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This article is a tribute to Oleksandr Mykolayovych Sharkovsky, a renowned mathematician and one of the founders of the mathematical concept of deterministic chaos. Sharkovsky's groundbreaking work, including his famous Sharkovsky theorem, has had a significant impact on the theory of dynamical systems. The article highlights Sharkovsky's early life, education, and career, as well as his contributions to the field of mathematics. It also mentions his achievements, such as publishing 10 monographs and over 250 research works, and receiving several prestigious awards. Sharkovsky's work continues to be influential in the study of nonlinear dynamics and chaos theory. [Extracted from the article]

Complex systems, as interwoven miscellaneous interacting entities that emerge and evolve through self-organization in a myriad of spiraling contexts, exhibit subtleties on global scale besides steering the way to understand complexity which has been under evolutionary processes with unfolding cumulative nature wherein order is viewed as the unifying framework. Indicating the striking feature of non-separability in components, a complex system cannot be understood in terms of the individual isolated constituents' properties per se, it can rather be comprehended as a way to multilevel approach systems behavior with systems whose emergent behavior and pattern transcend the characteristics of ubiquitous units composing the system itself. This observation specifies a change of scientific paradigm, presenting that a reductionist perspective does not by any means imply a constructionist view; and in that vein, complex systems science, associated with multiscale problems, is regarded as ascendancy of emergence over reductionism and level of mechanistic insight evolving into complex system. While evolvability being related to the species and humans owing their existence to their ancestors' capability with regards to adapting, emerging and evolving besides the relation between complexity of models, designs, visualization and optimality, a horizon that can take into account the subtleties making their own means of solutions applicable is to be entailed by complexity. Such views attach their germane importance to the future science of complexity which may probably be best regarded as a minimal history congruent with observable variations, namely the most parallelizable or symmetric process which can turn random inputs into regular outputs. Interestingly enough, chaos and nonlinear systems come into this picture as cousins of complexity which with tons of its components are involved in a hectic interaction with one another in a nonlinear fashion amongst the other related systems and fields. Relation, in mathematics, is a way of connecting two or more things, which is to say numbers, sets or other mathematical objects, and it is a relation that describes the way the things are interrelated to facilitate making sense of complex mathematical systems. Accordingly, mathematical modeling and scientific computing are proven principal tools toward the solution of problems arising in complex systems' exploration with sound, stimulating and innovative aspects attributed to data science as a tailored-made discipline to enable making sense out of voluminous (-big) data. Regarding the computation of the complexity of any mathematical model, conducting the analyses over the run time is related to the sort of data determined and employed along with the methods. This enables the possibility of examining the data applied in the study, which is dependent on the capacity of the computer at work. Besides these, varying capacities of the computers have impact on the results; nevertheless, the application of the method on the code step by step must be taken into consideration. In this sense, the definition of complexity evaluated over different data lends a broader applicability range with more realism and convenience since the process is dependent on concrete mathematical foundations. All of these indicate that the methods need to be investigated based on their mathematical foundation together with the methods. In that way, it can become foreseeable what level of complexity will emerge for any data desired to be employed. With relation to fractals, fractal theory and analysis are geared toward assessing the fractal characteristics of data, several methods being at stake to assign fractal dimensions to the datasets, and within that perspective, fractal analysis provides expansion of knowledge regarding the functions and structures of complex systems while acting as a potential means to evaluate the novel areas of research and to capture the roughness of objects, their nonlinearity, randomness, and so on. The idea of fractional-order integration and differentiation as well as the inverse relationship between them lends fractional calculus applications in various fields spanning across science, medicine and engineering, amongst the others. The approach of fractional calculus, within mathematics-informed frameworks employed to enable reliable comprehension into complex processes which encompass an array of temporal and spatial scales notably provides the novel applicable models through fractional-order calculus to optimization methods. Computational science and modeling, notwithstanding, are oriented toward the simulation and investigation of complex systems through the use of computers by making use of domains ranging from mathematics to physics as well as computer science. A computational model consisting of numerous variables that characterize the system under consideration allows the performing of many simulated experiments via computerized means. Furthermore, Artificial Intelligence (AI) techniques whether combined or not with fractal, fractional analysis as well as mathematical models have enabled various applications including the prediction of mechanisms ranging extensively from living organisms to other interactions across incredible spectra besides providing solutions to real-world complex problems both on local and global scale. While enabling model accuracy maximization, AI can also ensure the minimization of functions such as computational burden. Relatedly, level of complexity, often employed in computer science for decision-making and problem-solving processes, aims to evaluate the difficulty of algorithms, and by so doing, it helps to determine the number of required resources and time for task completion. Computational (-algorithmic) complexity, referring to the measure of the amount of computing resources (memory and storage) which a specific algorithm consumes when it is run, essentially signifies the complexity of an algorithm, yielding an approximate sense of the volume of computing resources and seeking to prove the input data with different values and sizes. Computational complexity, with search algorithms and solution landscapes, eventually points toward reductions vis à vis universality to explore varying degrees of problems with different ranges of predictability. Taken together, this line of sophisticated and computer-assisted proof approach can fulfill the requirements of accuracy, interpretability, predictability and reliance on mathematical sciences with the assistance of AI and machine learning being at the plinth of and at the intersection with different domains among many other related points in line with the concurrent technical analyses, computing processes, computational foundations and mathematical modeling. Consequently, as distinctive from the other ones, our special issue series provides a novel direction for stimulating, refreshing and innovative interdisciplinary, multidisciplinary and transdisciplinary understanding and research in model-based, data-driven modes to be able to obtain feasible accurate solutions, designed simulations, optimization processes, among many more. Hence, we address the theoretical reflections on how all these processes are modeled, merging all together the advanced methods, mathematical analyses, computational technologies, quantum means elaborating and exhibiting the implications of applicable approaches in real-world systems and other related domains. [ABSTRACT FROM AUTHOR]

• Traditional epidemiological models may struggle to accurately capture the chaotic nature of a pandemic/epidemic. • Chaos theory applied to epidemiology can capture and help to invistigate the non-linear dynamics underlying the progression of COVID-19 and other pandemics/epidemics. • Chaos theory can assist in analyzing complex interactions that can either exacerbate or mitigate the spread of a pathogen circulating in the human population. • Chaos theory might give information on the chances for an emerging virus to establish a long-lasting relationship with a new host. • Insights from chaos theory underscore the importance of flexibility and adaptability in response strategies. The chaos theory, a field of study in mathematics and physics, offers a unique lens through which to understand the dynamics of the COVID-19 pandemic. This theory, which deals with complex systems whose behavior is highly sensitive to initial conditions, can provide insights into the unpredictable and seemingly random nature of the pandemic's spread. In this review, we will discuss some literature data with the aim of showing how chaos theory could provide valuable perspectives in understanding the complex and dynamic nature of the COVID-19 pandemic. In particular, we will emphasize how the chaos theory can help in dissecting the unpredictable, non– linear progression of the disease, the importance of initial conditions, and the complex interactions between various factors influencing its spread. These insights are crucial for developing effective strategies to manage and mitigate the impact of the pandemic. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to expand our understanding of processes governing commercial property cycles, and to provide tools, which enable identification of property cycles' turning points' location. Design/methodology/approach: This paper is divided into three parts. The first looks at the demand-supply dynamics and the location of two characteristic cyclic points, the market bottom and the cycle commencement. In the second part a property relevant formula for entropy is derived, and its relation to the cycle overheated stage and the market peak is studied. In the third part, we discuss still another characteristic point of the cycle, which relates to the stage when developers elect to undertake new projects. This analysis is done by employing the chaos theory, and its relation to the cyclic evolution. Findings: It is found that some markets cycle, while others fluctuate only. A clear method for distinguishing among these is provided. The bottom of a cycle may overlap or be time separated from the start of a subsequent cycle. Market peaks are characterised by a sharp decrease in financial component to entropy for top quality building grades. A cycling market is characterised by crossing of a distinct vacancy rate during the cycle progression. Practical implications: The tools developed in the paper allow for clear characterisation of the market types and their cyclic behaviour. This in turn allows for timely characterisation of the market state and for short time-frame forecasting. The depth of a cycle may be calculated and the subsequent correction level estimated. Originality/value: The paper utilises cross-field approach by taking methods from both physics and mathematics and applying them to property markets. It breaks new ground both in property research and in applied mathematics by showing how the current frontier in pure mathematics may be applied to property. [ABSTRACT FROM AUTHOR]

Examines the classical, semiclassical and quantum mechanics of a chaotic Hamiltonian system. Adiabatic separation of the x and y motion; Connection between integrable approximation and the exact dynamics; Agreement between energy levels and energy spectrum.

Recently, there has been provided two chaotic models based on the twist-deformation of classical Henon-Heiles system. First of them has been constructed on the well-known, canonical space-time noncommutativity, while the second one on the Lie-algebraically type of quantum space, with two spatial directions commuting to classical time. In this article, we find the direct link between mentioned above systems, by synchronization both of them in the framework of active control method. Particularly, we derive at the canonical phase-space level the corresponding active controllers as well as we perform (as an example) the numerical synchronization of analyzed models. [ABSTRACT FROM AUTHOR]

In recent years, information security is an increasingly vital problem. In this paper, we introduce an audio encryption scheme based on the novel digital programmable multi-scroll chaotic system. The chaotic system is described by three differential equations with piecewise nonlinear functions. More interesting, the number of scrolls of the proposed chaotic system is programmable and can be changed real-time. The system is discretized by Euler method and the digital implementation is provided. Based on this, a chaos-based audio secure communication system is developed by using feedback drive-response synchronization. The number of scrolls changes randomly by extracting control signals from the encrypted data. A series of security analyses are applied, showing good performance of the method. The system is verified through experiment on an Altera Cyclone IV FPGA platform, which effectively confirms the theoretical analysis. [ABSTRACT FROM AUTHOR]

Equilibrium statistical mechanics allows the prediction of collective behaviors of large numbers of interacting objects from just a few system-wide properties; however, a similar theory does not exist for far-from-equilibrium systems exhibiting complex spatial and temporal behavior. We propose a method for predicting behaviors in a broad class of such systems and apply these ideas to an archetypal example, the spatiotemporal chaotic 1D complex Ginzburg-Landau equation in the defect chaos regime. Building on the ideas of Ruelle and of Cross and Hohenberg that a spatiotemporal chaotic system can be considered a collection of weakly interacting dynamical units of a characteristic size, the chaotic length scale, we identify underlying, mesoscale, chaotic units and effective interaction potentials between them. We find that the resulting equilibrium Takahashi model accurately predicts distributions of particle numbers. These results suggest the intriguing possibility that a class of far-from-equilibrium systems may be well described at coarse-grained scales by the wellestablished theory of equilibrium statistical mechanics. [ABSTRACT FROM AUTHOR]

Quantum entanglement in the Hénon-Heiles system is analyzed using the squeezed coherent state. Enhancement of quantum entanglement via squeezing is explored in connection with chaotic and regular dynamics of the system. It is found that the entanglement enhancement via squeezing is implicitly linked to the local structure of the classical phase-space and it shows a clear quantum-classical correspondence. In particular, the entanglement enhancement via squeezing is found to be negligible for a highly chaotic orbit compared to the regular and weakly chaotic orbits, and shows a clear correspondence to the degree of chaos present in the classical initial condition. We believe that these results might be useful to develop efficient strategies to enhance entanglement in quantum systems. [ABSTRACT FROM AUTHOR]

According to operator-sum representation theory, we have identified infinite-dimensional Kraus operators for describing a thermal channel with self-Kerr interaction after directly solving the corresponding master equation by virtue of thermo entangled state. Then we also prove in detail that Kraus operators hold the normalization. As an example, we exactly calculate the evolving result of a chaotic field in the thermal environment with the Kerr medium and find that the chaotic field evolves into a new chaotic field unaffected by the coupling factor with the Kerr medium. [ABSTRACT FROM AUTHOR]

Many methods have been proposed to synchronize chaotic systems. One of these methods is adaptive synchronization. Our purpose is the synchronization of two identical Chen systems using adaptive method and simulation of the results. The presented results show that these systems can synchronize with this method perfectly. [ABSTRACT FROM AUTHOR]

In this paper, we design a novel three-order autonomous system. Numerical simulations reveal the complex chaotic behaviors of the system. By applying the undetermined coefficient method, we find a heteroclinic orbit in the system. As a result, the Ši'lnikov criterion along with some other given conditions guarantees that the system has both Smale horseshoes and chaos of horseshoe type. [ABSTRACT FROM AUTHOR]

Abstract: The theory of chaos has emerged from a multidisciplinary encounter of mathematics, physics, and other sciences. The study of chaotic (microscopic) dynamics is now contributing in a major way to our understanding of nonequilibrium statistical mechanics. In particular one does not want to be restricted to situations close to equilibrium. We shall discuss some of the successes and challenges that have been met in this new development. [Copyright &y& Elsevier]

Projective synchronization in modulated time-delayed systems is studied by applying an active control method. Based on the Lyapunov asymptotical stability theorem, the controller and sufficient condition for projective synchronization are calculated analytically. We give a general method with which we can achieve projective synchronization in modulated time-delayed chaotic systems. This method allows us to adjust the desired scaling factor arbitrarily. The effectiveness of our method is confirmed by using the famous delay-differential equations related to optical bistable or hybrid optical bistable devices. Numerical simulations fully support the analytical approach. [ABSTRACT FROM AUTHOR]

The dynamics of indirectly coupled Lorenz circuits is investigated experimentally. The in-phase and anti-phase synchronization of indirectly coupled chaotic oscillators reported in Phys. Rev. E 81, 046216 (2010) is verified by physical experiments with electronic circuits. Two chaotic systems coupled through a common dynamic environment shows the verity of synchronization behaviours such as anti-phase synchronization, in-phase synchronization, identical synchronization, anti-synchronization, etc. [ABSTRACT FROM AUTHOR]

Abstract: The dynamics of fluid particles in the vicinity of a self-propagating vortex pair, embedded in a nonstationary shear flow, is studied. When the shear flow is steady, the vicinity of the pair, which is called as a vortex atmosphere, consists of closed stream-lines, which coincide with fluid particlesʼ trajectories. When the shear flow is nonstationary, the trajectoriesʼ behaviour changes drastically, then chaotic advection occurs. It is shown in the Letter that the vortex pair propagation velocity varies with the parameters (amplitude, and frequency) of the nonstationary shear flow. It is demonstrated, that changing of the mean velocity leads to changing of the size of the atmosphere. [Copyright &y& Elsevier]

With the abundance of chaotic systems that have now been identified and studied, it is prudent to establish a standard for the publication of new examples of such systems and to develop acceptable criteria for their characterization. [ABSTRACT FROM AUTHOR]

Abstract: The classical attractors of Lorenz, Rossler, Chua, Chen, and other widely-known attractors are those excited from unstable equilibria. From computational point of view this allows one to use numerical method, in which after transient process a trajectory, started from a point of unstable manifold in the neighborhood of equilibrium, reaches an attractor and identifies it. However there are attractors of another type: hidden attractors, a basin of attraction of which does not contain neighborhoods of equilibria. In the present Letter for localization of hidden attractors of Chuaʼs circuit it is suggested to use a special analytical–numerical algorithm. [Copyright &y& Elsevier]

This paper proposes a novel way to look at Chua's circuit and to investigate its chaotic behavior. We propose a systematic method to increase the dimension of Chua's circuit and to choose the element values. Our claims are confirmed by numerical simulations on Chua's circuit and on modified Chua's circuit and also supported by experimental results. [ABSTRACT FROM AUTHOR]

Abstract: We propose that the driven cold atomic system whose trap potential is periodically perturbed via parametric modulation of the magnetic field-gradien is a novel system to investigate the complex dynamics in nonlinear dynamical systems. We calculate the atomic trajectories and basins of attraction by varying the modulation amplitude and the modulation frequency. The calculation results show parametric resonance similar to those in Kim et. al.''s work [Opt. Commun. 236 (2004) 349] on cold atoms under parametric modulation of trap laser intensities in the case of small modulation amplitude or low modulation frequency. As the modulation amplitude or the modulation frequency is increased, the nonlinear effects are enhanced so that the dynamics of the system shows a wide variety of nonlinear behaviors, such as period doubling and chaos. We experimentally demonstrate the parametric resonance when the magnetic field gradient is parametrically modulated, which evidences the realization of the proposed system. We expect that this system is useful for understanding the stochastic phenomena occurring between complex basins of attraction, such as fluctuation induced transitions across the complex basin boundaries. [Copyright &y& Elsevier]

Abstract: We consider the role of fluid shear in maintaining anomalous “hotspot” solutions reported for a chaotically stirred bistable chemical reaction model [S.M. Cox, Persistent localized states for a chaotically mixed bistable reaction, Phys. Rev. E 74 (2006) 056206]. In the well-mixed regime, the chemical concentration is governed by a single autonomous ordinary differential equation with two stable equilibria. Whether the reaction goes to extinction or to an excited state depends on whether the initial concentration lies below or above some unstable threshold. By contrast, when the concentration varies spatially, and the chemical is stirred, the interplay between advection, diffusion and reaction is much more complicated, and the fate of the reaction depends sensitively on the initial conditions. It has previously been shown that, if the stirring is temporally periodic, a localised hotspot may form, and so the system never becomes fully extinct, nor fully excited. In this work, we first demonstrate that hotspots are in some sense generic, in that they are easily found by trial and error, by choosing reaction parameter values between those giving rise to global extinct and excited states. We also show that hotspots may be associated with hyperbolic, elliptic or even parabolic orbits associated with the underlying stirring. We observe that fluid shear is an important mechanism in localising a hotspot, and derive a reduced ordinary differential equation model which can predict the fate of a chemical stripe in a shear flow accurately. [Copyright &y& Elsevier]

Abstract: Combined breather–kink modes have been observed in several systems. Here we show that in the FPU lattice, small amplitude breathing–kink modes are three-soliton solutions of an associated mKdV equation. Since this is integrable, the modes can be constructed using the Bäcklund transform. As well as finding explicit solutions, we consider the stability of the combined mode using variational techniques and illustrate stability for some parameter values and instability for others. [Copyright &y& Elsevier]

A fundamental result in nonrelativistic quantum nonlinear dynamics is that the spectral statistics of quantum systems that possess no geometric symmetry, but whose classical dynamics are chaotic, are described by those of the Gaussian orthogonal ensemble (GOE) or the Gaussian unitary ensemble (GUE), in the presence or absence of time-reversal symmetry, respectively. For massless spin-half particles such as neutrinos in relativistic quantum mechanics in a chaotic billiard, the seminal work of Berry and Mondragon established the GUE nature of the level-spacing statistics, due to the combination of the chirality of Dirac particles and the confinement, which breaks the time-reversal symmetry. A question is whether the GOE or the GUE statistics can be observed in experimentally accessible, relativistic quantum systems. We demonstrate, using graphene confinements in which the quasiparticle motions are governed by the Dirac equation in the low-energy regime, that the level-spacing statistics are persistently those of GOE random matrices. We present extensive numerical evidence obtained from the tight-binding approach and a physical explanation for the GOE statistics. We also find that the presence of a weak magnetic field switches the statistics to those of GUE. For a strong magnetic field, Landau levels become influential, causing the level-spacing distribution to deviate markedly from the random-matrix predictions. Issues addressed also include the effects of a number of realistic factors on level-spacing statistics such as next nearest-neighbor interactions, different lattice orientations, enhanced hopping energy for atoms on the boundary, and staggered potential due to graphene-substrate interactions. [ABSTRACT FROM AUTHOR]

Abstract: The Kepler map was derived by as a tool for description of the long-term chaotic orbital behaviour of the comets in nearly parabolic motion. It is a two-dimensional area-preserving map, describing the motion of a comet in terms of energy and time. Its second equation is based on Kepler’s third law, hence the title of the map. Since 1980s the Kepler map has become paradigmatic in a number of applications in celestial mechanics and atomic physics. It represents an important kind of general separatrix maps. used refined methods of mathematical physics to derive analytical expressions for its single parameter. These methods became available only in the second half of the 20th century, and it may seem that the map is inherently a very modern mathematical tool. With the help of the Jacobi integral I show that the Kepler map, including analytical formulae for its parameter, can be derived by quite elementary methods. The prehistory and applications of the Kepler map are considered and discussed. [Copyright &y& Elsevier]

Abstract: We dissect Rössler equation and give a linear core, which can capture the essence of Rössler oscillator. The maximal transversal Lyapunov exponents calculated from the core are in good agreement with those obtained from Rössler equation. The idea is also tested for coupled Chua''s circuit systems. Numerical experiments are given. [Copyright &y& Elsevier]

Abstract: Connecting dynamical information of the nonlinear dynamical systems to the temporal evolution of complex networks, we analyze nonlinear dynamical systems with complex-network theory. We dynamically construct networks from attractors of nonlinear dynamical systems depending on their dynamical information and thereby we can evaluate the attractors through the evolution processes of constructed networks. As a result, we find that the constructed networks from periodic attractors correspond to the first-mover-advantage phase and those from chaotic attractors correspond to the fit-get-rich phase. We also show that the fit-get-rich property is originated to an essential mechanism for producing chaotic dynamics—stretching and folding. [Copyright &y& Elsevier]

The model of body–vortex interactions, where the fluid flow is planar, ideal and unbounded, and the vortex is a point vortex, is studied. The body may have a constant circulation around it. The governing equations for the general case of a freely moving body of arbitrary shape and mass density and an arbitrary number of point vortices are presented. The case of a body and a single vortex is then investigated numerically in detail. In this paper, the body is a homogeneous, elliptical cylinder. For large body–vortex separations, the system behaves much like a vortex pair regardless of body shape. The case of a circle is integrable. As the body is made slightly elliptic, a chaotic region grows from an unstable relative equilibrium of the circle-vortex case. The case of a cylindrical body of any shape moving in fluid otherwise at rest is also integrable. A second transition to chaos arises from the limit between rocking and tumbling motion of the body known in this case. In both instances, the chaos may be detected both in the body motion and in the vortex motion. The effect of increasing body mass at a fixed body shape is to damp the chaos. [ABSTRACT FROM AUTHOR]

Few reports have introduced chaotic attractors with both multiwing topological structure and hyperchaotic dynamics. A simple construction method, for designing chaotic system with multiwing attractors, is presented in this paper. The number of wings in the attractor was doubled on applying this method to an arbitrary smooth chaotic system. Moreover, the hyperchaotic property is preserved in the new system. A new hyperchaotic system with 16-wing attractors is constructed; the result system is not only verified via numerical simulation but also confirmed by a DSP-based experiment. [ABSTRACT FROM AUTHOR]

Abstract: Using the active control technique with Lyapunov stability theory and the Routh–Hurwitz criteria, control functions are designed to achieve complete synchronization between two identical Van der Pol oscillators (-VDPOs), two identical Duffing oscillators (-DOs), and two non-identical oscillators comprising -VDPO and -DO for the triple-well configuration of the potential. The coefficient matrix of the error dynamics between each pair of synchronized systems is chosen such that the number of active control functions reduces from two to one, thereby, significantly reducing controller complexity in the design. The designed controllers enable the state variables of the response system to synchronize with those of the drive system in both the identical and non-identical cases. The results are validated using numerical simulations. [Copyright &y& Elsevier]

Abstract: By employing a special feedback controlling scheme, a hyperchaotic Lorenz system with the structure of two time scales is constructed. Two kinds of bursting phenomena, symmetric fold/fold bursting and symmetric sub-Hopf/sub-Hopf bursting, can be observed in this system. Their respective dynamical behaviors are investigated by means of slow–fast analysis. In particular, symmetric fold/fold bursting is of focus–focus type, namely, both the up-state and the down-state are stable focus, which is different from the usual fold/fold bursting; Symmetric sub-Hopf/sub-Hopf bursting is also of focus–focus type, which has not been reported in previous work. Furthermore, phase plane analysis has been introduced to explore the evolution details of the fast subsystem for symmetric sub-Hopf/sub-Hopf bursting. With the variation of the parameter, symmetric sub-Hopf/sub-Hopf bursting can evolve to symmetric chaotic bursting or even hyperchaos. [Copyright &y& Elsevier]

Based on the three-dimensional Liu system with a nonlinear term of square, this paper appends a state variable to the system, and further adds a driving signal of the sine signal to construct a novel 4-demensional nonautonomous hyperchaotic Liu system. The appended variable is formed by the product of the nonlinear quadratic term of the original state variables and the driving signal. Through adjusting the frequency of the driving signal, the system can be controlled to show some different dynamic behaviors. By numerical simulations, the Lyapunov exponent spectrums, bifurcation diagrams and phase diagrams of the novel systems are analyzed. Furthermore, the corresponding hardware circuits are implemented. Both the experimental results and the simulation results confirm that the method is feasible. The method, which adjusts the frequency of the input sine signal to control the system to show different dynamic behaviors, can make the dynamic property of the system become more complex, but easier to be controlled accurately as well. [ABSTRACT FROM AUTHOR]

This paper studies how phase synchronization in complex networks depends on random shortcuts, using the piecewise-continuous chaotic Chua system as the nodes of the networks. It is found that for a given coupling strength, when the number of random shortcuts is greater than a threshold the phase synchronization is induced. Phase synchronization becomes evident and reaches its maximum as the number of random shortcuts is further increased. These phenomena imply that random shortcuts can induce and enhance the phase synchronization in complex Chua systems. Furthermore, the paper also investigates the effects of the coupling strength and it is found that stronger coupling makes it easier to obtain the complete phase synchronization. [ABSTRACT FROM AUTHOR]

Abstract: Recently, a cryptosystem based on two-dimensional discretized chaotic maps was proposed [T. Xiang, et al., Phys. Lett. A 364 (2007) 252]. In this Letter, we cryptanalyze the proposal using algebraic methods. We give three different attacks that yield all the secret parameters of the cryptosystem. [Copyright &y& Elsevier]

Abstract: We propose a direct and flexible implementation of logic operations using the dynamical evolution of a nonlinear system. The concept involves the observation of the state of the system at different times to obtain different logic outputs. We explicitly implement the basic NAND, AND, NOR, OR and XOR logic gates, as well as multiple-input XOR and XNOR logic gates. Further we demonstrate how the single dynamical system can do more complex operations such as bit-by-bit addition in just a few iterations. [Copyright &y& Elsevier]

Kicked Harper operators and on-resonance double kicked rotor operators model quantum systems whose semiclassical limits exhibit chaotic dynamics. Recent computational studies indicate a striking resemblance between the spectra of these operators. In this paper we apply C*-algebra methods to explain this resemblance. We show that each pair of corresponding operators belongs to a common rotation C*-algebra Bα, prove that their spectra are equal if α is irrational, and prove that the Hausdorff distance between their spectra converges to zero as q increases if α=p/q with p and q coprime integers. Moreover, we show that corresponding operators in Bα are homomorphic images of mother operators in the universal rotation C*-algebra Aα that are unitarily equivalent and hence have identical spectra. These results extend analogous results for almost Mathieu operators. We also utilize the C*-algebraic framework to develop efficient algorithms to compute the spectra of these mother operators for rational α and present preliminary numerical results that support the conjecture that their spectra are Cantor sets if α is irrational. This conjecture for almost Mathieu operators, called the ten Martini problem, was recently proven after intensive efforts over several decades. This proof for the almost Mathieu operators utilized transfer matrix methods, which do not exist for the kicked operators. We outline a strategy, based on a special property of loop groups of semisimple Lie groups, to prove this conjecture for the kicked operators. [ABSTRACT FROM AUTHOR]

We study the effect of static additive noise on the sample entropy (SampEn) algorithm [J. S. Richman and J. R. Moorman, Am. J. Physiol. Heart Circ. Physiol. 278, 2039 (2000); R. B. Govindan et al., Physica A 376, 158 (2007)] for analyzing time series. Using surrogate data tests, we empirically investigate the ability of the SampEn index to detect nonlinearity in simulated time series corrupted by increased amounts of noise. Discrete and continuous chaotic and nonchaotic systems are included in the numerical experiments. Both Gaussian and uniformly distributed noises are considered. The results indicate that the SampEn statistic is a robust index for detecting nonlinearity in time series corrupted by observational noise. [ABSTRACT FROM AUTHOR]

Abstract: The deterministic randomness not only can become a dominant approach in exploring the relationship between chaos and randomness, but also can be associated with some famous number theoretical concepts and open problems in number theory. Compared with chaotic sequences, asymptotic deterministic randomness sequences have the characteristic of multi-value correspondence, which makes those sequences unpredictable in short steps. In this Letter, we will propose the definition of the discrete asymptotic deterministic randomness, and then analyze the dynamical characteristics such as maximum-period and multi-value correspondence. Referring to the NIST800-22 statistical test suite, we will present and discuss two examples of PRBGs based on DADR, from the point of view of FPGA design and randomness quality. [Copyright &y& Elsevier]

In this paper we use a new scientific field called "DemoscopoPhysics" based on chaos theory to predict the Hellenic National election results in the form of time series for Hellenic political parties, New Democracy (ND), Panhellenic Social-istic Movement (PASOK), Hellenic Communistic Party (KKE), Coalition of the Radical Left (SYRIZA) and (Popular Orthodox Rally) LAOS". Using the results of our previous article we reconstruct of the corresponding strange attractor for each political party achieved up to a 30 time steps out of sample prediction of the public survey. Also reducing the degrees of freedom to 4 we have capture the polarization of voters. [ABSTRACT FROM AUTHOR]

An essay is presented on classical mechanics (CM) is presented. It explores the sensitive dependence on initial conditions (SDIC) in nonlinear models and the notions of SDIC and the chaotic behavior in system. It discusses the Lyapunov exponents and the relationship between nonlinear dynamics and chaos. It also examines the distinct relationship between quantum mechanics (QM) and CM.

Based on two modified Rösslor hyperchaotic systems, which are derived from the chaotic Rösslor system by introducing a state feedback controller, this paper proposes a new switched Rösslor hyperchaotic system. The switched system contains two different hyperchaotic systems and can change its behaviour continuously from one to another via a switching function. On the other hand, it presents a systematic method for designing the circuit of realizing the proposed hyperchaotic system. In this design, circuit state equations are written in normalized dimensionless form by rescaling the time variable. Furthermore, an analogous circuit is designed by using the proposed method and built for verifying the new hyperchaos and the design method. Experimental results show a good agreement between numerical simulations and experimental results. [ABSTRACT FROM AUTHOR]