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): U.A. Rozikov, I.A. Sattarov We consider a family of (2, 2)-rational functions given on the set of complex p-adic field C p . Each such function has a unique fixed point. We study p-adic dynamical systems generated by the (2, 2)-rational functions. We show that the fixed point is indifferent and therefore the convergence of the trajectories is not the typical case for the dynamical systems. Siegel disks of these dynamical systems are found. We obtain an upper bound for the set of limit points of each trajectory, i.e., we determine a sufficiently small set containing the set of limit points. For each (2, 2)-rational function on C p there are two points x ^ 1 = x ^ 1 ( f ) , x ^ 2 = x ^ 2 ( f ) ∈ C p which are zeros of its denominator. We give explicit formulas of radiuses of spheres (with the center at the fixed point) containing some points such that the trajectories (under actions of f) of the points after a finite step come to x ^ 1 or x ^ 2 . Moreover for a class of (2, 2)-rational functions we study ergodicity properties of the dynamical systems on the set of p-adic numbers Qp . For each such function we describe all possible invariant spheres. We show that if p ≥ 3 then the p-adic dynamical system reduced on each invariant sphere is not ergodic with respect to Haar measure. In case p = 2 under some conditions we prove non ergodicity and we show that there exists a sphere on which our dynamical system is ergodic.

Abstract: Publication date: December 2017 Source:Chaos, Solitons & Fractals, Volume 105 Author(s): Yan-Hong Yang, Wen-Jie Xie, Ming-Xia Li, Zhi-Qiang Jiang, Wei-Xing Zhou User activity fluctuations reflect the performance of online society. We investigate the statistical properties of 1 min user activity time series of simultaneously online users inhabited in 95 independent virtual worlds. The number of online users exhibits clear intraday and weekly patterns due to human’s circadian rhythms and weekly cycles. Statistical analysis shows that the distribution of absolute activity fluctuations has a power-law tail for 44 virtual worlds with an average tail exponent close to 2.15. The partition function approach unveils that the absolute activity fluctuations possess multifractal features for all the 95 virtual worlds. For the sample of 44 virtual worlds with power-law tailed distributions of the absolute activity fluctuations, the width of singularity Δα is negatively correlated with the maximum activity (p-value = 0.070) and the time to the maximum activity (p-value = 0.010). The negative correlations are not observed for neither the other 51 virtual worlds nor the whole sample of the 95 virtual worlds. In addition, numerical experiments indicate that both temporal structure and large fluctuations have influence on the multifractal spectrum. We also find that the temporal structure has a stronger impact on the singularity width than large fluctuations.

Abstract: Publication date: Available online 11 November 2017 Source:Chaos, Solitons & Fractals Author(s): Danyang Jia, Chen Shen, Hao Guo, Chen Chu, Jun Lu, Lei Shi Why would natural selection favor the prevalence of cooperation within the groups of selfish individuals' A fruitful framework to address this question is evolutionary game theory, the essence of which is captured in the so-called social dilemmas. Voluntary participation is considered as an effective approach to promote the persistence of cooperative behavior. Because three strategic types of players lead to a rock–scissor–paper dynamic with cyclic dominance. There is no doubt that loner has played a very important role. Thus we introduce a parameter p represents loners’ participation willingness in order to explore the impact of its on cooperation in voluntary prisoner's dilemma. Large quantities of simulations demonstrate that for traditional case ( p = 1 ), three strategies will coexist. However, with p decreases, loners will greatly increase, and cooperation level declines, because defectors can be suppressed very fast. More interesting, when defectors completely vanish, cooperation becomes the best strategy, and holds the whole system. Thus our work present a viable method of understand the ubiquitous cooperative behaviors in nature and hope that it will inspire further studies to resolve social dilemmas.

Abstract: Publication date: December 2017 Source:Chaos, Solitons & Fractals, Volume 105 Author(s): Shigen Gao, Yubing Wang, Hairong Dong, Bin Ning, Hongwei Wang An adaptive dynamic surface control method using nonlinear feedback is proposed for controlling the uncertain Genesio-Tesi chaotic system. The feature of the nonlinear feedback technique lies in that the feedback gains self-adjust under different amplitudes of system states. Based on the dynamic surface control technique, the complexity explosion problem existing in the backstepping-based chaotic controllers is circumvented. Moreover, the closed-loop stability is guaranteed with rigorous mathematical proof using Lyapunov stability theorem. Comparative results are given to verify the effectiveness and advantage of the proposed method with comparison to the existing linear feedback control.

Abstract: Publication date: December 2017 Source:Chaos, Solitons & Fractals, Volume 105 Author(s): Zu Gao, Xianhua Tang, Sitong Chen We consider the existence of ground state solutions for a class of nonlinear fractional Schrödinger-Poisson systems of the form { ( − Δ ) s u + u + ϕ u = f ( u ) , in R 3 , ( − Δ ) t ϕ = u 2 , in R 3 , where 0<s≤t<1 and 2 s + 2 t > 3 . By adopting a direct approach and the Pohozaev identity, we prove that this system possesses ground state solutions with a mild assumption on f with lim u → ∞ ∫ 0 u f ( t ) d t u 3 = ∞ .

Abstract: Publication date: December 2017 Source:Chaos, Solitons & Fractals, Volume 105 Author(s): Yingke Li, Zhidong Teng, Cheng Hu, Qing Ge The vaccination, latent and relapse period are three important factors affecting the whole disease development. In this paper, we propose an SVEIR epidemic model with continuous age-dependent vaccination, latency and relapse, at the same time, the nonlinear incidence rate is also considered. Uniform persistence of the model is proved by reformulating it as the so called Volterra integral equations. The basic reproduction number R 0 , which completely determines the global dynamics of the model, is derived. By using Lyapunov functionals, the global stability of the equilibria is obtained. Namely, the disease-free equilibrium is globally asymptotically stable if R 0 < 1 , while if R 0 > 1 the endemic equilibrium is globally asymptotically stable. Finally, two numerical examples support our main analytical results.

Abstract: Publication date: December 2017 Source:Chaos, Solitons & Fractals, Volume 105 Author(s): Sergiy Koshkin, Taylor Styers We introduce a natural generalization of the golden cryptography, which uses general unimodular matrices in place of the traditional Q matrices, and prove that it preserves the original error correction properties of the encryption. Moreover, the additional parameters involved in generating the coding matrices make this unimodular cryptography resilient to the chosen plaintext attacks that worked against the golden cryptography. Finally, we show that even the golden cryptography is generally unable to correct double errors in the same row of the ciphertext matrix, and offer an additional check number which, if transmitted, allows for the correction.

Abstract: Publication date: December 2017 Source:Chaos, Solitons & Fractals, Volume 105 Author(s): Adel Ouannas, Zaid Odibat, Tasawar Hayat Chaotic dynamics and synchronization of fractional-order systems have attracted much attention recently. Based on stability theory of fractional-order systems and stability theory of integer-order systems, this paper deals with the problem of coexistence of various types of synchronization between different dimensional fractional-order chaotic systems. To illustrate the capabilities of the novel schemes proposed herein, numerical and simulation results are given.

Abstract: Publication date: December 2017 Source:Chaos, Solitons & Fractals, Volume 105 Author(s): Jiaojiao Yang, Min Wu, Yiwei Zhang In this article, we discuss two important and related concepts in the studies of geometric dimension theory, e.g. the correlation dimension and the local dimension of measures. Our results can be summarized as the following two aspects: on one hand, we show that the correlation dimension of measures is invariant under the quasi-Lipschitz mapping, and also give a sufficient condition for the coincidence of the correlation dimension and the Hausdorff dimension of measures. On the other hand, we examine the local dimensions in the limit sets of Moran construction in abstract metric space, with reasonably weaker separation condition. These discussions generalized several known results by Mattila, Moran and Rey in [14] and Li, Lou and Wu in [10,11].

Abstract: Publication date: December 2017 Source:Chaos, Solitons & Fractals, Volume 105 Author(s): Vincent Fleury I present a simple model of vertebrate embryogenesis, coupling cell division, differentiation, and morphogenesis. The model relies on a gradient of cell cycle period, in a flat shell of tissue. When the cell cycle period varies linearly between two points, a cascade of cell division occurs in the tissue, which generates a staircase of cell sizes (the Angel's staircase). The variation in cell size is associated to a stepwise variation of mechanical properties, which induces a deterministic pattern of folds. The folded shell is recognized as an animal.

Abstract: Publication date: December 2017 Source:Chaos, Solitons & Fractals, Volume 105 Author(s): Guangxi Cao, Yingying Shi The two-component autoregressive fractionally integrated moving average (ARFIMA) process and mix-correlated ARFIMA(MC-ARFIMA) are applied in this paper to generate artificial sequence with H xy = 1/2(H x + H y ) and H xy < 1/2(H x + H y ) respectively and simulate the results of multifractal detrended cross-correlation analysis (MFXDFA), multifractal detrending moving average cross-correlation analysis (MFXDMA), MFDCCA based on maximum overlap wavelet transform (MFDCCA-MODWT), and multifractal detrended partial cross-correlation analysis(MF-DPXA). The advantages and disadvantages of MFXDFA, MFXDMA(θ = 0,0.5,1), and MFDCCA-MODWT are compared to the long-memory of sequences. In the case of H xy < 1/2(H x + H y ), these three methods keep around the theoretical value with small fluctuations in a variety of sequence lengths. In the case of H xy = 1/2(H x + H y ), the curves are significantly stable and are slightly smaller than the theoretical value. The precision of these estimators may be influenced by the relationship between H xy and 1/2(H x + H y ). Multifractal features is detected and the result shows that MFXDMA-0 and MFXDMA-1 is optimal to detect the multifractality. An interesting finding is that MFDCCA-MODWOT performs best in both case of Hxy = 1/2(Hx + Hy) and Hxy < 1/2(Hx + Hy), but it performs worst to detect the multifractality. When Gaussian noise is added to the sequences with different long-memory levels, MFDPXA can eliminate the noise interference compared with MFXDFA, thereby verifying the effectiveness of this method.

Abstract: Publication date: December 2017 Source:Chaos, Solitons & Fractals, Volume 105 Author(s): Liping Chen, Wei Pan, Kunpeng Wang, Ranchao Wu, J. A. Tenreiro Machado, António M. Lopes An unified method to yield a family of fractional-order (FO) hyper-chaotic multi-scroll (HCMS) systems in Rn is proposed. Firstly, a new simple 3-dimensional (3-D) FO unstable linear system is introduced. Afterwards, additional variables are added and one nonlinear controller with adjustable parameters is included to generate HCMS attractors. A guideline to construct HCMS systems of any dimension is presented, that is verified along within the dynamics of three examples, namely 4-D, 5-D and 10-D FO HCMS systems. Phase portraits, Poincaré maps and two positive Lyapunov exponents are calculated. Moreover, a circuit of 0.96-order is also designed to realize one 4-D FO HCMS system. Numerical simulations and circuit simulation results show the feasibility of the novel approach.

Abstract: Publication date: December 2017 Source:Chaos, Solitons & Fractals, Volume 105 Author(s): Xiaotong Niu, Jiwei Xu, Zhenghong Deng Social punishment, a mechanism that cooperative individual spends a little cost to penalize defector, is verified to be an effective mechanism for promoting the evolution of cooperation. In this paper, we introduce conditional punishment, the willingness to punish p, which decides whether to carry out penalty. It is shown that cooperative behavior is significantly enhanced when punishers are taken into account and the frequency of cooperation increases with p. In addition, we find out the protective effect of punishers on evolution of cooperation from a micro point of view. We hope our work may shed light on understanding of cooperative behavior in society.

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): Yingchao Zhang, Hongyun Ning, Juan Wang, Chengyi Xia In this paper, we propose a novel public goods game model with preferential learning mechanism on two-layered lattices, and two different tunable parameters w and α have been introduced into the model to denote the extent of preferential learning and coupling strength among corresponding players, respectively. Huge quantities of Monte Carlo simulations indicate that, on the one hand, the fraction of cooperators at the stationary state can be greatly enhanced if the preference parameter w > 0, but the cooperation level can be reduced if w < 0; On the other hand, the interdependency between two-layered lattices will further enrich the evolution of cooperation, and the role of promotion becomes much more obvious when coupling strength w is zero or a small positive constant; while the interdependency will play a minor role when w ≥ 1 since the preference mechanism has driven the cooperation to arrive at a very high level. All these results can help us to further analyze and understand the evolution of cooperation within many real-world systems.

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.

Abstract: Publication date: December 2017 Source:Chaos, Solitons & Fractals, Volume 105 Author(s): Haijun Wen, Shiwang Hou, Zhaohua Liu, Yongjiang Liu It is known that production planning and scheduling are mutual influence and restriction. In this paper, we aim to obtain the minimum remanufacturing time of recycling parts by use of birandom variables and further optimize an integrated remanufacturing production planning and scheduling system under uncertain conditions. An integrated production planning and scheduling optimization model with birandom variable restraints is firstly established. Then we develop a hybrid intelligent algorithm including random simulation technique, neural network, and genetic algorithms to optimize an integrated remanufacturing production planning and scheduling system. Furthermore, we generate a random variable samples matrix through random simulation technique and a trained neural network is embedded into genetic algorithm. Finally, this hybrid intelligent algorithm is applied to optimize an integrated remanufacturing production planning and scheduling system through a case study.

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): 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): 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.