Particularly, two types of dynamical phenomena of quasi-periodic behavior and point-cycle chaotic bursting that are further identified by using 0–1 test are observed in such a third-order autonomous memristive oscillator, which have been rarely reported in the previous literatures. Additionally, hardware experiments are implemented and the quasi-periodic behavior and point-cycle chaotic bursting are well confirmed.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): R.R. Nigmatullin, H.C. Budnikov, A.V. Sidelnikov The general mesoscopic theory pretending on the quantitative description of the interfacial surface and the self-similar structure of the double electric layer in the vicinity of solid electrodes is suggested. It takes into account the fact that the fractal dimension can be complex and depends on the applied potential. In the frame of the suggested theory, the fitting function pretending on description of the VAGs was found. It was applied for the fitting of original experimental data related to detection of the Cd+ ions in different concentrations in the KCl solution. The assumptions and possible applications of the suggested theory to description of other measured data are discussed.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): B. Nana, S.B. Yamgoué, I. Kemajou, R. Tchitnga, P. Woafo Inductive devices with ferromagnetic core are widely used in many electronic circuits to store magnetic energy. They should be treated as nonlinear devices, and the nonlinearity of their characteristics arises from the dependence of inductance on current. Such inductors display saturation and hysteresis behaviors. In the present paper, we report a new mathematical model based on the experimental data of hysteresis for ferromagnetic core inductors. We used the model to determine analytically the expression of current in a RLC series circuit forced by an alternating source. Multi periodic and high amplitude chaotic signals are observed and good agreement is found between theoretical and experimental results.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): J. Kengne, S.M. Njikam, V.R. Folifack Signing In the present contribution, the dynamics of a simple autonomous jerk system with hyperbolic tangent nonlinearity is considered. The system consists of a linear transformation of Model MO13 previously introduced in [Sprott, 2010]. The form of nonlinearity is interesting in the sense that with the variation of a control parameter, saturation may be approached gradually obeying hyperbolic tangent function, as in the case of magnetization in ferromagnetic system, non ideal op. amplifier, solar-wind-driven magnetosphere-ionosphere system, and activation function in neural network. The fundamental properties of the model are discussed including equilibria and stability, phase portraits, Poincaré sections, bifurcation diagrams and Lyapunov exponents’ spectrum. Period doubling bifurcation, antimonotonicity (i.e. concurrent creation an annihilation of periodic orbits), chaos, hysteresis, and coexisting bifurcations are reported. As a major outcome of this paper, a window in the parameter space is revealed in which the jerk system experiences the unusual phenomenon of multiple coexisting attractors (i.e. coexistence of two, four or six disconnected periodic and chaotic self excited attractors) resulting from the simultaneous presence of three families of parallel bifurcation branches and hysteresis. To the best of the authors’ knowledge, no example of such a simple and ‘elegant’ 3D autonomous system capable of six different strange attractors is reported in the relevant literature. Some PSpice simulations based on a physical implementation of the system are carried out to support the theoretical analysis.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Z. Lajmiri, R. Khoshsiar Ghaziani, Iman Orak In this paper, we study the bifurcation and stability of a ratio-dependent predator-prey model with nonconstant predator harvesting rate. The analysis is carried out both analytically and numerically. We determine stability and dynamical behaviours of the equilibria of this system and characterize codimension 1 and codimension 2 bifurcations of the system analytically. Our bifurcation analysis indicates that the system exhibits numerous types of bifurcation phenomena, including Fold, Hopf, Cusp, and Bogdanov–Takens bifurcations. We use the numerical software MATCONT, to compute curves of equilibria and to compute several bifurcation curves. We especially approximate a family of limit cycles emanating from a Hopf point. Our results generalize and improve some known results and show that the model has more rich dynamics than the ratio-dependent predator-prey model without harvesting rate.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Saravanan M, Jagtap Ameya D, Vasudeva Murthy A.S. The nonlinear molecular deformation of the ferronematic liquid crystal in the presence of external applied magnetic field intensity is investigated in view of solitons for the director axis. The Frank’s free energy density of the nematic liquid crystal comprising the basic elastic deformations, molecular deformation associated with the nematic molecules and the suspended ferromagnetic particles and their interactions with magnetic field intensity is deduced to a sine-Gordon like equation using the classical Euler–Lagrange’s equation. Using the small angle approximation we establish the Ginzburg–Landau (GL) equation and a class of solutions are obtained. In the normal condition of large angle oscillation of the director axis, we constructed a damped sine-Gordon (sG) equation with the additional perturbation appears in the form of cosine function. The sG equation is solved using numerical simulation and kink excitations were obtained as the molecular deformation for the case of constant damping and distorted kink to a planar configuration transition as we increase the damping.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Tuan D. Pham, Hong Yan Measures of statistical regularity or complexity for time series are pervasive in many fields of research and applications, but relatively little effort has been made for image data. This paper presents a method for quantifying the statistical regularity in images. The proposed method formulates the entropy rate of an image in the framework of a stationary Markov chain, which is constructed from a weighted graph derived from the Kullback–Leibler divergence of the image. The model is theoretically equal to the well-known approximate entropy (ApEn) used as a regularity statistic for the complexity analysis of one-dimensional data. The mathematical formulation of the regularity statistic for images is free from estimating critical parameters that are required for ApEn.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Xianghua Li, Jingyi Guo, Chao Gao, Leyan Zhang, Zili Zhang Network immunization is an effective strategy for restraining virus spreading in computer networks and rumor propagation in social networks. Currently, lots of strategies are proposed based on topological structures of networks, such as degree-based and betweenness-based network immunization strategies. However, these studies assume that nodes in a network are homogeneous, i.e., each node has the same characteristic. However, more and more studies have revealed the heterogeneous characteristic of a network. For example, the activities of individual in a computer and social network play an important role in virus spreading and rumor propagation. Some active individuals can promote the outbreak of virus and the spread of a rumor. In this paper, a new network immunization strategy is proposed through combining the characteristics of network structure with node activities. Comprehensive experiments in both benchmark and synthetic networks show that our proposed strategy can restrain virus prorogation effectively.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Xiuli Cen, Jaume Llibre, Meirong Zhang In the present paper we study periodic solutions and their stability of the m-order differential equations of the form x ( m ) + f n ( x ) = μ h ( t ) , where the integers m, n ≥ 2, f n ( x ) = δ x n or δ x n with δ = ± 1 , h(t) is a continuous T-periodic function of non-zero average, and μ is a positive small parameter. By using the averaging theory, we will give the existence of T-periodic solutions. Moreover, the instability and the linear stability of these periodic solutions will be obtained.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): S. Eskandar, S.M. Hoseini The Inverse Scattering Transform (IST) method is applied to find soliton solutions for a higher-order nonlinear Schrödinger (NLS) equation. Eigenfunctions of linearized operator which have a central role in soliton perturbation theory are explicitly found.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Debajyoti Saha, Sabuj Ghosh, Pankaj Kumar Shaw, M.S. Janaki, A.N.S. Iyengar Interplay of transition of floating potential fluctuations in a glow discharge plasma in the toroidal vacuum vessel of SINP tokamak has been observed. With variation in the strength of the vertical and toroidal magnetic fields, regular and inverted relaxation oscillations as well as sinusoidal oscillations are observed with the slow and fast time scale of the relaxation oscillations reversing their nature at a high value of vertical magnetic field strength. However for small value of toroidal magnetic field the transitions follow relaxation → chaotic oscillations with the chaotic nature prevailing at higher values of toroidal magnetic field. Evolution of associated anode fireball dynamics under the action of increasing vertical, toroidal as well as increasing vertical field at a fixed toroidal field (mixed field) of different strength has been studied. Estimation of phase coherence index for each case has been carried out to examine the evidence of finite nonlinear interaction. A comprehensive study of the dynamics of the fireball is found to be associated with the values of phase coherence index. The index is found to take maximum values for the case of toroidal, mixed field when there is an existence of power/energy concentration in a large region of frequency band. A detailed study of the scaling region using detrended fluctuation analysis (DFA) by estimating the scaling exponent has been carried out for increasing values of discharge voltage (DV), vertical, toroidal as well as the mixed field (toroidal plus vertical). A persistence long range behaviour associated with the nature of the anode glow has been investigated in case of higher values of toroidal, mixed field whereas increasing DV, vertical magnetic field lead to a perfectly correlated dynamics with values of scaling exponent greater than unity.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Khalid K. Ali, R.I. Nuruddeen, K.R. Raslan In this paper, we constructed new solitary structures for the space-time fractional simplified modified Camassa-Holm (MCH) equation and space-time fractional symmetric regularized long wave (SRLW) equation using the modified extended tanh method. The space-time fractional derivatives are defined in the sense of the new conformable fractional derivative. Further, with the help of Mathematica software, the set of over-determined algebraic equations obtained after reducing the equations to ordinary differentials equations are treated. We finally provide graphical illustrations for some structures.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Srđan Kostić, Nebojša Vasović, Kristina Todorović, Igor Franović In present paper, authors examine the dynamics of a spring-slider model, considered as a phenomenological setup of a geological fault motion. Research is based on an assumption of delayed interaction between the two blocks, which is an idea that dates back to original Burridge–Knopoff model. In contrast to this first model, group of blocks on each side of transmission zone (with delayed interaction) is replaced by a single block. Results obtained indicate predominant impact of the introduced time delay, whose decrease leads to transition from steady state or aseismic creep to seismic regime, where each part of the seismic cycle (co-seismic, post-seismic and inter-seismic) could be recognized. In particular, for coupling strength of order 102 observed system exhibit inverse Andronov–Hopf bifurcation for very small value of time delay, τ≈0.01, when long-period (T = 12) and high-amplitude oscillations occur. Further increase of time delay, of order 10−1, induces an occurrence of a direct Andronov–Hopf bifurcation, with short-period (T = 0.5) oscillations of approximately ten times smaller amplitude. This reduction in time delay could be the consequence of the increase of temperature due to frictional heating, or due to decrease of pressure which follows the sudden movement along the fault. Analysis is conducted for the parameter values consistent with previous laboratory findings and geological observations relevant from the seismological viewpoint.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Ignacio S. Gomez We present a calculus of the Kolmogorov–Sinai entropy for quantum systems having a mixing quantum phase space. The method for this estimation is based on the following ingredients: i) the graininess of quantum phase space in virtue of the Uncertainty Principle, ii) a time rescaled KS–entropy that introduces the characteristic time scale as a parameter, and iii) a mixing condition at the (finite) characteristic time scale. The analogy between the structures of the mixing level of the ergodic hierarchy and of its quantum counterpart is shown. Moreover, the logarithmic time scale, characteristic of quantum chaotic systems, is obtained.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): A.R. Dzhanoev, I.M. Sokolov The diffusion in the comb structures is a popular model of geometrically induced anomalous diffusion. In the present work we concentrate on the diffusion along the backbone in a system where sidebranches are planes, and the diffusion thereon is anomalous and described by continuous time random walks (CTRW). We show that the mean squared displacement (MSD) in the backbone of the comb behaves differently depending on whether the waiting time periods in the sidebranches are reset after the step in the backbone is done (a rejuvenating junction model), or not (a non-rejuvenating junction model). In the rejuvenating case the subdiffusion in the sidebranches only changes the prefactor in the ultra-slow (logarithmic) diffusion along the backbone, while in the non-rejuvenating case the ultraslow, logarithmic subdiffusion is changed to a much faster power-law subdiffusion (with a logarithmic correction) as it was found earlier by Iomin and Mendez [25]. Moreover, in the first case the result does not change if the diffusion in the backbone is itself anomalous, while in the second case it does. Two of the special cases of the considered models (the non-rejuvenating junction under normal diffusion in the backbone, and rejuvenating junction for the same waiting time distribution in the sidebranches and in junction points) were also investigated within the approach based on the corresponding generalized Fokker–Planck equations.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Xin Sui, Liang Li In this paper, we construct a dynamic guarantee network model. Based on the constructed model, the dynamic evolution of risk contagion is researched by means of simulation methods. The risk contagion research is carried out from three aspects:guarantee mechanism, partner selection mechanism, and production parameter. The research shows that: (1) Firm size distribution takes on a power-law tail. (2) Guarantee network provides a channel for risk contagion and aggravates risk contagion among firms. (3) The type of partner selection mechanism has an impact on risk contagion. Risk cognation in the net worth mechanism is more serious in comparison to the random mechanism. (4) Risk contagion among firms is the increasing function of the production parameter φ.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Yuki Ida, Jun Tanimoto Recently, a new vaccination game model was proposed, where an intermediate defense measure besides two fundamental strategies; committing vaccination that leads to a perfect immunity and not committing vaccination, was introduced as third strategy. We explore what happens if both effectiveness and cost of an intermediate defense measure stochastically perturbing on the viewpoint of whether or not the third strategy helping to improve total social payoff. We found that unlike resonance effect by adding noise to payoff matrix in case of spatial prisoner's dilemma (SPD) games, adding time-varying noise on both effectiveness and cost does not make difference from the default setting without perturbation to the third strategy. However, if the noise initially given to each agent is frozen, we found the third strategy becoming robust to survive. In particular, if the strategy updating rule allows a more advantageous third strategy can be more commonly shared among agents through copying, the total social payoff is significantly improved.

Abstract: Publication date: December 2017 Source:Chaos, Solitons & Fractals, Volume 105 Author(s): Uğur Erkin Kocamaz, Barış Cevher, Yılmaz Uyaroğlu In this study, a new reaching rule in the Sliding Mode Control (SMC) is proposed for chaos control and synchronization. It is applied on the nonlinear Chua's circuit. The SMC signals are provided with the Lyapunov stability theory. Classical, cubic and partial cubic variants of SMC signals are constructed. Numerical simulations are performed to compare the effectiveness of the SMC signals and the results are discussed.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Jaume Llibre, Clàudia Valls Recently several works have studied the following model of finance x ˙ = z + ( y − a ) x , y ˙ = 1 − b y − x 2 , z ˙ = − x − c z , where a, b and c are positive real parameters. We study the global dynamics of this polynomial differential system, and in particular for a one–dimensional parametric subfamily we show that there is an equilibrium point which is a global attractor.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Wenchen Han, Junzhong Yang Recently, the synchronization on multi-layer networks has drawn a lot of attention. In this work, we study the stability of complete synchronization on duplex networks. We first numerically investigate the effects of coupling functions on complete synchronization on duplex networks. Then, we propose two approximation methods to deal with the stability of complete synchronization on duplex networks. In the first method, we introduce a modified master stability function and, in the second method, we only take into consideration the contributions of a few most unstable transverse modes to the stability of complete synchronization. We find that both methods work well for predicting the stability of complete synchronization for small networks. For large networks, the second method still works pretty well.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Swalpa Kumar Roy, Siddharth Kumar, Bhabatosh Chanda, Bidyut B. Chaudhuri, Soumitro Banerjee This paper presents a novel approach to calculate the affine parameters of fractal encoding, in order to reduce its computational complexity. A simple but efficient approximation of the scaling parameter is derived which satisfies all properties necessary to achieve convergence. It allows us to substitute to the costly process of matrix multiplication with a simple division of two numbers. We have also proposed a modified horizontal-vertical (HV) block partitioning scheme, and some new ways to improve the encoding time and decoded quality, over their conventional counterparts. Experiments on standard images show that our approach yields performance similar to the state-of-the-art fractal based image compression methods, in much less time.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Salim Lahmiri, Stelios Bekiros Since its inception, the digital currency market is considerably growing, especially in the most recent years. The main purpose of this paper is to investigate, assess and detect chaos, randomness, and multi-scale temporal correlation structure in prices and returns of this specific virtual and speculative market throughout two distinct time periods; namely under a low-level regime period during which prices slowly increased, and during a high and turbulent regime time period whereby they exponentially increased. We found that chaos is only present in prices during both periods, whilst the level of uncertainty in returns has significantly increased during the high-price time period. Furthermore, both prices and returns exhibit long-range correlations and multi-fractality. The fat-tailed probability distributions are the main source of multi-fractality in the time series of prices and returns. Finally, short (long) fluctuations in returns are dominant during low (high) price-regime time period, respectively. Overall, the high-price regime phase has profoundly revealed consistent nonlinear dynamical patterns in the Bitcoin market.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): J. Palanivel, K. Suresh, D. Premraj, K. Thamilmaran In this paper, we report the effect of fractional order, time delay and noisy parameter on slow passage phenomenon in a nonlinear oscillator. We consider a second order LCR based nonlinear electronic circuit with a time varying resistor and use sinusoidal modulation on the resistor to change the resistance value. The time dependent parameter of a dynamical system causes slow passage effect which leads to bifurcation delay in the system dynamics and leaving the actual bifurcation point unpredictable. We find that the fractional order of the system significantly changes the magnitude of bifurcation delay and brings the system to oscillatory state. While the time delay in dynamical systems destroys the stable steady state leading it to oscillatory state. We study both these fractional order and time delay and their combined effect on the slow passage effect. We have also included the noise with the sinusoidal periodic modulation on the resistor to understand the effect of noise on the slow passage effect and found that the noise enhances the oscillatory behaviour of the system.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Isa Abdullahi Baba, Evren Hincal In this paper we consider three strains of influenza (I1, I2, and I3) where we have vaccine for strain1 (V1) only, and population has enough awareness of strain 2. There is neither vaccine nor awareness for strain 3. Our main aim is to mathematically analyze the effect of the vaccine for strain 1 and awareness of strain 2 on the dynamics of strain 3. It is also in our aim to study the coexistence of these three strains. Six equilibrium points were obtained and their global stability using Lyapunov functions was shown to depend on the magnitude of a threshold quantity, called basic reproduction ratio. It was shown that the coexistence of strain 1 and strain 2 is not possible and the coexistence of the three strains was shown numerically. It can be observed from the numerical simulations that, although vaccine curtail the spread of strain 1, awareness curtail the spread of strain 2, but they both have negative effect on strain 3. This tells the relevant authorities whenever there is influenza epidemic to investigate thoroughly the possibilities of the existence of multiple strains, so as to provide vaccines and enough awareness on all the strains present.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Françoise Pène We study the mixing of observables of Z d -extensions of probability preserving dynamical systems. We explain how this question is directly linked to the local limit theorem and establish a scaling rate for dynamically continuous observables of the Z 2 -periodic Sinai billiard. We compare our approach with the induction method.

Abstract: Publication date: January 2018 Source:Chaos, Solitons & Fractals, Volume 106 Author(s): Saptarshi Ghosh, Anna Zakharova, Sarika Jalan We present the emergence of chimeras, a state referring to coexistence of partly coherent, partly incoherent dynamics in networks of identical oscillators, in a multiplex network consisting of two non-identical layers which are interconnected. We demonstrate that the parameter range displaying the chimera state in the homogeneous first layer of the multiplex networks can be tuned by changing the link density or connection architecture of the same nodes in the second layer. We focus on the impact of the interconnected second layer on the enlargement or shrinking of the coupling regime for which chimeras are displayed in the homogeneous first layer. We find that a denser homogeneous second layer promotes chimera in a sparse first layer, where chimeras do not occur in isolation. Furthermore, while a dense connection density is required for the second layer if it is homogeneous, this is not true if the second layer is inhomogeneous. We demonstrate that a sparse inhomogeneous second layer which is common in real-world complex systems, can promote chimera states in a sparse homogeneous first layer.

Abstract: Publication date: December 2017 Source:Chaos, Solitons & Fractals, Volume 105 Author(s): Mukta Garg, Ruchi Das In this article, we introduce the notion of almost average shadowing property (ALASP) for continuous semi-flows and study its properties. We give a class of continuous semi-flows possessing the ALASP. We also investigate the relationship of the ALASP with various known dynamical properties for continuous semi-flows, for instance, topological transitivity, weak mixing, strong ergodicity, equicontinuity, sensitivity, Li-Yorke chaos and Takens and Ruelle chaos.

Abstract: Publication date: December 2017 Source:Chaos, Solitons & Fractals, Volume 105 Author(s): Jose Castro, Joaquin Alvarez, Fernando Verduzco, Juan E. Palomares-Ruiz In this paper the chaotic behavior of second-order, discontinuous systems with a pseudo-equilibrium point on a discontinuity surface is analyzed. The discontinuous system is piecewise linear and approximated to a non-smooth continuous system. The discontinuous term is represented by a sign function that is replaced by a saturation function with high slope. Some of the conditions that determine the chaotic behavior of the approximate system are formally established. Besides, the convergence of its chaotic solutions to those of the discontinuous system is shown. Several bifurcation diagrams of both systems show the similarity of their dynamical behavior in a wide parameter range, and particularly for a parameter region determined from the application of the Melnikov technique to non-smooth systems, where a chaotic behavior can be displayed.

Abstract: Publication date: December 2017 Source:Chaos, Solitons & Fractals, Volume 105 Author(s): Aikaterini Mandilara, Zoran Ivić, Dalibor Čevizović, Željko Pržulj We show that the self-induced transparency of optical phonons may appear in a systems consisting of a two level atoms interacting with elastic waves. The presence of the gap in phonon spectrum substantially enhances the pulse delay in respect to the acoustic self induced transparency phenomena. One of the main characteristics of the predicted phenomenon is the appearance of the critical velocity of the self-induced transparency pulse which, in the absorbing media, represents the upper limit which pulse may reach. Its magnitude is determined by the ratio of the phonon gap and the energy difference of the two level system. This feature opens a new way for the control of the speed of elastic waves. We believe that, in the view of the emerging new quantum technologies relying on creation and trapping of the coherent phonons interacting with artificial atoms, some practical implementations of interest for the storage and manipulation of quantum information may be realised on the basis of our work.

Abstract: Publication date: December 2017 Source:Chaos, Solitons & Fractals, Volume 105 Author(s): Yang Wang, Binfeng Pan, Yue Zheng, Xiang Lu The chaotic transport in Earth-Moon three-body system has been demonstrated to be a novel approach to explore the Moon at a low energy cost. However, existing targeting methods require sufficient experience to construct Earth-Moon chaotic transfer orbits, which is not an easy task to the untrained eye. In this paper, the elitist teaching-learning-based optimization (ETLBO) based optimal targeting method is presented to provide a systematic approach to design the chaotic transfer orbits in the Earth-Moon planar circular restricted three-body problem, without any requirement of prior experience. Unlike the existing targeting methods, the chaotic transfer orbits design problem is treated as a class of multi-constraints fuel-optimal problem with multi-dimensional decision variables. A discrete chaotic dynamical model is formulated according to the Poincaré map, and several consecutive control steps of small bounded thrusts are made to direct the chaotic series towards the desired invariant torus near the Moon. The suboptimal consecutive control thrusts are obtained by a state-of-art numerical optimization algorithm ETLBO, which does not require any algorithm-specific parameters with less computational effort. Numerical demonstrates are provided to illustrate the applications of the ETLBO based optimal targeting method, which reveal that several potential chaotic transfer orbits can be easily obtained by this method.

Abstract: Publication date: December 2017 Source:Chaos, Solitons & Fractals, Volume 105 Author(s): Salim Lahmiri, Stelios Bekiros Recent works in econophysics have quantitatively shown that the latest global financial crisis has considerably affected nonlinear dynamics in markets worldwide. In the current study, we focus on complexity in volatility time series during pre-crisis, crisis, and post-crisis time periods. In this regard, a large set of international stock and commodity markets as well as economic uncertainty indices is considered in our work. The main finding is that empirical distributions of long memory parameter, Kolmogorov complexity and Shannon entropy, have all varied across pre-crisis, crisis, and post-crisis time periods. In other words, all three complexity measures are informative and suitable in order to characterize nonlinear dynamics in volatility series throughout the examined sample periods. Indeed, it was found that complexity increased during crisis period, yet diminished during the pre-crisis period. Overall, the latest financial crisis has truly affected complexity revealed in the volatility time series of the world major markets.

Abstract: Publication date: December 2017 Source:Chaos, Solitons & Fractals, Volume 105 Author(s): Raffaela Capitanelli, Cristina Pocci This paper concerns the periodic homogenization of the stationary heat equation in a domain with two connected components, separated by an oscillating interface defined on prefractal Koch type curves. The problem depends both on the parameter ε that defines the periodic structure of the interface and on n, which is the index of the prefractal iteration. First, we study the limit as ε vanishes, showing that the homogenized problem is strictly dependent on the amplitude of the oscillations and the parameter appearing in the transmission condition. Finally, we perform the asymptotic behaviour as n goes to infinity, giving rise to a limit problem defined on a domain with fractal interface.

Abstract: Publication date: December 2017 Source:Chaos, Solitons & Fractals, Volume 105 Author(s): Edgardo Comas, Walter Legnani This paper aims to identify the chaotic behavior of plasma flow in hypersonic flights from the measurements made in a vector of atmospheric sounding. To this end, we analyzed the data obtained from the flight of the SONDA vector for the study of the upper atmosphere (GRADICOM II Project), which was launched on 11 July 2011 by the Instituto de Investigaciones Científicas y Técnicas para la Defensa from the Ministerio de Defensa de la Nación. The vector reached an apogee of 92 km approximately and a maximum speed of 6.56 Mach, presenting plasma formation immediately after entering hypersonic flight. Based on the inertial measurements in the vector's body, we proceeded to solve the equations of navigation, obtaining the trajectory and speed of the vector together with the transformation matrix between the reference systems, Earth Centered Earth Fixed “ECEF” and East North Up “ENU”. After the identification of the hypersonic flight zone, and from the point of view of the accelerations in ENU reference system, we computed several indexes and verified which were compatible with chaotic behavior using various methods. These results led us to continue our research work in order to better understand this characteristic of hypersonic flight.

Abstract: Publication date: December 2017 Source:Chaos, Solitons & Fractals, Volume 105 Author(s): Badr-eddine Berrhazi, Mohamed El Fatini, Aziz Laaribi, Roger Pettersson, Regragui Taki In this paper, we establish the existence of a unique global positive solution for a stochastic epidemic model, incorporating media coverage and driven by Lévy noise. We also investigate the dynamic properties of the solution around both disease-free and endemic equilibria points of the deterministic model. Furthermore, we present some numerical results to support the theoretical work.

Abstract: Publication date: December 2017 Source:Chaos, Solitons & Fractals, Volume 105 Author(s): Z.T. Njitacke, J. kengne, L. Kamdjeu Kengne This paper focuses on the dynamics of a modified jerk circuit obtained via replacing the diode bridge memristor in the original jerk circuit introduced in [24] with a first-order hybrid diode circuit. Both memristive diode bridge and first order hybrid diode are frequency dependent component even though the later device doesn't has a pinched hysteresis loop. The analysis is carried out in terms of bifurcation diagrams, graph of Lyapunov exponents, phase portraits, Poincaré section, time series and frequency spectra. The results indicate that, the new circuit exhibits rich dynamic behaviors including multiple coexisting self-excited attractors (e.g. coexistence of two, four or six disconnected periodic and chaotic attractors) and antimonotonicity (i.e. concurrent creation and annihilation of periodic orbits) compared to the original memrisitve jerk circuit. Basins of attraction of various coexisting attractors display extremely complex structures thus justifying jumps between coexisting attractors in experiment. Both PSpice simulations and laboratory experimental measurements are carried out to support the theoretical analyses.