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This paper is concerned with the relaxation dynamics of a large population of identical phase oscillators, each of which interacts with all the others through random couplings whose parameters obey the same Gaussian distribution with the average equal to zero and are mutually independent. The results obtained by numerical simulation suggest that for the infinite-size system, the absolute value of Kuramoto's order parameter exhibits superslow relaxation, i.e., 1/ln t as time t increases. Moreover, the statistics on both the transient time T for the system to reach a fixed point and the absolute value of Kuramoto's order parameter at t = T are also presented together with their distribution densities over many realizations of the coupling parameters. [ABSTRACT FROM AUTHOR]

Non-smooth dynamics induced by switches, impacts, sliding, and other abrupt changes are pervasive in physics, biology, and engineering. Yet, systems with non-smooth dynamics have historically received far less attention compared to their smooth counterparts. The classic "Bristol book" [di Bernardo et al., Piecewise-smooth Dynamical Systems. Theory and Applications (Springer-Verlag, 2008)] contains a 2008 state-of-the-art review of major results and challenges in the study of non-smooth dynamical systems. In this paper, we provide a detailed review of progress made since 2008. We cover hidden dynamics, generalizations of sliding motion, the effects of noise and randomness, multi-scale approaches, systems with time-dependent switching, and a variety of local and global bifurcations. Also, we survey new areas of application, including neuroscience, biology, ecology, climate sciences, and engineering, to which the theory has been applied. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the synchronization problem of coupled dynamical networks with nonidentical Duffing-type oscillators without or with coupling delays. Different from cluster synchronization of nonidentical dynamical networks in the previous literature, this paper focuses on the problem of complete synchronization, which is more challenging than cluster synchronization. By applying an impulsive controller, some sufficient criteria are obtained for complete synchronization of the coupled dynamical networks of nonidentical oscillators. Furthermore, numerical simulations are given to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we present a study on a network-based susceptible-infected-recovered (SIR) epidemic model with a saturated treatment function. It is well known that treatment can have a specific effect on the spread of epidemics, and due to the limited resources of treatment, the number of patients during severe disease outbreaks who need to be treated may exceed the treatment capacity. Consequently, the number of patients who receive treatment will reach a saturation level. Thus, we incorporated a saturated treatment function into the model to characterize such a phenomenon. The dynamics of the present model is discussed in this paper. We first obtained a threshold value R 0 , which determines the stability of a disease-free equilibrium. Furthermore, we investigated the bifurcation behavior at R 0 = 1. More specifically, we derived a condition that determines the direction of bifurcation at R 0 = 1. If the direction is backward, then a stable disease-free equilibrium concurrently exists with a stable endemic equilibrium even though R 0 < 1. Therefore, in this case, R 0 < 1 is not sufficient to eradicate the disease from the population. However, if the direction is forward, we find that for a range of parameters, multiple equilibria could exist to the left and right of R 0 = 1. In this case, the initial infectious invasion must be controlled to a lower level so that the disease dies out or approaches a lower endemic steady state. [ABSTRACT FROM AUTHOR]

This paper provides some historical comments on the study of the Fermi, Pasta, and Ulam (FPU) paper and its influence on the development of the theory of chaos. We also discuss some problems raised in the FPU paper and the links of these problems to such contemporary notions in chaos theory as ergodicity, mixing, recurrences, pseudochaos, kinetics, intermittency, etc. [ABSTRACT FROM AUTHOR]

This paper studies the evolution process of competitive dynamics on triplex complex networks. We propose a new triplex network model in which the state of the node in each layer is affected by its neighbors as well as inter-layer competition. Through this model, we combine the opinion diffusion model, the Ising model, and the signed network and extend their application from single-layer to multi-layer networks. We derive the evolution process and dynamical equations of the model and carry out a series of numerical simulations to discuss the influence of several factors on the evolution process and the competitiveness of the network. First, we find that the increase of global transition threshold p or the proportion of initial active nodes will lead to more surviving layers and more active nodes in each layer. In addition, we summarize the similarities and differences of the evolution curves under different conditions. Second, we discuss the influence of initial active nodes and the average degree on the competitiveness of the network and find the correlations between them. Finally, we study the relationship between network topology and network competitiveness and conclude the conditions for the best competitiveness of the network. Based on the simulation results, we give specific suggestions on how to improve the competitiveness of the platform in reality. [ABSTRACT FROM AUTHOR]

This paper investigates consensus dynamics in a dynamical system with additive stochastic disturbances that is characterized as network coherence by using the Laplacian spectrum. We introduce a class of weighted networks based on a complete graph and investigate the first- and second-order network coherence quantifying as the sum and square sum of reciprocals of all nonzero Laplacian eigenvalues. First, the recursive relationship of its eigenvalues at two successive generations of Laplacian matrix is deduced. Then, we compute the sum and square sum of reciprocal of all nonzero Laplacian eigenvalues. The obtained results show that the scalings of first- and second-order coherence with network size obey four and five laws, respectively, along with the range of the weight factor. Finally, it indicates that the scalings of our studied networks are smaller than other studied networks when 1 d < r ≤ 1. [ABSTRACT FROM AUTHOR]

Safe sets are a basic ingredient in the strategy of partial control of chaotic systems. Recently we have found an algorithm, the sculpting algorithm, which allows us to construct them, when they exist. Here we define another type of set, an asymptotic safe set, to which trajectories are attracted asymptotically when the partial control strategy is applied. We apply all these ideas to a specific example of a Duffing oscillator showing the geometry of these sets in phase space. The software for creating all the figures appearing in this paper is available as supplementary material. [ABSTRACT FROM AUTHOR]

We introduce the concept of this special focus issue on solitons in nonintegrable systems. A brief overview of some recent developments is provided, and the various contributions are described. The topics covered in this focus issue are the modulation of solitons, bores, and shocks, the dynamical evolution of solitary waves, and existence and stability of solitary waves and embedded solitons. [ABSTRACT FROM AUTHOR]

In recent years, fractional(-order) differential equations have attracted increasing interests due to their applications in modeling anomalous diffusion, time dependent materials and processes with long range dependence, allometric scaling laws, and complex networks. Although an autonomous system cannot define a dynamical system in the sense of semigroup because of the memory property determined by the fractional derivative, we can still use the Lyapunov exponents to discuss its dynamical evolution. In this paper, we first define the Lyapunov exponents for fractional differential systems then estimate the bound of the corresponding Lyapunov exponents. For linear fractional differential system, the bounds of its Lyapunov exponents are conveniently derived which can be regarded as an example for the theoretical results established in this paper. Numerical example is also included which supports the theoretical analysis. [ABSTRACT FROM AUTHOR]

The paper considers the neuron model of Hindmarsh–Rose and studies in detail the system dynamics which controls the transition between the spiking and bursting regimes. In particular, such a passage occurs in a chaotic region and different explanations have been given in the literature to represent the process, generally based on a slow-fast decomposition of the neuron model. This paper proposes a novel view of the chaotic spiking-bursting transition exploiting the whole system dynamics and putting in evidence the essential role played in the phenomenon by the manifolds of the equilibrium point. An analytical approximation is developed for the related crucial elements and a subsequent numerical analysis signifies the properness of the suggested conjecture. [ABSTRACT FROM AUTHOR]

In this paper, the problem of generalized outer synchronization between two completely different complex dynamical networks is investigated. With a nonlinear control scheme, a sufficient criterion for this generalized outer synchronization is derived based on Barbalat’s lemma. Two corollaries are also obtained, which contains the situations studied in two lately published papers as special cases. Numerical simulations further demonstrate the feasibility and effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

The present paper points out a novel scenario for the formation of chaotic attractors in a class of models of excitable cell membranes near an Andronov–Hopf bifurcation (AHB). The mechanism underlying chaotic dynamics admits a simple and visual description in terms of the families of one-dimensional first-return maps, which are constructed using the combination of asymptotic and numerical techniques. The bifurcation structure of the continuous system (specifically, the proximity to a degenerate AHB) endows the Poincaré map with distinct qualitative features such as unimodality and the presence of the boundary layer, where the map is strongly expanding. This structure of the map in turn explains the bifurcation scenarios in the continuous system including chaotic mixed-mode oscillations near the border between the regions of sub- and supercritical AHB. The proposed mechanism yields the statistical properties of the mixed-mode oscillations in this regime. The statistics predicted by the analysis of the Poincaré map and those observed in the numerical experiments of the continuous system show a very good agreement. [ABSTRACT FROM AUTHOR]

We study projective-anticipating, projective, and projective-lag synchronization of time-delayed chaotic systems on random networks. We relax some limitations of previous work, where projective-anticipating and projective-lag synchronization can be achieved only on two coupled chaotic systems. In this paper, we realize projective-anticipating and projective-lag synchronization on complex dynamical networks composed of a large number of interconnected components. At the same time, although previous work studied projective synchronization on complex dynamical networks, the dynamics of the nodes are coupled partially linear chaotic systems. In this paper, the dynamics of the nodes of the complex networks are time-delayed chaotic systems without the limitation of the partial linearity. Based on the Lyapunov stability theory, we suggest a generic method to achieve the projective-anticipating, projective, and projective-lag synchronization of time-delayed chaotic systems on random dynamical networks, and we find both its existence and sufficient stability conditions. The validity of the proposed method is demonstrated and verified by examining specific examples using Ikeda and Mackey–Glass systems on Erdös–Rényi networks. [ABSTRACT FROM AUTHOR]

In this paper, we study chaotic dynamics of a class of three-dimensional Glass networks with different decay constants, illustrate how the horseshoe is generated, and present a rigorous computer-assisted verification of chaoticity by virtue of interval analysis and topological horseshoe theory. [ABSTRACT FROM AUTHOR]

The pre-sliding–pre-rolling phase of friction behavior is dominated by rate-independent hysteresis. Many machine elements in common engineering use exhibit, therefore, the characteristic of “hysteresis springs,” for small displacements at least. Plain and rolling element bearings that are widely used in motion guidance of machine tools are typical examples. While the presence of a hysteresis element may mark the character of the resulting dynamics, little is to be found about this topic in the literature. The study of the nonlinear dynamics caused by such elements becomes imperative if we wish to achieve accurate control of such machines. In this Part I of the investigation, we examine a single-degree-of-freedom mass-hysteresis-spring system and show that, while the free response case is amenable to an exact solution, the more important case of forced response has no closed form solution and requires other methods of treatment. We consider harmonic-balance analysis methods (which are common analysis tools in engineering) suitable for frequency-domain treatment, in particular the approximate describing function (DF) method, and compare those results with “exact” numerical simulations. The DF method yields basically a linear equation with amplitude-dependent modal parameters. We find that agreement in the frequency response function, between DF and exact solution, is good for small excitation amplitudes and for very large amplitudes. Intermediate values, however, show high sensitivity to amplitude variations and, consequently, no regular solution is obtainable by either approach. This appears to be an inherent property of the system pointing to the need for developing further analysis methods. Experimental verification of the analysis outlined in this Part I is given in Part II of the paper. [ABSTRACT FROM AUTHOR]

Delay differential equations (DDEs) recently have been used in models of cardiac electrophysiology, particularly in studies focusing on electrical alternans, instabilities, and chaos. A number of processes within cardiac cells involve delays, and DDEs can potentially represent mechanisms that result in complex dynamics both at the cellular level and at the tissue level, including cardiac arrhythmias. However, DDE-based formulations introduce new computational challenges due to the need for storing and retrieving past values of variables at each spatial location. Cardiac tissue simulations that use DDEs may require over 28 GB of memory if the history of variables is not managed carefully. This paper addresses both computational and dynamical issues. First, we present new methods for the numerical solution of DDEs in tissue to mitigate the memory requirements associated with the history of variables. The new methods exploit the different time scales of an action potential to dynamically optimize history size. We find that the proposed methods decrease memory usage by up to 95% in cardiac tissue simulations compared to straightforward history-management algorithms. Second, we use the optimized methods to analyze for the first time the dynamics of wave propagation in two-dimensional cardiac tissue for models that include DDEs. In particular, we study the effects of DDEs on spiral-wave dynamics, including wave breakup and chaos, using a canine myocyte model. We find that by introducing delays to the gating variables governing the calcium current, DDEs can induce spiral-wave breakup in 2D cardiac tissue domains. [ABSTRACT FROM AUTHOR]

In a network of high-dimensionality, it is not feasible to measure every single node. Thus, an important goal is to define the optimal choice of sensor nodes that provides a reliable state reconstruction of the network system state-space. This is an observability problem. In this paper, we propose a particle filtering (PF) framework as a way to assess observability properties of a dynamical network, where each node is composed of an individual dynamical system. The PF framework is applied to two benchmarks, networks of Kuramoto and Rössler oscillators, to investigate how the interplay between dynamics and topology impacts the network observability. Based on the numerical results, we conjecture that, when the network nodal dynamics are heterogeneous, better observability is conveyed for sets of sensor nodes that share some dynamical affinity to its neighbourhood. Moreover, we also investigate how the choice of an internal measured variable of a multidimensional sensor node affects the PF performance. The PF framework effectiveness as an observability measure is compared with a well-consolidated nonlinear observability metric for a small network case and some chaotic system benchmarks. [ABSTRACT FROM AUTHOR]

Random number generation is a fundamental technology behind information security. Recently, physical random number generators (RNGs), which especially harness optical chaos such as in delay-feedback lasers, have been studied intensively. Although these are promising technologies for future information security, there is little theoretical foundation. In this paper, we newly introduce a mathematical formulation of physical RNGs based on a model of chaotic dynamics and give the first rigorous results. In particular, by combining ergodic theory, information theory, and response theory of statistical physics, our theory guarantees, for the model of chaotic dynamics, the coexistence of two crucial properties necessary for physical RNGs: fast random number generation and robustness. Compared with other types of physical RNGs, our theoretical findings highlight an unnoticed advantage of chaotic dynamics utilized for physical RNGs. [ABSTRACT FROM AUTHOR]

In this paper, the interactions of some given chaotic properties between a non-autonomous discrete system (X , f 0 , ∞) and its induced set-valued system (K (X) , f ¯ 0 , ∞) are obtained. It is proved that the specification property, property P, topological mixing, mild mixing, and topological exactness of (X , f 0 , ∞) are equivalent to those of (K (X) , f ¯ 0 , ∞) , respectively. It is shown that Robinson chaos (resp. Kato chaos) between (X , f 0 , ∞) and (K (X) , f ¯ 0 , ∞ ) are equivalent under certain conditions. Furthermore, Li-Yorke chaos and distributional chaos of (X , f 0 , ∞) imply those of (K (X) , f ¯ 0 , ∞) , respectively. Topological equi-conjugacy between two systems is proved to be preserved by their induced set-valued systems. Topological entropy of (X , f 0 , ∞) is guaranteed to be no larger than that of (K (X) , f ¯ 0 , ∞). Two examples are finally provided with computer simulations for illustration. [ABSTRACT FROM AUTHOR]

To reduce parameter error caused by human factors and ensure the accuracy of the largest Lyapunov exponent (LLE) obtained from chaotic equations, this paper proposes a simple method based on two nearby pseudo-orbits. First, a point is selected from a solution trajectory of chaotic equation by the roundoff error. Second, the selected point is used as an initial condition to solve the same equation to obtain another solution trajectory. Third, the evolution distance of the two solution trajectories is calculated. Finally, the LLE is the slope of the linear region in the curve of the track distance of the natural algorithm. Our method has been successfully applied to simulate five well-known chaotic systems and some non-chaotic systems. The results show that, compared with other traditional methods, the proposed method is efficient, simple, and robust without reconstructing phase space and computing the Jacobian matrix. [ABSTRACT FROM AUTHOR]

This paper considers a recently introduced D -dimensional generalized Kuramoto model for many (N ≫ 1) interacting agents, in which the agent states are D -dimensional unit vectors. It was previously shown that, for even (but not odd) D , similar to the original Kuramoto model (D = 2), there exists a continuous dynamical phase transition from incoherence to coherence of the time asymptotic attracting state (time t → ∞) as the coupling parameter K increases through a critical value which we denote K c (+) > 0. We consider this transition from the point of view of the stability of an incoherent state, where an incoherent state is defined as one for which the N → ∞ distribution function is time-independent and the macroscopic order parameter is zero. In contrast with D = 2 , for even D > 2 , there is an infinity of possible incoherent equilibria, each of which becomes unstable with increasing K at a different point K = K c. Although there are incoherent equilibria for which K c = K c (+) , there are also incoherent equilibria with a range of possible K c values below K c (+) , (K c (+) / 2) ≤ K c < K c (+) . How can the possible instability of incoherent states arising at K = K c < K c (+) be reconciled with the previous finding that, at large time (t → ∞) , the state is always incoherent unless K > K c (+) ? We find, for a given incoherent equilibrium, that, if K is rapidly increased from K < K c to K c < K < K c (+) , due to the instability, a short, macroscopic burst of coherence is observed, in which the coherence initially grows exponentially, but then reaches a maximum, past which it decays back into incoherence. Furthermore, after this decay, we observe that the equilibrium has been reset to a new equilibrium whose K c value exceeds that of the increased K. Thus, this process, which we call "Instability-Mediated Resetting," leads to an increase in the effective K c with continuously increasing K , until the equilibrium has been effectively set to one for which K c ≈ K c (+) . Thus, instability-mediated resetting leads to a unique critical point of the t → ∞ time asymptotic state (K = K c (+) ) in spite of the existence of an infinity of possible pretransition incoherent states. [ABSTRACT FROM AUTHOR]

In this paper, we present detailed analyses of the dynamics of a number of embodied neuromechanical systems of a class that has been shown to efficiently exploit chaos in the development and learning of motor behaviors for bodies of arbitrary morphology. This class of systems has been successfully used in robotics, as well as to model biological systems. At the heart of these systems are neural central pattern generating (CPG) units connected to actuators which return proprioceptive information via an adaptive homeostatic mechanism. Detailed dynamical analyses of example systems, using high resolution largest Lyapunov exponent maps, demonstrate the existence of chaotic regimes within a particular region of parameter space, as well as the striking similarity of the maps for systems of varying size. Thanks to the homeostatic sensory mechanisms, any single CPG "views" the whole of the rest of the system as if it was another CPG in a two coupled system, allowing a scale invariant conceptualization of such embodied neuromechanical systems. The analysis reveals chaos at all levels of the systems; the entire brain-body-environment system exhibits chaotic dynamics which can be exploited to power an exploration of possible motor behaviors. The crucial influence of the adaptive homeostatic mechanisms on the system dynamics is examined in detail, revealing chaotic behavior characterized by mixed mode oscillations (MMOs). An analysis of the mechanism of the MMO concludes that they stems from dynamic Hopf bifurcation, where a number of slow variables act as "moving" bifurcation parameters for the remaining part of the system. [ABSTRACT FROM AUTHOR]

This paper investigates feedback pinning control of synchronization behaviors aroused by epidemic spread on complex networks. Based on the quenched mean field theory, epidemic control synchronization models with the inhibition of contact behavior are constructed, combined with the epidemic transmission system and the adaptive dynamical network carrying active controllers. By the properties of convex functions and the Gerschgorin theorem, the epidemic threshold of the model is obtained, and the global stability of disease-free equilibrium is analyzed. For individual's infected situation, when an epidemic disease spreads, two types of feedback control strategies depending on the diseases' information are designed: the first one only adds controllers to infected individuals, and the other adds controllers to both infected and susceptible ones. By using the Lyapunov stability theory, under designed controllers, some criteria that guarantee the epidemic controlled synchronization system achieving behavior synchronization are also derived. Several numerical simulations are performed to show the effectiveness of our theoretical results. As far as we know, this is the first work to address the controlled behavioral synchronization induced by epidemic spread under the pinning feedback mechanism. It is hopeful that we may have deeper insights into the essence between the disease's spread and collective behavior under active control in complex dynamical networks. [ABSTRACT FROM AUTHOR]

The Random Coupling Model (RCM) is a statistical approach for studying the scattering properties of linear wave chaotic systems in the semi-classical regime. Its success has been experimentally verified in various over-moded wave settings, including both microwave and acoustic systems. It is of great interest to extend its use in nonlinear systems. This paper studies the impact of a nonlinear port on the measured statistical electromagnetic properties of a ray-chaotic complex enclosure in the short wavelength limit. A Vector Network Analyzer is upgraded with a high power option, which enables calibrated scattering (S) parameter measurements up to + 43 dBm. By attaching a diode to the excitation antenna, amplitude-dependent S-parameters and Wigner reaction matrix (impedance) statistics are observed. We have systematically studied how the key components in the RCM are affected by this nonlinear port, including the radiation impedance, short ray orbit corrections, and statistical properties. By applying the newly developed radiation efficiency extension to the RCM, we find that the diode admittance increases with the excitation amplitude. This reduces the amount of power entering the cavity through the port so that the diode effectively acts as a protection element. As a result, we have developed a quantitative understanding of the statistical scattering properties of a semi-classical wave chaotic system with a nonlinear coupling channel. [ABSTRACT FROM AUTHOR]

In this paper, a size-dependent viscoelastic pipe model is developed to investigate the size effects on flutter and divergence instability of functionally graded viscoelastic nanotubes conveying fluid. The nonlocal strain gradient theory and the Kelvin-Voigt model are used to consider the significance of nonlocal field, strain gradient field, and viscoelastic damping effects. The dimensionless equation of transverse motion and related classical and non-classical boundary conditions are derived using the variational approach. The partial differential equations are discretized to a system of ordinary differential equations by the use of Galerkin's method. The frequency equation is obtained as a function of dimensionless flow velocity, small-scale parameters, damping coefficient, and power-law parameter. Numerical results are presented to study the dynamical behavior of the system and are compared with experimental and theoretical results reported by other researchers. Coupled and single mode fluttering related to higher vibration modes of fluid-conveying nanotubes supported at both ends are studied for the first time. It is found that coupled mode fluttering can be seen for different vibration modes by increasing the flow velocity in the absence of structural damping. Structural damping changes the dynamical behavior of the system, in which by increasing the flow velocity, single mode fluttering occurs instead of coupled mode fluttering. In addition, the presence of structural damping increases the critical flow velocity and, as a result, increases the stability of the system. The results also show that increasing the nonlocal parameter will have a stiffness-softening effect, while increasing the strain gradient length scale has an opposing effect. [ABSTRACT FROM AUTHOR]

Many systems in biology, physics, and chemistry can be modeled through ordinary differential equations (ODEs), which are piecewise smooth, but switch between different states according to a Markov jump process. In the fast switching limit, the dynamics converges to a deterministic ODE. In this paper, we suppose that this limit ODE supports a stable limit cycle. We demonstrate that a set of such oscillators can synchronize when they are uncoupled, but they share the same switching Markov jump process. The latter is taken to represent the effect of a common randomly switching environment. We determine the leading order of the Lyapunov coefficient governing the rate of decay of the phase difference in the fast switching limit. The analysis bears some similarities to the classical analysis of synchronization of stochastic oscillators subject to common white noise. However, the discrete nature of the Markov jump process raises some difficulties: in fact, we find that the Lyapunov coefficient from the quasi-steady-state approximation differs from the Lyapunov coefficient one obtains from a second order perturbation expansion in the waiting time between jumps. Finally, we demonstrate synchronization numerically in the radial isochron clock model and show that the latter Lyapunov exponent is more accurate. [ABSTRACT FROM AUTHOR]

A model-based approach to forecasting chaotic dynamical systems utilizes knowledge of the mechanistic processes governing the dynamics to build an approximate mathematical model of the system. In contrast, machine learning techniques have demonstrated promising results for forecasting chaotic systems purely from past time series measurements of system state variables (training data), without prior knowledge of the system dynamics. The motivation for this paper is the potential of machine learning for filling in the gaps in our underlying mechanistic knowledge that cause widely-used knowledge-based models to be inaccurate. Thus, we here propose a general method that leverages the advantages of these two approaches by combining a knowledge-based model and a machine learning technique to build a hybrid forecasting scheme. Potential applications for such an approach are numerous (e.g., improving weather forecasting). We demonstrate and test the utility of this approach using a particular illustrative version of a machine learning known as reservoir computing, and we apply the resulting hybrid forecaster to a low-dimensional chaotic system, as well as to a high-dimensional spatiotemporal chaotic system. These tests yield extremely promising results in that our hybrid technique is able to accurately predict for a much longer period of time than either its machine-learning component or its model-based component alone. [ABSTRACT FROM AUTHOR]

This paper focuses on the intricate relationship between smooth and nonsmooth phenomena in an impacting system. In particular a boundary saddle-point solution, that is born in a nonsmooth fold, is analysed. Accessible boundary saddle-point solutions play a key role in determining the global dynamics of a system and here we will show how grazing bifurcations can affect their existence. [ABSTRACT FROM AUTHOR]

This paper presents the nonlinear dynamics and bifurcations of optically injected semiconductor lasers in the frame of relative high injection strength. The behavior of the system is explored by means of bifurcation diagrams; however, the exact nature of the involved dynamics is well described by a detailed study of the dynamics evolutions as a function of the effective gain coefficient. As results, we notice the different types of symmetry chaotic attractors with the riddled basins, supercritical pitchfork and Hopf bifurcations, crisis of attractors, instability of chaos, symmetry breaking and restoring bifurcations, and the phenomena of the bursting behavior as well as two connected parts of the same chaotic attractor which merge in a periodic orbit. [ABSTRACT FROM AUTHOR]

In this paper, we study cluster synchronization in networks of coupled nonidentical dynamical systems. The vertices in the same cluster have the same dynamics of uncoupled node system but the uncoupled node systems in different clusters are different. We present conditions guaranteeing cluster synchronization and investigate the relation between cluster synchronization and the unweighted graph topology. We indicate that two conditions play key roles for cluster synchronization: the common intercluster coupling condition and the intracluster communication. From the latter one, we interpret the two cluster synchronization schemes by whether the edges of communication paths lie in inter- or intracluster. By this way, we classify clusters according to whether the communications between pairs of vertices in the same cluster still hold if the set of edges inter- or intracluster edges is removed. Also, we propose adaptive feedback algorithms to adapting the weights of the underlying graph, which can synchronize any bi-directed networks satisfying the conditions of common intercluster coupling and intracluster communication. We also give several numerical examples to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper dynamical networks with community structure and nonidentical nodes and with identical local dynamics for all individual nodes in each community are considered. The cluster synchronization of these networks with or without time delay is studied by using some feedback control schemes. Several sufficient conditions for achieving cluster synchronization are obtained analytically and are further verified numerically by some examples with chaotic or nonchaotic nodes. In addition, an essential relation between synchronization dynamics and local dynamics is found by detailed analysis of dynamical networks without delay through the stage detection of cluster synchronization. [ABSTRACT FROM AUTHOR]

We show that under suitable assumptions, Poincaré recurrences of a dynamical system determine its topology in phase space. Therefore, dynamical systems with the same recurrences are dynamically equivalent. This conclusion can be drawn from a theorem proved in this paper which states that the recurrence matrix determines the topology of closed sets. The theorem states that if a set of points M is mapped onto another set N, such that two points in N are closer than some prescribed fixed distance if and only if the corresponding points in M are closer than some, in general different, prescribed fixed distance, then both sets are homeomorphic, i.e., identical up to a continuous change in the coordinate system. The theorem justifies a range of methods in nonlinear dynamics which are based on recurrence properties. [ABSTRACT FROM AUTHOR]

It is shown, under weak conditions, that the dynamical evolution of large systems of globally coupled phase oscillators with Lorentzian distributed oscillation frequencies is, in an appropriate physical sense, time-asymptotically attracted toward a reduced manifold of the system states. This manifold was previously known and used to facilitate the discovery of attractors and bifurcations of such systems. The result of this paper establishes that attractors for the order parameter dynamics obtained by restriction to this reduced manifold are, in fact, the only such attractors of the full system. Thus all long time dynamical behaviors of the order parameters of these systems can be obtained by restriction to the reduced manifold. [ABSTRACT FROM AUTHOR]

In this paper, generalized synchronization of two chaotic systems is investigated. The auxiliary system approach, which is suggested by H. Abarbanel, N. Rulkov, and M. Sushchik [Phys. Rev. E 53, 4528 (1996)], is used to detect and study generalized synchronization. Based on the Lyapunov method and matrix measure, some less restrictive criteria are obtained to guarantee the asymptotical stability of the error system between the response system and the auxiliary system, which indicates the drive-response systems are synchronized in a general sense. It is shown that the feedback gain can be reduced by means of the matrix measure approach, compared to the norm method. All theoretical results are illustrated by analytical and numerical examples. [ABSTRACT FROM AUTHOR]

This paper studies the effect of parameter mismatch on anticipating synchronization of chaotic systems with time delay in the framework of the master-slave configuration. The convergence criteria for the error dynamical system under study are established by means of model transformation incorporated with Lyapunov functional and linear matrix inequality. The error bound of anticipating synchronization is estimated by rigorous theoretical analysis. Its accuracy is confirmed by numerical simulation results. [ABSTRACT FROM AUTHOR]

This paper introduces a new three-dimensional quadratic autonomous system, which can generate a pair of double-wing chaotic attractors. More importantly, this new system can generate three-wing and four-wing chaotic attractors with very complicated topological structures over a large range of parameters. Several issues, such as some basic dynamical behaviors, bifurcations, and the dynamical structure of the new chaotic system, are investigated either analytically or numerically. [ABSTRACT FROM AUTHOR]

This paper investigates the synchronization problem for a class of complex delayed dynamical networks by pinning periodically intermittent control. Based on a general model of complex delayed dynamical networks, using the Lyapunov stability theory and periodically intermittent control method, some simple criteria are derived for the synchronization of such dynamical networks. Furthermore, a Barabási–Albert network consisting of coupled delayed Chua oscillators is finally given as an example to verify the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the pinning control problem of fractional-order weighted complex dynamical networks. The well-studied integer-order complex networks are the special cases of the fractional-order ones. The network model considered can represent both directed and undirected weighted networks. First, based on the eigenvalue analysis and fractional-order stability theory, some local stability properties of such pinned fractional-order networks are derived and the valid stability regions are estimated. A surprising finding is that the fractional-order complex networks can stabilize itself by reducing the fractional-order q without pinning any node. Second, numerical algorithms for fractional-order complex networks are introduced in detail. Finally, numerical simulations in scale-free complex networks are provided to show that the smaller fractional-order q, the larger control gain matrix D, the larger tunable weight parameter β, the larger overall coupling strength c, the more capacity that the pinning scheme may possess to enhance the control performance of fractional-order complex networks. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new method for detecting regular behavior of time series: this method is based on the Poincaré section and the Higuchi fractal dimension. The new method aims to distinguish random signals from deterministic signals. In fact, our method provides a pattern for decision making about whether a signal is random or deterministic. We apply this method to different time series, such as chaotic signals, random signals, and periodic signals. We apply this method to examples from all types of route to chaotic signals. This method has also been applied to data about iris tissues. The results show that the new method can distinguish different types of signals. [ABSTRACT FROM AUTHOR]

The parameter perturbation methods (the most known being the OGY method) apply small wisely chosen swift kicks to the system once per cycle, to maintain it near the desired unstable periodic orbit. Thus, one can consider that a new attractor is finally generated. Another class of methods which allow the attractors born, imply small perturbations of the state variable [see, e.g., J. Güémez and M. A. Matías, Phys. Lett. A 181, 29 (1993)]. Whatever technique is utilized, generating any targeted attractor starting from a set of two or more of any kind of attractors (stable or not) of a considered dissipative continuous-time system cannot be achieved with these techniques. This kind of attractor synthesis [introduced in M.-F. Danca, W. K. S. Tang, and G. Chen, Appl. Math. Comput. 201, 650 (2008) and proved analytically in Y. Mao, W. K. S. Tang, and M.-F. Danca, Appl. Math. Comput. (submitted)] which starts from a set of given attractors, allows us, via periodic parameter-switching, to generate any of the set of all possible attractors of a class of continuous-time dissipative dynamical systems, depending linearly on the control parameter. In this paper we extend this technique proving empirically that even random manners for switching can be utilized for this purpose. These parameter-switches schemes are very easy to implement and require only the mathematical model of the underlying dynamical system, a convergent numerical method to integrate the system, and the bifurcation diagram to choose specific attractors. Relatively large parameter switches are admitted. As a main result, these switching algorithms (deterministic or random) offer a new perspective on the set of all attractors of a class of dissipative continuous-time dynamical systems. [ABSTRACT FROM AUTHOR]

The paper first illustrates how multilayered tori can arise through one or more pitchfork bifurcations of the saddle cycle on an ordinary resonance torus. The paper hereafter describes three different scenarios by which a multilayered torus can be destructed. One scenario involves a saddle-node bifurcation in which the middle layer of a three-layered torus disappears in an abrupt transition to chaos while the outer-layer manifolds and their associated saddle and unstable-focus cycles continue to exist and to control the transient dynamics. In a second scenario, the unstable focus cycles of the intermediate layers in a five-layered torus turn into unstable nodes, and closed loop connections are established between the unstable nodes and the points of the stable resonance node on the torus. Finally, a third scenario describes a transition in which homoclinic bifurcations destroy first the outer layers and thereafter also the inner layer. The paper also illustrates how the formation and destruction of multilayered tori can occur in the cluster dynamics of an ensemble of globally coupled maps. This leads to three additional scenarios for the destruction of multilayered tori. [ABSTRACT FROM AUTHOR]

Although the sensitivity of sensors can be significantly enhanced using chaotic dynamics due to its extremely sensitive dependence on initial conditions and parameters, how to reconstruct the measured signal from the distorted sensor response becomes challenging. In this paper we suggest an effective method to reconstruct the measured signal from the distorted (chaotic) response of chaos sensors. This measurement signal reconstruction method applies the neural network techniques for system structure identification and therefore does not require the precise information of the sensor’s dynamics. We discuss also how to improve the robustness of reconstruction. Some examples are presented to illustrate the measurement signal reconstruction method suggested. [ABSTRACT FROM AUTHOR]

This paper is focused on the problem of complete synchronization in arrays of externally driven identical or slightly different oscillators. These oscillators are coupled by common driving which makes an occurrence of generalized synchronization between a driving signal and response oscillators possible. Therefore, the phenomenon of generalized synchronization is also analyzed here. The research is concentrated on the cases of an irregular (chaotic or stochastic) driving signal acting on continuous-time (Duffing systems) and discrete-time (Henon maps) response oscillators. As a tool for quantifying the robustness of the synchronized state, response (conditional) Lyapunov exponents are applied. The most significant result presented in this paper is a novel method of estimation of the largest response Lyapunov exponent. This approach is based on the complete synchronization of two twin response subsystems via additional master-slave coupling between them. Examples of the method application and its comparison with the classical algorithm for calculation of Lyapunov exponents are widely demonstrated. Finally, the idea of effective response Lyapunov exponents, which allows us to quantify the synchronizability in case of slightly different response oscillators, is introduced. [ABSTRACT FROM AUTHOR]

Parameter identification of dynamical systems from time series has received increasing interest due to its wide applications in secure communication, pattern recognition, neural networks, and so on. Given the driving system, parameters can be estimated from the time series by using an adaptive control algorithm. Recently, it has been reported that for some stable systems, in which parameters are difficult to be identified [Li et al., Phys Lett. A 333, 269–270 (2004); Remark 5 in Yu and Cao, Physica A 375, 467–482 (2007); and Li et al., Chaos 17, 038101 (2007)], and in this paper, a brief discussion about whether parameters can be identified from time series is investigated. From some detailed analyses, the problem of why parameters of stable systems can be hardly estimated is discussed. Some interesting examples are drawn to verify the proposed analysis. [ABSTRACT FROM AUTHOR]

When a radio frequency signal is radiated by a transmitter, the properties of the transmitter itself affect the properties of the signal. These transmitter-induced changes are known as unintentional modulation, to differentiate them from intentional modulation used to add information to the signal. The unintentional modulation can be used to identify which transmitter produced a signal. This paper shows how phase space analysis based on nonlinear dynamics ideas can be used to determine which of several amplifiers produced a signal. [ABSTRACT FROM AUTHOR]

In this paper we demonstrate how to construct signals (time series) of continuous-time dynamical systems that exhibit a given symbolic dynamics. This is achieved without construction of the ordinary differential equations that generate the flow. This construction is of theoretical interest and is useful as a source of dynamical data that can be used to test various data analysis algorithms. [ABSTRACT FROM AUTHOR]

The industrial demand for machine tools with ever increasing speed and accuracy calls for a closer look at the physical phenomena that are present at small movements of those machine’s slides. One of these phenomena, and probably the most dominant one, is the dependence of the friction force on displacement that can be described by a rate-independent hysteresis function with nonlocal memory. The influence of this highly nonlinear effect on the dynamics of the system has been theoretically analyzed in Part I of this paper. This part (II) aims at verifying these theoretical results on three experimental setups. Two setups, consisting of linearly driven rolling element guideways, have been built to specifically study the hysteretic friction behavior. The experiments performed on these specially designed setups are then repeated on one axis of an industrial pick-and-place device, driven by a linear motor and guided by commercial guideways. The results of the experiments on all the setups agree qualitatively well with the theoretically predicted ones and point to the inherent difficulty of accurate quantitative identification of the hysteretic behavior. They further show that the hysteretic friction behavior has a direct bearing on the dynamics of machine tools and its presence should therefore be carefully considered in the dynamic identification process of these systems. [ABSTRACT FROM AUTHOR]

In this paper different topologies of populations of FitzHugh–Nagumo neurons have been introduced in order to investigate the role played by the noise in the network. Each neuron is subjected to an independent source of noise. In these conditions the behavior of the population depends on the connection among the elements. By analyzing several kinds of topology (ranging from regular to random) different behaviors have been observed. Several topologies behave in an optimal way with respect to the range of noise level leading to an improvement in the stimulus response coherence, while others with respect to the maximum values of the performance index. However, the best results in terms of both the suitable noise level and high stimulus response coherence have been obtained when a diversity in neuron characteristic parameters has been introduced and the neurons have been connected in a small-world topology. [ABSTRACT FROM AUTHOR]

The objective of the present paper is to review some developments that have occurred in detonation theory over the last ten years. They concern nonlinear dynamics of detonation fronts, namely patterns of pulsating and/or cellular fronts, selection of the cell size, dynamical self-quenching, direct (blast) or spontaneous initiation, and transition from deflagration to detonation. These phenomena are all well documented by experiments since the sixties but remained unexplained until recently. In the first part of the paper, the patterns of cellular detonations are described by an asymptotic solution to nonlinear hyperbolic equations (reactive Euler equations) in the form of unsteady (sometime chaotic) and multidimensional traveling-waves. In the second part, turning points of quasi-steady solutions are shown to correspond to critical conditions of fully unsteady problems, either for (direct or spontaneous) initiation or for spontaneous failure (self-quenching). Physical insights are tentatively presented rather than technical aspects. The challenge is to identify the physical mechanisms with their relevant parameters, and more specifically to explain how the length-scales involved in detonation dynamics are larger by two order of magnitude (at least) than the length-scale involved in the steady planar traveling-wave solution (detonation thickness). © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]