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The study of damage spreading in cellular automata (CA) is essential for understanding chaos and phase transitions in CA and complex systems in general. It helps us make sense of the dynamics and emergent properties of complex systems and informs various practical applications in science and engineering. In this study, we present a novel and comprehensive perspective on damage spreading in CA and Boolean networks. We introduce a novel concept, the tangent space of a CA, enabling us to introduce a methodology for computing the Lyapunov spectrum of both CA and Boolean networks. This approach mirrors the well-established method employed in continuous-state dynamical systems, hence facilitating the application of established theorems from dynamical systems theory. Additionally, our approach reveals how the existing notions related to damage spreading and Lyapunov exponents of CA are related to their configuration space and tangent space, thereby bridging seemingly unrelated approaches. We illustrate the versatility of our approach through the analysis of Lyapunov spectra for Elementary CA (ECA) and the Domany–Kinzel CA, showcasing its effectiveness in scenarios involving probabilistic update rules and network structures. Finally, we explore the relationship between the Lyapunov spectrum and the Kolmogorov–Sinai entropy, affirming our approach by checking the validity of Pesin's theorem. This work contributes to the comprehensive understanding of damage spreading in CA and Boolean networks, unraveling connections between chaos theory, computational systems, and informing various real-world applications across scientific and engineering domains. • We introduce a method to compute the Lyapunov spectrum of cellular automata (CA). • Our method extends to CA on a network and with probabilistic update rules. • With our approach, we connect existing notions related to damage spreading in CA. • Lyapunov spectra of elementary CA and the Domany–Kinzel CA on a network are analyzed. [ABSTRACT FROM AUTHOR]

A new problem at the intersection of multi-agent systems, chaotic time series prediction, and flow map learning is formulated in this paper. The problem involves agents collaborating to track moving targets in chaotic dynamic systems by communicating. Inspired by the multi-scale hierarchical time-stepper (HiTS), a novel Distributed Prediction Network based on Multi-scale Attention (DPNMA) is proposed to fuse predictions from agents at different scales through an enhanced self-attention mechanism. The experimental evaluation demonstrates that DPNMA effectively mitigates cumulative errors and enhances the accuracy and robustness of the predictions, which has important implications for the scenarios where the agents have heterogeneous and constrained capabilities. [ABSTRACT FROM AUTHOR]

The exploration of the nonlinear dynamics related to the new coupled Konno–Oono equation, which determines the propagation of magnetic fields, is the focus of this work. Through the employing of Lie group analysis, the bifurcation phase portraits, and chaos theory, the project will investigate symmetry reductions in dynamical systems and examine the dynamic behavior of perturbed dynamical systems. The 3D phase portrait, 2D phase portrait, Lyapunov exponent, time series analysis, sensitivity analysis, and an examination of the existence of multistability in the autonomous system under various initial conditions constitute a few of the methods used for recognizing chaotic behavior. Furthermore, the investigation constructs general solutions for solitary wave solutions, such as exponential and hyperbolic function, singular, dark, and bright soliton solutions, by using the new Kudryashov methodology to determine the investigated equation analytically. These solutions are shown graphically as 2D, 3D, and contour plots with specifically selected values. They include as well with the related constraint circumstances. Additionally, a discussion and a visual illustration of the considered equation's sensitivity analysis are presented. The observations demonstrate that the aforementioned approach is an effective procedure for treating a variety of nonlinear PDE systems that arise in nonlinear physics analytically. The plot of the Lyapunov exponents is employed to validate the chaotic dynamics of the studied model. Additionally, the multiplier method is employed to determine the conserved vectors for the analyzed problem. • Coupled Konno–Oono equation within the domain of magnetic fields is considered. • The Lie invariance condition is applied to identify symmetry generators. • Soliton solutions are derived and phase portrait analysis has been done. • Chaotic behaviors are examined with different characteristic settings. • Conserved vectors for the studied equation are calculated. [ABSTRACT FROM AUTHOR]

This paper is subjected to study the nonlinear integrable model which is the (3+1)-dimensional Boussinesq equation which has a lot of applications in engineering and modern sciences. To find and examine the analytical exact solitary wave solutions of (3+1)-dimensional Boussinesq equation, a modified generalized exponential rational functional method is exerted. As a result, waves, singular periodic, hyperbolic, and trigonometric type solutions are obtained. These acquired solutions are more innovative and encouraging to researchers in their endeavor to study physical marvels. To illustrate how some selected exact solutions propagate, the graphical representation in 2D, Contour, and 3D of those solutions is provided with various parametric values. The considered equation is additionally transformed into the planar dynamical structure by applying the Galilean transformation. All potential phase portraits of the dynamical system are investigated using the theory of bifurcation. The Hamiltonian function of the dynamical system of differential equations is established to see that, the system is conservative over time. The presentation of energy levels through graphics provides valuable insights, and it demonstrates that the model has solutions that can be expressed in closed form. The periodic, quasi-periodic, and chaotic behaviors of the 2D, 3D, and time series are also observable once the dynamical system is subjected to an external force. Meanwhile, the sensitivity of the derived solutions is carefully examined for a range of initial conditions. [ABSTRACT FROM AUTHOR]

The objective of this manuscript is to examine the non-linear characteristics of the modified equal width-Burgers equation, known as the generalized Kadomtsive–Petviashvili equation, and its ability to generate a long-wave with dispersion and dissipation in a nonlinear medium. We employ the Lie symmetry approach to reduce the dimension of the equation, resulting in an ordinary differential equation. Utilizing the newly developed generalized logistic equation method, we are able to derive solitary wave solutions for the aforementioned ordinary differential equation. In order to gain a deeper understanding of the physical implications of these solutions, we present them using various visual representations, such as 3D, 2D, density, and polar plots. Following this, we conduct a qualitative analysis of the dynamical systems and explore their chaotic behavior using bifurcation and chaos theory. To identify chaos within the systems, we utilize various chaos detection tools available in the existing literature. The results obtained from this study are novel and valuable for further investigation of the equation, providing guidance for future researchers. • Application of the generalized logistic equation method to address the dynamics of solitary wave solutions for the generalized KP–MEW-Burgers equation. • Computation of the invariant transformations and symmetry reductions using Lie symmetry analysis. • Bifurcation and phase portraits of the unperturbed dynamical system using the idea of bifurcation theory of dynamical systems. • Chaos behavior of the perturbed dynamical system is identified through different chaos detecting tools using chaos theory of dynamical systems. [ABSTRACT FROM AUTHOR]

A Duffing-Holmes system containing nonlinear damping is used to investigate some dynamical properties of the system under the combined action of constant excitation and simple harmonic excitation. The harmonic balance method is used to find the main vibration equation of the system and obtain the amplitude-frequency response relationship. In the analysis of the global characteristics of the system, the Melnikov method is applied to analyze the necessary, insufficient conditions for the analysis of the system chaos, and the correctness of the analytical solution is verified by numerical calculations; the effect of the constant excitation amplitude on the global characteristics of the system is analyzed by the cell mapping method. The paper shows: gradually increasing the constant excitation amplitude causes the system amplitude-frequency response to appear rigidly asymptotic phenomenon; the boundary curve of the system generating chaos in the sense of Smale's horseshoe, with the increase of excitation frequency ω, the curve first decreases, then rises, finally tends to infinity, and the possibility of chaos occurs most near the resonance frequency of the system; with the change of the constant excitation amplitude, The number of attractor domains and attractors in the global attraction domain of the system changes substantially with the change of the constant excitation amplitude, and the attractor domains are entangled with each other. • The harmonic balance method is used to solve the main resonance equation of the system, and the amplitude-frequencyresponse relationship is obtained. • In the analysis of the global characteristics of the system, the necessary conditions for the chaos of the system are studied by Melnikov method, and the stability of the system is studied by cell mapping method. [ABSTRACT FROM AUTHOR]

In this paper, the modified Kadomtsev Petviashvili (mKP) equation is derived by considering the dynamics of the dust-acoustic waves (DAWs) with kappa-distributed hot and cold ions and Boltzmannian electrons. The reductive perturbation technique is applied for deriving mKP. The bifurcation theory of the planar dynamical system is used to obtain the phase portrait of the DAWs in the framework of mKP equation. The formation of dust-acoustic solitary waves (DASWs) is analyzed for different plasma parameters in the Saturn's magnetosphere. In the current plasma model, the external periodic force is introduced to study the quasiperiodic and chaotic behavior of the DAWs. It is noted that periodic initial conditions lead to the emergence of many outlandish features in the system by comparison with the solitary initial conditions. The frequency of the external periodic force ω plays an important role in the transition from quasiperiodic to chaotic behavior. Moreover, the strength of the external periodic force, the value of the nonlinear coefficient of the mKP equation, and the nonlinear acoustic speed are also found to have a significant effect on the chaotic behavior of the system. • Dynamical system and phase portrait analysis of the mKP Equation. • The bifurcation theory is applied to obtain the phase portrait of the dust-acoustic waves (DAWs). • Dust-acoustic solitons are investigated for Saturn's magnetosphere plasma parameters. • External periodic force (EPF) is introduced for studying the chaotic behavior of DAWs. • The EPF frequency has a great role in the transition from quasiperiodic to chaotic behavior. [ABSTRACT FROM AUTHOR]

Highlights • A new parameter to describe chaotic motion identification (NPSI) was defined. • Probability density function to supplement information of the Poincaré cross-section. • Dynamics of a quasi-zero stiffness vibration isolator was investigated. • Bifurcations leading to chaotic vibrations were identified. • The evolution of a geometrical structure of strange attractors was characterized. Abstract The aim of this work is to evaluate the dynamics of the quasi-zero stiffness vibration isolator, with particular emphasis on the density distribution of Poincaré cross-section points for different conditions of the system's excitation. The research was mainly focused on model tests carried out to examine the structures representing the geometry of strange attractors of a quasi-zero stiffness vibration isolator. The analysed mechanical system consists of one main spring and two compensation springs. Energy losses are caused by friction at the connection points of the compensation springs and vibroisolated mass. On the basis of a formulated non-linear mathematical model, the ranges of variation of physical parameters of an external dynamic input for which the system motion is chaotic are identified. Taking into account three selected values of the dynamic input amplitude, a bifurcation diagram and a diagram defining the number of phase stream intersections (NPSI) with the abscissa of the phase plane are generated. Based on the diagrams, a solution is selected. The diagram proposed by us (NPSI) is an alternative method of identifying areas in which the system moves chaotically. The classic Poincaré cross-section combined by us with information about the density of point distribution on the trajectory intersection with the control plane serve as a basis for the assessment of evolution of the geometric structure of strange attractors. The evaluation of the density distribution of Poincaré cross-section points provides important information regarding the evolution of geometrical structures of strange attractors. It has been shown that in relation to large ranges of changes in the control parameter, the geometric structure of the strange attractor is stretched and curved. However, in the area of small changes in the control parameter, the evolution of the attractor' geometric structure can only be observed by analysing the density of point distribution on the Poincaré cross-section by the probability density function (PDF). The areas with the highest density of the Poincaré cross-section are usually located in places where the strange attractor is curved. Taking into account the practical application of research results, operational guidelines are formulated. [ABSTRACT FROM AUTHOR]

Highlights • The new system has a hyperbolic cosine nonlinearity which is physically implemented by two semiconductor diodes instead of using multipliers. • The system presents unusual and very rare behaviors such as quasi-periodic behaviors. • Hidden 2-torus and 3-torus (quasi-periodic behaviors) can coexist in the system. • System is asymmetrical throughout the phase space, a rare and unique property causing complex phenomena. Abstract Recently, the study of systems with hidden attractors has become one of the most followed topics owing to their fundamental and technological importance. This contribution is focused on a new simple 4-D chaotic system (whose nonlinearity is a hyperbolic function) inspired by the quadratic system introduced by [Jay and Roy Nonlinear Dyn (2017) 89:1845–1862]. Basic properties of the new system are discussed and its complex behaviors are characterized using dynamic systems analysis tools. This system exhibits a rich repertoire of dynamic behaviors including chaos, chaos 2-torus, and quasi-periodic. Other interesting phenomena such as multistability, antimonotonicity, and torus-doubling bifurcations are also reported. Moreover, the hyperbolic cosine nonlinearity is easily implemented by using a pair of semiconductor diodes (no analog multiplier is involved). We confirm the feasibility of the proposed theoretical model using PSpice simulations and a physical realization based on an electronic analog implementation of the model. [ABSTRACT FROM AUTHOR]

Highlights • This article considers the interplay among four financial factors. • We use integer order and fractional order differential equation systems to model a financial system. • Both mathematical analyses and numerical simulations are carried out to illustrate the characteristic of the model. • The model displays a variety of rich dynamic behaviors including chaos over a wide range of system parameters. Abstract In this paper, we use integer order and fractional order differential equation systems to model a financial system. Based on the interaction among several financial factors, a model is constructed. Both mathematical analyses and numerical simulations are carried out to illustrate the characteristic of the model. We find that the system displays a variety of rich dynamic behaviours including chaos over a wide range of system parameters. Our investigation indicates that the interplay among several financial factors lead to chaos under some circumstances. We then design control laws to synchronization two integer order financial systems and two fractional order financial systems. Numerical simulations are presented to verify the effectiveness of the designed control laws. [ABSTRACT FROM AUTHOR]

Abstract This paper presents a practical implementation of a memristor based Duffing oscillator. We replace a diode based nonlinear element by a simple flux-controlled memristor in an existing circuit early proposed.We model the current-voltage characteristic and show experimentally that our circuit presents, for the same forcing amplitude, a route to chaos on the falling branch of the hysteresis loop, while its dynamics remains regular on the raising branch. This observation clearly highlights the memory effect of the memristor. [ABSTRACT FROM AUTHOR]

Mineral extraction activities and the commercialisation of mineral-based products in later stages are fundamental for economic, technological and social development, so understanding their long-term market behaviour in order to forecast prices is crucial for governments, companies and society. Mineral commodity markets are multidimensional as they are driven by the dynamic evolution of financial, technological, psychological and geopolitical factors. Thus, identifying the key features governing mineral commodity prices and modelling their long-term behaviour is an intricate problem that becomes even more complex when the relationship with time is also considered. Although several techniques are available to represent the behaviour and reduce the dimensionality of data sets, these techniques often neglect the temporal relationship and evolution of variables. In many cases, the selection of key variables is solely based on the correlation with each other, relevance and degree of variance, and the cause and effect behaviour of the time-relation of variables, is neglected. This is a critical issue in assessing dynamic systems, especially those involving human behaviour such as social sciences and biology where learning and cognition capacities evolve through time. In this sense, before any attempt to forecast mineral commodity prices, a proper understanding of the embedding dimension and time delay of the variables governing the system is fundamental to determining the number of key features driving the system and the extension of the delayed effects of changes in the initial conditions, respectively. Determining these parameters may become even more complicated because of data scarcity, since the high cost, technical or temporal limitations often hamper obtaining large data sets related to complex dynamic systems that occur in nature and the social sciences. Daily and monthly quotations have been commonly used to represent the dynamics of mineral commodity markets and predict prices, yet they are not sufficient to capture and understand the dynamic evolution of prices in the long-term. These quotations are too short as the delayed effects that reflect on the entire economy and arises from changes in macroeconomic variables can normally be perceived about a year ahead, and may last for up to six years. However, chaos theory has provided valuable tools to assess the dynamics of complex systems allowing the proper determination of the embedding parameters (dimension and time delay), even when using small data sets. This paper examines the long-term dynamics of mineral commodity prices via chaos theory based on a short data set of annual copper prices observed between 1900 and 2015. Copper was used as a study case because it is one of the most competitive markets and representative of mineral commodities traded worldwide. It was found that the dynamic-deterministic behaviour of annual copper prices is embedded in a high-dimensional space that fluctuates between five and seven, and a relatively short time delay between two and three periods. Our finding argues for the use of chaos theory as an important technique to assess the long-term behaviour of mineral commodity prices using short data sets because it improves the understanding of those complex dynamic systems and provides important guidelines that remarkably simplifies the forecasting task. [ABSTRACT FROM AUTHOR]

In this paper we consider Hausdorff dimension of the sets of Li-Yorke pairs for some chaotic dynamical systems including A -coupled expanding systems. We prove that Li-Yorke pairs of A -coupled-expanding system under some conditions have full Hausdorff dimension in the invariant set. Moreover we give a generalization of the result of [5] which is on the Hausdorff dimension of Li-Yorke pairs of dynamical systems topologically conjugate to the full shift and have a self-similar invariant set, to the case of the dynamical systems topologically semi-conjugate to some kinds of subshifts. Furthermore Hausdorff dimension of “chaotic invariant set” for some kinds of A-coupled-expanding maps is shown. [ABSTRACT FROM AUTHOR]

In this paper, a novel fourth-order autonomous hyperjerk circuit is proposed and the corresponding dynamics is systematically analyzed. Two anti-parallel semiconductor diodes form the nonlinear component necessary for chaotic oscillations. The mathematical model of the novel circuit consists of a fourth-order (“elegant”) autonomous hyperjerk system with (a single) hyperbolic sine nonlinearity. The fundamental dynamic properties of the model are investigated including fixed points and stability, phase portraits, bifurcation diagrams, and Lyapunov exponent plots. Period-doubling bifurcation, periodic windows, coexisting bifurcations, symmetry recovering crises, and antimonotonicity (i.e. concurrent creation and annihilation of periodic orbit) are reported when monitoring the systems parameters. One of the main findings in this work is the presence of various windows in the parameter space in which the novel 4D-hyperjerk system develops the interesting property of multiple coexisting attractors (e.g. coexistence of two, three, four, five, six, seven height or nine disconnected periodic and chaotic attractors). To the best of the authors’ knowledge, this striking phenomenon is unique and has not yet been reported previously in a hyperjerk circuit, and thus represents a significant contribution to the understanding of the behavior of nonlinear dynamical systems in general. Laboratory experiments of the oscillator are carried out to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

In ecological modeling, seasonality can be represented as an alternation between environmental conditions. This concept of alternation holds common ground between ecologists and chemists, who design time-dependent settings for chemical reactors to influence the yield of a desired product. In this study and for a variety of maps, we consider a switching strategy that alternates between two undesirable dynamics that yields a stable desirable dynamic behavior. By comparing bifurcation diagrams of a map and its alternate version, we can easily find parameter values, which, on their own, yield chaotic orbits. When alternated, however, the parameter values yield a stable periodic orbit. Our analysis of the two-dimensional (2-D) maps is an extension of our previous work with one-dimensional (1-D) maps. In the case of 2-D maps, we consider the Beddington, Free, and Lawton and Udwadia and Raju maps. For these 2-D maps, we not only show that we can find “chaotic” parameters for the so-called “chaos” + “chaos” = “periodic” case, but we find two new “desirable” dynamic situations: “quasiperiodic” + “quasiperiodic” = “periodic” and “chaos” + “chaos” = “periodic coexistence.” In the former case, the alternation of chaotic dynamics yield two different periodic stable orbits implying the coexistence of attractors. [ABSTRACT FROM AUTHOR]

We present a calculus of the Kolmogorov–Sinai entropy for quantum systems having a mixing quantum phase space. The method for this estimation is based on the following ingredients: i) the graininess of quantum phase space in virtue of the Uncertainty Principle, ii) a time rescaled KS–entropy that introduces the characteristic time scale as a parameter, and iii) a mixing condition at the (finite) characteristic time scale. The analogy between the structures of the mixing level of the ergodic hierarchy and of its quantum counterpart is shown. Moreover, the logarithmic time scale, characteristic of quantum chaotic systems, is obtained. [ABSTRACT FROM AUTHOR]

In this work, we present another chaotic (or hyperchaotic) complex nonlinear framework. This chaotic (or hyperchaotic) complex framework can be considered as a speculation of ”Guan” framework [1]. The new framework is a 5-dimensional nonstop real independent chaotic (or hyperchaotic) framework. The fundamental properties and elements of our framework are examined. Once with the all parameters are real and the other with one of these parameters is a complex parameter. When we examine the dynamics of the new framework and the parameters in real form, the conduct of the new framework is chaotic. While when one of these parameters is complex, our new framework is hyperchaotic. On the premise of Lyapunov capacity and dynamic control method, a scheme is designed to accomplish the complex anti-synchronization of two identical chaotic (or hyperchaotic) attractors of these frameworks. The effectiveness of the acquired outcomes will be delineated by a simulation case. Numerical outcomes are schemed to indicate state variables and errors of these chaotic attractors after synchronization to show that synchronization is accomplished. The above outcomes will give hypothetical establishment to the secure communication applications in light of the proposed scheme. In this secure communication scheme, synchronization between transmitter and receiver is accomplished and message signals will be recuperated. The encryption and rebuilding of the signals will be simulated numerically. [ABSTRACT FROM AUTHOR]

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 FROM AUTHOR]

In the context of chaotic dynamical systems with exponential divergence of nearby trajectories in phase space, hyperchaos is defined as a state where there is divergence or stretching in at least two directions during the evolution of the system. Hence the detection and characterization of a hyperchaotic attractor is usually done using the spectrum of Lyapunov Exponents (LEs) that measure this rate of divergence along each direction. Though hyperchaos arise in different dynamical situations and find several practical applications, a proper understanding of the geometric structure of a hyperchaotic attractor still remains an unsolved problem. In this paper, we present strong numerical evidence to suggest that the geometric structure of a hyperchaotic attractor can be characterized using a multifractal spectrum with two superimposed components. In other words, apart from developing an extra positive LE, there is also a structural change as a chaotic attractor makes a transition to the hyperchaotic phase and the attractor changes from a simple multifractal to a dual multifractal, equivalent to two inter-mingled multifractals. We argue that a cross-over behavior in the scaling region for computing the correlation dimension is a manifestation of such a structure. In order to support this claim, we present an illustrative example of a synthetically generated set of points in the unit interval (a Cantor set with a variable iteration scheme) displaying dual multifractal spectrum. Our results are also used to develop a general scheme to generate both hyperchaotic as well as high dimensional chaotic attractors by coupling two low dimensional chaotic attractors and tuning a time scale parameter. [ABSTRACT FROM AUTHOR]

Chaotic Set/Reset (RS) flip-flop circuits are investigated once again in the context of discrete planar dynamical system models of the threshold voltages, but this time starting with simple bilinear (minimal) component models derived from first principles. The dynamics of the minimal model is described in detail, and shown to exhibit some of the expected properties, but not the chaotic regimes typically found in simulations of physical realizations of chaotic flip-flop circuits. Any electronic physical realization of a chaotic logical circuit must necessarily involve small perturbations from the ideal - usually with large or even nonexistent derivatives in small diameter subsets of the phase space. Therefore, perturbed forms of the minimal model are also analyzed in considerable detail. It is proved that very slightly perturbed minimal models can exhibit chaotic regimes, sometimes associated with chaotic strange attractors, as well as some of the bifurcations present in most of the differential equations models for similar physical circuit realizations. In essence, this work is a mathematical exploration of simple models that reproduce the qualitative behavior of threshold control units of a chaotic RS flip-flop design. It is also shown that this method can be extended to other similar circuits. Validation of the approach developed is provided by some comparisons with (mainly simulated) dynamical results obtained from more traditional investigations. [ABSTRACT FROM AUTHOR]

In a seminal paper, Grigorenko and Grigorenko [15], numerically studied fractional order dynamical systems (FODS) of the form D α i x i = f i ( x 1 , x 2 , x 3 ) , 0 ≤ α i ≤ 1 , ( i = 1 , 2 , 3 ) ; and showed the existence of chaos in case of fractional Lorenz system when Σ = α 1 + α 2 + α 3 ≤ 3 . Since then voluminous numerical work has been done to explore various FODS, in this regard. It is now an established fact that Σ acts as a chaos controlling parameter. In the present article we take a survey of present literature on chaotic behavior of fractional order dynamical systems. Further we numerically explore fractional Chen, Rössler, Bhalekar–Gejji, Lorenz and Liu systems and observe that chaos always disappear if Σ ≤ 2. The existing examples in the literature along with the systems that we have analyzed lead us to conjecture non-existence of chaos if Σ ≤ 2; which in some sense is a generalization of classical Poincaré–Bendixon theorem for FODS. [ABSTRACT FROM AUTHOR]

We study the interaction of some dynamical properties of a nonautonomous discrete dynamical system ( X, f ∞ ) and its induced nonautonomous discrete dynamical system ( K ( X ) , f ∞ ¯ ) , where K ( X ) is the hyperspace of non-empty compact sets in X , endowed with the Vietoris topology. We consider properties like transitivity, weakly mixing, points with dense orbit, density of periodic points, among others. We also present examples of nonautonomous discrete dynamical systems showing that transitivity, density of periodic points and sensitive dependence on initial conditions are independents on the unit interval, i.e., unlike autonomous discrete dynamical systems, in definition of Devaney chaotic there are not redundant conditions for NDS on the interval. Actually, our examples give an even more precise conclusion: the classical result stating that transitivity is a sufficient condition for an autonomous discrete dynamical system on the interval to be Devaney chaotic fails to be true for nonautonomous dynamical systems. [ABSTRACT FROM AUTHOR]

While local models of dynamical systems have been highly successful in terms of using extensive data sets observing even a chaotic dynamical system to produce useful forecasts, there is a typical problem as follows. Specifically, with k -near neighbors, kNN method, local observations occur due to recurrences in a chaotic system, and this allows for local models to be built by regression to low dimensional polynomial approximations of the underlying system estimating a Taylor series. This has been a popular approach, particularly in context of scalar data observations which have been represented by time-delay embedding methods. However such local models can generally allow for spatial discontinuities of forecasts when considered globally, meaning jumps in predictions because the collected near neighbors vary from point to point. The source of these discontinuities is generally that the set of near neighbors varies discontinuously with respect to the position of the sample point, and so therefore does the model built from the near neighbors. It is possible to utilize local information inferred from near neighbors as usual but at the same time to impose a degree of regularity on a global scale. We present here a new global perspective extending the general local modeling concept. In so doing, then we proceed to show how this perspective allows us to impose prior presumed regularity into the model, by involving the Tikhonov regularity theory, since this classic perspective of optimization in ill-posed problems naturally balances fitting an objective with some prior assumed form of the result, such as continuity or derivative regularity for example. This all reduces to matrix manipulations which we demonstrate on a simple data set, with the implication that it may find much broader context. [ABSTRACT FROM AUTHOR]

The trajectory of a chaotic dynamical system is sensitive to the initial condition, while its statistics normally do not have sensitive dependence on small disturbances. Such a kind of chaos is called normal-chaos. Recently, a new concept named ultra-chaos has been proposed for the first time. Unlike normal-chaos, the small disturbances lead to large deviation even in statistics for ultra-chaos. However, few ultra-chaos have been reported up to now. In this work, we investigate the influences of small disturbances on the chaotic navigation system and the motion of mobile robot, respectively. It is illustrated that the chaotic navigation system of the mobile robot belongs to normal-chaos. However, the motion of chaotic mobile robot belongs to ultra-chaos. Tiny disturbances would cause huge deviation in the motion of chaotic mobile robot, even in the statistical meaning. Thus, the ultra-chaos is indeed a higher-level disorder compared with the normal-chaos and it widely exists in practice. Hopefully, the new concept, i.e., ultra-chaos, could deepen our understandings of chaos and open a new door to explore chaos theory. • The navigation system of chaotic mobile robot belongs to "normal-chaos". • The motion of chaotic mobile robot belongs to "ultra-chaos". • "Ultra-chaos" has indeed a higher disorder than "normal-chaos". [ABSTRACT FROM AUTHOR]

Memristor-based chaotic systems have complex dynamical behaviors, which are characterized as nonlinear and hysteresis characteristics. Modeling and identification of their nonlinear model is an important premise for analyzing the dynamical behavior of the memristor-based chaotic systems. This paper presents a novel nonlinear Wiener adaptive filtering identification approach to the memristor-based chaotic systems. The linear part of Wiener model consists of the linear transversal adaptive filters, the nonlinear part consists of nonlinear adaptive filters based on the backslash operator for the hysteresis characteristics of the memristor. The weight update algorithms for the linear and nonlinear adaptive filters are derived. Final computer simulation results show the effectiveness as well as fast convergence characteristics. Comparing with the adaptive nonlinear polynomial filters, the proposed nonlinear adaptive filters have less identification error. [ABSTRACT FROM AUTHOR]

The dynamical behavior of an autonomous simplest electronic circuit that consists of three elements connected in series governed by three ordinary differential equations was investigated. The circuit elements are a linear passive inductor, a linear passive capacitor, and a nonlinear active memristor. Under variation of two parameters we observed a rich variety of bifurcation phenomena, including periodic, quasi-periodic, intermittent, and chaotic behaviors associated with this very simple nonlinear system. One and two-parameter bifurcation diagrams were studied. Route of period-doubling to chaos through different structures was observed. Our theoretical results are compared with the available experimental results and are found to be in good agreement with these results. [ABSTRACT FROM AUTHOR]

Toy model dynamical systems, such as the baker maps, are useful to shed light on some of the conditions verified by deterministic models in non-equilibrium statistical physics. We investigate a 2D dynamical system, enjoying a weak form of reversibility, with peculiar basins of attraction and steady states. In particular, we test the conditions required for the validity of the transient Fluctuation Relation. Our analysis illustrates by means of concrete examples why ergodicity of the equilibrium dynamics (also known as “ergodic consistency”) seems to be a necessary condition for the transient Fluctuation Relation. This investigation then leads to the numerical verification of a kind of transient relation which, differently from the usual transient Fluctuation Relation, holds only asymptotically. At the same time, this relation is not a steady state Fluctuation Relation, because the steady state is a fixed point without fluctuations. [ABSTRACT FROM AUTHOR]

Spatially distributed systems are rather difficult to investigate due to two distinct problems which can be sometimes combined. First, the spatial extension is taken into account by monitoring the system evolution at different locations. Second, the dynamics cannot always be continuously tracked in time, and segments of data – sometimes recorded at different places – are only available. When the dynamics underlying a single marker is under consideration – as for instance the normalized difference vegetation index which can be used for assessing the vegetation canopy of a given area – a global model can be obtained from a single scalar time series built by aggregating the available time series recorded at different places and/or associating the segments of data recorded at different times (and possibly at different locations). We investigated how these two data preprocessing – common in environmental studies – may affect the model dynamics by using a system of spatially distributed Rössler systems which are phase synchronized or not. [ABSTRACT FROM AUTHOR]

Jibin Li studied “exact explicit travelling wave solutions for (n+1)-dimensional Klein–Gordon–Zakharov equations.” By using the approach of dynamical system, the author claimed that they had obtained five classes of exact explicit travelling wave solutions. However, we checked the results and found two typographical technical errors in the main results. Furthermore, the author fails to note, for a = 0 , b < 0 , h ∈ ( 0 , ∞ ) and a < 0 , b > 0 , h = 0 , there is a family of periodic wave solutions and a series of breaking wave solutions, respectively. [ABSTRACT FROM AUTHOR]

Accurately detecting a homoclinic cycle and chaos in a concrete system takes a huge challenge. This paper gives a sufficient condition of coexisting homoclinic cycles (degenerate) with respect to a saddle-focus in a class of three-dimensional piecewise systems with two switching manifolds. Furthermore, by constructing the Poincaré map, it is rigorously shown that chaotic behavior is induced by the coexisting cycles without necessary symmetry in the considered system, which is evidently different from the usual dynamics in smooth system theory that a pair of symmetrical homoclinic cycles can generate chaos but the asymmetric ones cannot. It finally provides an example to illustrate the effectiveness of the results established. • Obtain a criterion for accurately detecting coexisting homoclinic cycles (degenerate and non-degenerate) in a class of piecewise systems. • Propose existence conditions of chaos, which generalizes the Shilnikov theorem of smooth systems to non-smooth systems. • Rigorously prove the coexisting cycles possibly without symmetry can give rise to topological horseshoes, which is in sharp contrast with the smooth systems that need symmetrical homoclinic cycles to produce chaos. • The new findings and theoretical results contribute to better understanding the geometry and generation mechanism of chaotic attractors. • The analysis method in this paper is also applicable to a investigation on similar objects caused by other types of coexisting singular cycles in non-smooth dynamical systems. [ABSTRACT FROM AUTHOR]

In this paper, a dynamic system, the fractional-order hyper-chaotic complex Lü system is proposed for the first time and its rich dynamics are investigated through numerical simulations. It is found that the variation of both the system parameters and the fractional order can induce bifurcations, and the parameter ranges of chaos are different. Furthermore, based on the Open-Plus-Closed-Loop (OPCL) technique and the stability theory of fractional-order dynamical system, a scheme of modified projective synchronization (MPS) for the fractional-order hyper-chaotic complex Lü system is presented, which can effectively achieve modified projective synchronization in a class of high dimensional fractional-order systems. Examples of numerical simulations are presented to show the validity of the proposed control strategy. [ABSTRACT FROM AUTHOR]

We introduce a map to describe the systematics of orbit creation and annihilation in Lorenz-like dynamical systems. This map, y ′ = b - | y | , has a singular maximum and is useful for describing flows that undergo a tear-and-squeeze route to chaos. We call this map the Lorenz map. We find: much of the dynamics is determined by the bifurcations of the period-one and period-two orbits; orbits are created in explosions (singular saddle-node bifurcations) based on two symbols s 0 , s 1 , and later removed in inverse processes that are implosions. The order in which direct and inverse explosions occur generally follows the inverse order shown by the logistic map. In the entire parameter range only one regular saddle-node bifurcation and one period-doubling bifurcation occurs. [ABSTRACT FROM AUTHOR]

The principal signatures of deterministic dynamics in the probabilistic properties of extremes are identified. Analytical expressions for n -fold cumulative distributions and their associated densities are derived. Substantial differences from the classical statistical theory of extreme values are found and illustrated on generic classes of dynamical systems giving rise to fully developed chaos and to quasi-periodic behavior. [ABSTRACT FROM AUTHOR]

Record statistics is the study of how new highs or lows are created and sustained in any dynamical process. The study of the highest or lowest records constitute the study of extreme values. This paper represents an exploration of record statistics for certain aspects of the classical and quantum standard map. For instance the momentum square or energy records is shown to behave like that of records in random walks when the classical standard map is in a regime of hard chaos. However different power laws is observed for the mixed phase space regimes. The presence of accelerator modes are well-known to create anomalous diffusion and we notice here that the record statistics is very sensitive to their presence. We also discuss records in random vectors and use it to analyze the quantum standard map via records in their eigenfunction intensities, reviewing some recent results along the way. [ABSTRACT FROM AUTHOR]

In this paper, chaotic dynamics of the vibro-impact system under bounded noise excitation is investigated by an extended Melnikov method. Firstly, the Melnikov method in the deterministic vibro-impact system is extended to the stochastic case. Then, a typical stochastic Duffing vibro-impact system is given to application. The analytic conditions for occurrence of chaos are derived by using the random Melnikov process in the mean-square-value sense. In addition, the numerical simulations confirm the validity of analytic results. Also, the influences of interesting system parameters on the chaotic dynamics are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we consider characteristic polynomials of n -dimensional systems that determine a segment of polynomials. One parameter is used to characterize this segment of polynomials in order to determine the maximal interval of dissipativity and unstability. Then we apply this result to the generation of a family of attractors based on a class of unstable dissipative systems (UDS) of type affine linear systems. This class of systems is comprised of switched linear systems yielding strange attractors. A family of these chaotic switched systems is determined by the maximal interval of perturbation of the matrix that governs the dynamics for still having scroll attractors. [ABSTRACT FROM AUTHOR]

We investigate the collective behavior of a system of chaotic Rössler oscillators indirectly coupled through a common environment that possesses its own dynamics and which in turn is modulated by the interaction with the oscillators. By varying the parameter representing the coupling strength between the oscillators and the environment, we find two collective states previously not reported in systems with environmental coupling: (i) nontrivial collective behavior, characterized by a periodic evolution of macroscopic variables coexisting with the local chaotic dynamics; and (ii) dynamical clustering, consisting of the formation of differentiated subsets of synchronized elements within the system. These states are relevant for many physical and biological systems where interactions with a dynamical environment are frequent. [ABSTRACT FROM AUTHOR]

In studying a dynamical system, we know that there frequently exist some kinds of disturbance or false phenomenon, namely that some chaotic sets are included in the Borel sets with absolute measure zero. From the viewpoint of ergodic theory, a Borel set with absolute measure zero is negligible. In order to remove these disturbance and false phenomenon, reference Zhou (1993) introduced a concept of measure center. This paper is concerned with distributional chaos occurring on measure center. We draw three conclusions: (i) the one-sided shift has an uncountable distributionally scrambled set which is included in the set of all weakly almost periodic points but not in the set of almost periodic points; (ii) we give a sufficient condition for the compact dynamic system ( X , f ) to exhibit distributional chaos on measure center via semiconjugacy; (iii) the one-sided shift on the measure center is distributionally chaotic in sequence. [ABSTRACT FROM AUTHOR]

It is known that the whole space can be a Li–Yorke scrambled set in a compact dynamical system, but this does not hold for distributional chaos. In this paper we construct a noncompact weekly mixing dynamical system, and prove that the whole space is a transitive extremal distributionally scrambled set in this system. [ABSTRACT FROM AUTHOR]

Recurrence quantification analysis (RQA) is used to study dynamical systems and to identify the underlying physics when a system exhibits a transition due to changes in some control parameter. The tendency of reoccurrence of different states after certain interval reflects and reveals the hidden patterns of a complex time series data. The present work involves the study of the floating potential fluctuations of a glow discharge plasma obtained by using a Langmuir probe. Determinism, entropy and Lmax are important measures of RQA that show an increasing and decreasing trend with variation in the values of discharge voltages and indicate an order-chaos transition in the dynamics of the fluctuations. Statistical analysis techniques represented by skewness and kurtosis are also supportive of a similar phenomenon occurring in the system. [ABSTRACT FROM AUTHOR]

By investigating the topology of chaotic solutions to the generalized Moore and Spiegel equations, we address the question how the solutions of nonlinear dynamical systems are dependent on the nature of the nonlinearity. In these generalized jerk equations, the single nonlinear term has a parity which depends on u n u ̇ . The system has an inversion symmetry when n is even and no symmetry property when n is odd. It is shown that the topology of chaotic solutions only depends on the parity of n , that is, on the symmetry properties and not on the degree of nonlinearity. The value of n only affects the possibility to develop the chaotic solution, that is, to increase the number of unstable periodic orbits within the attractor. [ABSTRACT FROM AUTHOR]

For a topological dynamical system (X, f) we consider the structure of the set F (f) of asymptotic distributions of the distances between pairs of trajectories. If f has the weak specification property then F (f) is closed and convex, and it can contain all nondecreasing functions [0, diam X] → [ 0, 1]. We discuss relations to distributional chaos. [ABSTRACT FROM AUTHOR]

This note obtains some characteristics of accessibility and Kato’s chaos. Applying these results, an accessible dynamical system whose product system is not accessible is constructed, giving a negative answer to a question in [Li R, Wang H, Zhao Y. Kato’s chaos in duopoly games. Chaos Solit Fract 2016;84:69–72]. Besides, it is proved that every transitive interval self-map is accessible. [ABSTRACT FROM AUTHOR]

Highlights: [•] Chaos and nonlinear dynamical analysis are applied for mental-disorder detection. [•] Experimental results show significant detection improvement with feature synergy. [•] Proposed approach is effective for analysis of photoplethysmographic signals. [•] Proposed approach is promising for developing automated mental-health systems. [Copyright &y& Elsevier]

Abstract: In this paper, an extremely accurate numerical algorithm, namely the “clean numerical simulation” (CNS), is proposed to accurately simulate the propagation of micro-level inherent physical uncertainty of chaotic dynamic systems. The chaotic Hamiltonian Hénon–Heiles system for motion of stars orbiting in a plane about the galactic center is used as an example to show its basic ideas and validity. Based on Taylor expansion at rather high-order and MP (multiple precision) data in very high accuracy, the CNS approach can provide reliable trajectories of the chaotic system in a finite interval t ∈[0, T c ], together with an explicit estimation of the critical time T c . Besides, the residual and round-off errors are verified and estimated carefully by means of different time-step Δt, different precision of data, and different order M of Taylor expansion. In this way, the numerical noises of the CNS can be reduced to a required level, i.e. the CNS is a rigorous algorithm. It is illustrated that, for the considered problem, the truncation and round-off errors of the CNS can be reduced even to the level of 10−1244 and 10−1000, respectively, so that the micro-level inherent physical uncertainty of the initial condition (in the level of 10−60) of the Hénon–Heiles system can be investigated accurately. It is found that, due to the sensitive dependence on initial condition (SDIC) of chaos, the micro-level inherent physical uncertainty of the position and velocity of a star transfers into the macroscopic randomness of motion. Thus, chaos might be a bridge from the micro-level inherent physical uncertainty to the macroscopic randomness in nature. This might provide us a new explanation to the SDIC of chaos from the physical viewpoint. [Copyright &y& Elsevier]

Abstract: This work analyzes the problem of community structure in real-world networks based on the synchronization of nonidentical coupled chaotic Rössler oscillators each one characterized by a defined natural frequency, and coupled according to a predefined network topology. The interaction scheme contemplates an uniformly increasing coupling force to simulate a society in which the association between the agents grows in time. To enhance the stability of the correlated states that could emerge from the synchronization process, we propose a parameterless mechanism that adapts the characteristic frequencies of coupled oscillators according to a dynamic connectivity matrix deduced from correlated data. We show that the characteristic frequency vector that results from the adaptation mechanism reveals the underlying community structure present in the network. [Copyright &y& Elsevier]

• The nonlinear chaotic complex system are studied. • The ultimate bound of the nonlinear chaotic complex systems are presented. • The obtained solutions are presented by 2D figures. The competitive modes for a nonlinear chaotic complex system are studied in this paper. In hyperchaotic, chaotic, and periodic cases, we examined competitive modes that are a tool for detecting chaos in a system. Also, using an analytical method and Lagrange optimization, we were able to calculate the ultimate bound of the nonlinear chaotic complex systems. We have presented is simpler and more accurate than other methods that implicitly calculate the ultimate bound. The estimation of the explicit ultimate bound can be used to study chaos control and chaos synchronization. Numerical simulations illustrate the analytical results. [ABSTRACT FROM AUTHOR]

• We propose a spatiotemporal chaotic system, a novel pseudo-random CML (PRCML) system, with the pseudo-random coupling method based on the elementary cellular automata (ECA), and add different perturbations to lattices in each iteration according to the ECA. In addition, the innovations of the proposed system are listed as follows: • First of all, the pseudo-random coupling is utilized in proposed system. Rather than employing the adjacent lattices for coupling as a conventional CML, the PRCML's choice of lattices for coupling is innovatively dependent on the iterative result of the ECA which is in chaos. Furthermore, the choice of lattices for coupling is always changing in each iteration because the ECA is iterated with the system. Thus, the coupling is pseudo-random that enhance the complexity of the system. • Another innovation is that the iterative result of the ECA is employed for disturbing the PRCML. This novel scheme leads to the following advantages: Firstly, the periodic windows in bifurcation diagram fade out obviously. Secondly, the ECA is essentially a discrete dynamical system, and the finite computing precision has no influence on it. Thus, employing iterative results of the ECA as the perturbation alleviates dynamical degradation efficiently. Thirdly, owing to the different pseudo-random perturbation for each lattice, the correlation between any two lattices is significantly reduced. • Theory analysis and simulation test indicate that the new system has better performance in complexity, ergodic and unpredictability than other CML systems such as adjacent CML and nonlinear CML based on fractional order logistic equation, etc. Furthermore, the correlation coefficient between any two lattices in proposed system is significantly lower than other systems. The coupled map lattices (CML) is a spatiotemporal chaotic system with complex dynamic behavior. In this paper, we propose a spatiotemporal chaotic system with a novel pseudo-random coupling method based on elementary cellular automata (ECA), and introduce different perturbations into lattices in each iteration according to ECA. We investigate the spatiotemporal dynamic properties and chaotic behaviors of the proposed system such as bifurcation diagrams, Kolmogorov Sinai entropy, and uniformity. Moreover, the randomness of sequences generated by the proposed system and the correlation between any two lattices are discussed. Theory analyses and simulations indicate that the new system has better performance in complexity, ergodic and unpredictability than other CML systems such as adjacent CML and nonlinear CML based on fractional order logistic equation, etc. Furthermore, the correlation coefficient between any two lattices in the proposed system is significantly lower than other systems, and another advantage of the proposed system is utilizing the output of ECA to perturb the chaotic system which can effectively alleviate the dynamical degradation in digital system. The excellent performance of the proposed system demonstrates that it has great potential for crypto-system. [ABSTRACT FROM AUTHOR]

• Dynamics of a mid size firm model is studied. • Analytical stability criteria is obtained. • Analytical localization of attractors is provided. • Lyapunov dimension is computed. Forecasting and analyses of the dynamics of financial and economic processes such as deviations of macroeconomic aggregates (GDP, unemployment, and inflation) from their long-term trends, asset markets volatility, etc., are challenging because of the complexity of these processes. Important related research questions include, first, how to determine the qualitative properties of the dynamics of these processes, namely, whether the process is stable, unstable, chaotic (deterministic), or stochastic; and second, how best to estimate its quantitative indicators including dimension, entropy, and correlation characteristics. These questions can be studied both empirically and theoretically. In the empirical approach, researchers consider real data expressed in terms of time series, identify the patterns of their dynamics, and then forecast the short- and long-term behavior of the process. The second approach is based on postulating the laws of dynamics for the process, deriving mathematical dynamical models based on these laws, and conducting subsequent analytical investigation of the dynamics generated by the models. To implement these approaches, either numerical or analytical methods can be used. While numerical methods make it possible to study dynamical models, the possibility of obtaining reliable results using them is significantly limited due to the necessity of performing calculations only over finite time intervals, rounding-off errors in numerical methods, and the unbounded space of initial data sets. Analytical methods allow researchers to overcome these limitations and to identify the exact qualitative and quantitative characteristics of the dynamics of the process. However, effective analytical applications are often limited to low-dimensional models (in the literature, two-dimensional dynamical systems are most often studied). In this paper, we develop analytical methods for the study of deterministic dynamical systems based on the Lyapunov stability theory and on chaos theory. These methods make it possible not only to obtain analytical stability criteria and to estimate limiting behavior (to localize self-excited and hidden attractors and identify multistability), but also to overcome difficulties related to implementing reliable numerical analysis of quantitative indicators such as Lyapunov exponents and the Lyapunov dimension. We demonstrate the effectiveness of the proposed methods using the mid-size firm model suggested by Shapovalov. [ABSTRACT FROM AUTHOR]