Abstract: Publication date: Available online 27 December 2019Source: Chaos, Solitons & Fractals: XAuthor(s): Aleksandra V. Tutueva, Erivelton G. Nepomuceno, Artur I. Karimov, Valery S. Andreev, Denis N. ButusovAbstractChaos-based stream ciphers form a prospective class of data encryption techniques. Usually, in chaos-based encryption schemes, the pseudo-random generators based on chaotic maps are used as a source of randomness. Despite the variety of proposed algorithms, nearly all of them possess many shortcomings. While sequences generated from single-parameter chaotic maps can be easily compromised using the phase space reconstruction method, the employment of multi-parametric maps requires a thorough analysis of the parameter space to establish the areas of chaotic behavior. This complicates the determination of the possible keys for the encryption scheme. Another problem is the degradation of chaotic dynamics in the implementation of the digital chaos generator with finite precision. To avoid the appearance of quasi-chaotic regimes, additional perturbations are usually introduced into the chaotic maps, making the generation scheme more complex and influencing the oscillations regime. In this study, we propose a novel technique utilizing the chaotic maps with adaptive symmetry to create chaos-based encryption schemes with larger parameter space. We compare pseudo-random generators based on the traditional Zaslavsky map and the new adaptive Zaslavsky web map through multi-parametric bifurcation analysis and investigate the parameter spaces of the maps. We explicitly show that pseudo-random sequences generated by the adaptive Zaslavsky map are random, have a weak correlation and possess a larger parameter space. We also present the technique of increasing the period of the chaotic sequence based on the variability of the symmetry coefficient. The speed analysis shows that the proposed encryption algorithm possesses a high encryption speed, being compatible with the best solutions in a field. The obtained results can improve the chaos-based cryptography and inspire further studies of chaotic maps as well as the synthesis of novel discrete models with desirable statistical properties.

Abstract: Publication date: Available online 17 December 2019Source: Chaos, Solitons & Fractals: XAuthor(s): Ilhan Ozturk, Fatma OzkoseAbstractIn this paper, a fractional-order model of tumor-immune system interaction has been considered. In modeling dynamics, the total population of the model is divided into three subpopulations: macrophages, activated macrophages and tumor cells. The effects of fractional derivative on the stability and dynamical behaviors of the solutions are investigated by using the definition of the Caputo fractional operator that provides convenience for initial conditions of the differential equations. The existence and uniqueness of the solutions for the fractional derivative is examined and numerical simulations are presented to verify the analytical results. In addition, our model is used to describe the kinetics of growth and regression of the B-lymphoma BCL1 in the spleen of mice. Numerical simulations are given for different choices of fractional order α and the obtained results are compared with the experimental data. The best approach to reality is observed around α=0.80. One can conclude that fractional model best fit experimental data better than the integer order model.

Abstract: Publication date: Available online 9 December 2019Source: Chaos, Solitons & Fractals: XAuthor(s): A.C. Fowler, M.J. McGuinnessAbstractWe provide an analytic estimate for the size of the bulbs adjoining the main cardioid of the Mandelbrot set. The bulbs are approximate circles, and are associated with the stability regions in the complex parameter μ-space of period-q orbits of the underlying map z→z2−μ. For the (p, q) orbit with winding number p/q, the associated stability bulb is an approximate circle with radius 1q2sinπpq.

Abstract: Publication date: Available online 2 November 2019Source: Chaos, Solitons & Fractals: XAuthor(s): Ned J. CorronAbstractA linear partial differential equation is shown to exhibit three properties often used to define chaotic dynamics. The system comprises a one-dimensional wave equation with gain that operates on a semi-infinite line. A boundary condition enforces that the waves remain finite. It is shown that the resulting solution set is dense with periodic orbits, contains transitive orbits, and exhibits extreme sensitivity to initial conditions. Definitions of chaos are considered in light of such linear chaos.

Abstract: Publication date: Available online 11 October 2019Source: Chaos, Solitons & Fractals: XAuthor(s): M.I. Syam, Mohammed Al-RefaiAbstractWe study linear and nonlinear fractional differential equations of order 0

Abstract: Publication date: Available online 3 October 2019Source: Chaos, Solitons & Fractals: XAuthor(s): P.L. Tubaro, G.B. MindlinAbstractIn this work we build a convolutional neural network capable of identifying individual birds by their songs. Since the actual data available from each individual is very limited, we use a dynamical system capable of synthesizing realistic songs, to generate surrogate-training data. The different synthetic songs are the result of integrating the dynamical system with slightly varied parameters. We show that a data set built in this way allows us to train the network to successfully identify the different individuals in our study. In this way, we present a novel way to perform data augmentation using dynamical systems.

Abstract: Publication date: Available online 19 September 2019Source: Chaos, Solitons & Fractals: XAuthor(s): Guy KatrielAbstractWe explore simple models aimed at the study of social contagion, in which contagion proceeds through two stages. When coupled with demographic turnover, we show that two-stage contagion leads to nonlinear phenomena which are not present in the basic ‘classical’ models of mathematical epidemiology. These include: bistability, critical transitions, endogenous oscillations, and excitability, suggesting that contagion models with stages could account for some aspects of the complex dynamics encountered in social life. These phenomena, and the bifurcations involved, are studied by a combination of analytical and numerical means.

Abstract: Publication date: Available online 20 July 2019Source: Chaos, Solitons & Fractals: XAuthor(s): Junhai Ma, Aili Hou, Yi TianAbstractNowadays, with more and more people choosing energy-saving product, green supply chain is increasingly becoming popular. In this paper, we build a dynamic Stackelberg game model that contains government, two green innovation enterprise which is a leader and produces two kinds of green product with similar function but different quality and a retailer selling two kinds of green product meanwhile. We use classical backward induction to solve the model. Price decision is considered first and is divided into two stages. We analyze the equilibrium price of green innovation enterprise and retailer and the stable region. Next, we analyze energy-saving index of two kinds of green product. Through numerical simulation, we further analyze system's stability conditions and system's dynamic evolution process when different parameter is adopted. In the end, we consider green products’ baseline energy-saving index made by government and get the optimal baseline energy-saving index by numerical simulation. We find that too stringent subsidy standards may can't induce manufacturer to invest greener technology. Besides, we use the delay feedback control to effectively control the chaotic phenomenon.

Abstract: Publication date: Available online 18 June 2019Source: Chaos, Solitons & Fractals: XAuthor(s): E. Bonyah, A. Atangana, Mehar ChandAbstractA mathematical model providing an asymptotic description of macro-economic system is considered in this work. The system in general deals with performance, behaviour, decision-making of an economy as a whole and also the structure. Due to the complexities of this system, a more complex mathematical model is requested. In this work, we considered the extension of the model using some non-local differential operators and the stochastic approach where the given parameters are converted to normal distributions. We have presented the conditions of existence of uniquely exact solutions of the system using the fixed-point theorem approach. Each model is solved numerical via a newly introduced modified Adams-Bashforth for fractional differential equations. We presented numerical simulations for different values of fractional order. The models with the Atangana-Baleanu and Caputo differential operators provided us with new attractors.

Abstract: Publication date: March 2019Source: Chaos, Solitons & Fractals: X, Volume 1Author(s): P.R.L. AlvesAbstractFrom the Diagram Accuracy-Deviation and the new quantifier of chaos, this paper presents time series analysis for historical prices, volatilities and returns. The study cases are financial price series of United States Brent Oil (BNO), Wipro Limited (WIT), Nasdaq, Inc. (NDAQ) and SPDR S&P 500 ETF (SPY). Detection of chaos and randomness cover the period from November 2010 to November 2018. This work introduces the chaoticity in the Lorenz’s sense, a new measure for comparison between time series. The set of results enlights the underlying dynamics of the time evolution observed in economic indexes nowadays.

Abstract: Publication date: March 2019Source: Chaos, Solitons & Fractals: X, Volume 1Author(s): Alexander N. Pisarchik, Parth Chholak, Alexander E. HramovAbstractWe consider the brain as an autonomous stochastic system, whose fundamental frequencies are locked to an external periodic stimulation. Taking into account that phase synchronization between brain response and stimulating signal is affected by noise, we propose a novel method for experimental estimation of brain noise by analyzing neurophysiological activity during perception of flickering visual stimuli. Using magnetoencephalography (MEG) we evaluate steady-state visual evoked fields (SSVEF) in the occipital cortex when subjects observe a square image with modulated brightness. Then, we calculate the probability distribution of the SSVEF phase fluctuations and compute its kurtosis. The higher kurtosis, the better the phase synchronization. Since kurtosis characterizes the distribution’s sharpness, we associate inverse kurtosis with brain noise which broadens this distribution. We found that the majority of subjects exhibited leptokurtic kurtosis (K > 3) with tails approaching zero more slowly than Gaussian. The results of this work may be useful for the development of efficient and accurate brain-computer interfaces to be adapted to individual features of every subject in accordance with his/her brain noise.

Abstract: Publication date: March 2019Source: Chaos, Solitons & Fractals: X, Volume 1Author(s): Alexander P. Kartun-Giles, Ginestra BianconiAbstractIn Network Science, node neighbourhoods, also called ego-centered networks, have attracted significant attention. In particular the clustering coefficient has been extensively used to measure their local cohesiveness. In this paper, we show how, given two nodes with the same clustering coefficient, the topology of their neighbourhoods can be significantly different, which demonstrates the need to go beyond this simple characterization. We perform a large scale statistical analysis of the topology of node neighbourhoods of real networks by first constructing their clique complexes, and then computing their Betti numbers. We are able to show significant differences between the topology of node neighbourhoods of real networks and the stochastic topology of null models of random simplicial complexes revealing local organisation principles of the node neighbourhoods. Moreover we observe that a large scale statistical analysis of the topological properties of node neighbourhoods is able to clearly discriminate between power-law networks, and planar road networks.

Abstract: Publication date: March 2019Source: Chaos, Solitons & Fractals: X, Volume 1Author(s): Zoran Ivić, Nikos Lazarides, G.P. TsironisAbstractWe introduce a quantum superconducting metamaterial design constituted of flux qubits that operate as artificial atoms and analyze the dynamics of an injected electromagnetic pulse in the system. Qubit-photon interaction affects dramatically the nonlinear photon pulse propagation. We find analytically that the well known atomic phenomenon of self induced transparency may occur in this metamaterial as well and may lead to significant control over the optical pulse propagating properties. Specifically, the pulse may be slowed down substantially or even be stopped. These pulse properties depend crucially on the inhomogeneous broadening of the levels of the artificial atoms.

Abstract: Publication date: March 2019Source: Chaos, Solitons & Fractals: X, Volume 1Author(s): A. IominAbstractFractional time (non-unitary) quantum mechanics is discussed for both Schrödinger and Heisenberg representations of quantum mechanics. A correct form of the fractional Schrödinger equation is elucidated. A generic regularization procedure for the fractional Heisenberg equations of motion is suggested that makes it possible to solve these nonlinear equations in the framework of photonic and spin coherent states consideration.