ALBERT

Description: The duplication-divergence network model is generally thought to incorporate key ingredients underlying the growth and evolution of protein-protein interaction networks. Properties of the model have been elucidated through numerous simulation studies. However, a comprehensive theoretical study of the model is lacking. Here, we derived analytic expressions for quantities describing key characteristics of the network—the average degree, the degree distribution, the clustering coefficient, and the neighbor connectivity—in the mean-field, large- N limit of an extended version of the model, duplication-divergence complemented with heterodimerization and addition. We carried out extensive simulations and verified excellent agreement between simulation and theory except for one partial case. All four quantities obeyed power-laws even at moderate network size ( N ∼ 10 4 ), except the degree distribution, which had an additional exponential factor observed to obey power-law. It is shown that our network model can lead to the emergence of scale-free property and hierarchical modularity simultaneously, reproducing the important topological properties of real protein-protein interaction networks.

Description: Chimera states are complex spatio-temporal patterns in which domains of synchronous and asynchronous dynamics coexist in coupled systems of oscillators. We examine how the character of the individual elements influences chimera states by studying networks of nonlocally coupled Van der Pol oscillators. Varying the bifurcation parameter of the Van der Pol system, we can interpolate between regular sinusoidal and strongly nonlinear relaxation oscillations and demonstrate that more pronounced nonlinearity induces multi-chimera states with multiple incoherent domains. We show that the stability regimes for multi-chimera states and the mean phase velocity profiles of the oscillators change significantly as the nonlinearity becomes stronger. Furthermore, we reveal the influence of time delay on chimera patterns.

Description: We present the design of an autonomous time-delay Boolean network realized with readily available electronic components. Through simulations and experiments that account for the detailed nonlinear response of each circuit element, we demonstrate that a network with five Boolean nodes displays complex behavior. Furthermore, we show that the dynamics of two identical networks display near-instantaneous synchronization to a periodic state when forced by a common periodic Boolean signal. A theoretical analysis of the network reveals the conditions under which complex behavior is expected in an individual network and the occurrence of synchronization in the forced networks. This research will enable future experiments on autonomous time-delay networks using readily available electronic components with dynamics on a slow enough time-scale so that inexpensive data collection systems can faithfully record the dynamics.

Description: Fractal and multifractal characteristics of self-affine surfaces of BaF 2 thin films, deposited on crystalline Si ⟨1 1 1⟩ substrate at room temperature, were studied. Self-affine surfaces were prepared by irradiation of 120 MeV Ag 9+ ions which modified the surface morphology at nanometer scale. The surface morphology of virgin thin film and those irradiated with different ion fluences are characterized by atomic force microscopy technique. The surface roughness (interface width) shows monotonic decrease with ion fluences, while the other parameters, such as lateral correlation length, roughness exponent, and fractal dimension, did not show either monotonic decrease or increase in nature. The self-affine nature of the films is further confirmed by autocorrelation function. The power spectral density of thin films surfaces exhibits inverse power law variation with spatial frequency, suggesting the existence of fractal component in surface morphology. The multifractal detrended fluctuation analysis based on the partition function approach is also performed on virgin and irradiated thin films. It is found that the partition function exhibits the power law behavior with the segment size. Moreover, it is also seen that the scaling exponents vary nonlinearly with the moment, thereby exhibiting the multifractal nature.

Description: A Lagrangian flow network is constructed for the atmospheric blocking of Eastern Europe and Western Russia in summer 2010. We compute the most probable paths followed by fluid particles, which reveal the Omega -block skeleton of the event. A hierarchy of sets of highly probable paths is introduced to describe transport pathways when the most probable path alone is not representative enough. These sets of paths have the shape of narrow coherent tubes flowing close to the most probable one. Thus, even when the most probable path is not very significant in terms of its probability, it still identifies the geometry of the transport pathways.

Description: The visibility graph method is used to transform time series into complex networks. In this letter, a fast transform algorithm is proposed for obtaining a visibility graph. Based on the strategy of “ divide & conquer ,” the time complexity of the proposed algorithm is raised to O ( n   log   n ) , which is more efficient than the previous basic algorithm whose time complexity is O ( n 2 ).

Description: We construct a two-parameter family of moon-shaped billiard tables with boundary made of two circular arcs. These tables fail the defocusing mechanism and other known mechanisms that guarantee ergodicity and hyperbolicity. We analytically study the stability of some periodic orbits and prove there is a class of billiards in this family with elliptic periodic orbits. These moon billiards can be viewed as generalization of annular billiards, which all have Kolmogorov-Arnold-Moser islands. However, the novelty of this paper is that by varying the parameters, we numerically observe a subclass of moon-shaped billiards with a single ergodic component.

Description: A scheme is proposed to classify the basins for attractors of dynamical systems in arbitrary dimensions. There are four basic classes depending on their size and extent, and each class can be further quantified to facilitate comparisons. The calculation uses a Monte Carlo method and is applied to numerous common dissipative chaotic maps and flows in various dimensions.

Description: Random rectangular graphs (RRGs) represent a generalization of the random geometric graphs in which the nodes are embedded into hyperrectangles instead of on hypercubes. The synchronizability of RRG model is studied. Both upper and lower bounds of the eigenratio of the network Laplacian matrix are determined analytically. It is proven that as the rectangular network is more elongated, the network becomes harder to synchronize. The synchronization processing behavior of a RRG network of chaotic Lorenz system nodes is numerically investigated, showing complete consistence with the theoretical results.

Description: We rigorously show that dissipatively driven Frenkel-Kontorova models with either uniform or time-periodic driving asymptotically synchronize for a wide range of initial conditions. The main tool is a new Lyapunov function, as well as a 2D representation of the attractor. We then characterize dynamical phase transitions and outline new algorithms for determining them.

Description: It is a wide-spread convention to identify repelling Lagrangian Coherent Structures (LCSs) with ridges of the forward finite-time Lyapunov exponent (FTLE) field and to identify attracting LCSs with ridges of the backward FTLE. However, we show that, in two-dimensional incompressible flows, also attracting LCSs appear as ridges of the forward FTLE field. This raises the issue of the characterization of attracting LCSs using a forward finite-time Lyapunov analysis. To this end, we extend recent results regarding the relationship between forward and backward maximal and minimal FTLEs, to both the whole finite-time Lyapunov spectrum and to stretch directions. This is accomplished by considering the singular value decomposition (SVD) of the linearized flow map. By virtue of geometrical insights from the SVD, we provide characterizations of attracting LCSs in forward time for two geometric approaches to hyperbolic LCSs. We apply these results to the attracting FTLE ridge of the incompressible saddle flow.

Description: We consider dynamics of magnetic billiards with curved boundaries and strong inhomogeneous magnetic field. We investigate a violation of adiabaticity of charged particle motion in this system. The destruction of the adiabatic invariance is due to the change of type of the particle trajectory: particles can drift along the boundary reflecting from it or rotate around the magnetic field at some distance from the boundary without collisions with it. Trajectories of these two types are demarcated in the phase space by a separatrix. Crossings of the separatrix result in jumps of the adiabatic invariant. We derive an asymptotic formula for such a jump and demonstrate that an accumulation of these jumps leads to the destruction of the adiabatic invariance.

Description: While more rigorous and sophisticated methods for identifying Lagrangian based coherent structures exist, the finite-time Lyapunov exponent (FTLE) field remains a straightforward and popular method for gaining some insight into transport by complex, time-dependent two-dimensional flows. In light of its enduring appeal, and in support of good practice, we begin by investigating the effects of discretization and noise on two numerical approaches for calculating the FTLE field. A practical method to extract and refine FTLE ridges in two-dimensional flows, which builds on previous methods, is then presented. Seeking to better ascertain the role of a FTLE ridge in flow transport, we adapt an existing classification scheme and provide a thorough treatment of the challenges of classifying the types of deformation represented by a FTLE ridge. As a practical demonstration, the methods are applied to an ocean surface velocity field data set generated by a numerical model.

Description: We present a novel computational method to estimate the topological entropy and Lyapunov exponent of nonlinear maps using a shift transform. Unlike the computation of periodic orbits or the symbolic dynamical approach by the Markov partition, the method presented here does not require any special techniques in computational and mathematical fields to calculate these quantities. In spite of its simplicity, our method can accurately capture not only the chaotic region but also the non-chaotic region (window region) such that it is important physically but the (Lebesgue) measure zero and usually hard to calculate or observe. Furthermore, it is shown that the Kolmogorov-Sinai entropy of the Sinai-Ruelle-Bowen measure (the physical measure) coincides with the topological entropy.

Description: Switching dynamics among saddles in a network of nonlinear oscillators can be exploited for information encoding and processing (hence computing), but stable attractors in the system can terminate the switching behavior. An effective control strategy is presented to sustain switching dynamics in networks of pulse-coupled oscillators. The support for the switching behavior is a set of saddles, or unstable invariant sets in the phase space. We thus identify saddles with a common property, localize the system in the vicinity of them, and then guide the system from one metastable state to another to generate desired switching dynamics. We demonstrate that the control method successfully generates persistent switching trajectories and prevents the system from entering stable attractors. In addition, there exists correspondence between the network structure and the switching dynamics, providing fundamental insights on the development of a computing paradigm based on the switching dynamics.

Description: Many complex networks possess a scale-free vertex-degree distribution in a power-law form of c k − γ , where k is the vertex-degree variable and c and γ are constants. To better understand the mechanism of the power-law formation in scale-free networks, it is important to understand and analyze their vertex-degree sequences. We had shown before that, for a scale-free network of size N , if its vertex-degree sequence is k 1 〈 k 2 〈 ⋯ 〈 k l , where { k 1 , k 2 , ... , k l } is the set of all non-equal vertex degrees in the network, and if its power exponent satisfies γ 〉 1 , then the length l of the vertex-degree sequence is of order log   N . In the present paper, we further study complex networks with a more general vertex-degree distribution, not restricted to the power-law, and prove that the same conclusion holds as well. In addition, we verify the new result by real data from a large number of real-world examples. We finally discuss some potential applications of the new finding in various fields of science, technology, and society.

Description: In stem cell differentiation, a pluripotent stem cell becomes progressively specialized and generates specific cell types through a series of epigenetic processes. How cells can precisely determine their fate in a fluctuating environment is a currently unsolved problem. In this paper, we suggest an abstract gene regulatory network to describe mathematically the differentiation phenomenon featuring stochasticity, divergent cell fates, and robustness. The network consists of three functional motifs: an upstream chaotic motif, a buffering motif of incoherent feed forward loop capable of generating a pulse, and a downstream motif which is bistable. The dynamic behavior is typically a transient chaos with fractal basin boundaries. The trajectories take transiently chaotic journeys before divergently settling down to the bistable states. The ratio of the probability that the high state is achieved to the probability that the low state is reached can maintain a constant in a population of cells with varied molecular fluctuations. The ratio can be turned up or down when proper parameters are adjusted. The model suggests a possible mechanism for the robustness against fluctuations that is prominently featured in pluripotent cell differentiations and developmental phenomena.

Description: Analytical solutions of the period-four orbits exhibited by a classical family of n -dimensional quadratic maps are presented. Exact expressions are obtained by applying harmonic balance and Gröbner bases to a single-input single-output representation of the system. A detailed study of a generalized scalar quadratic map and a well-known delayed logistic model is included for illustration. In the former example, conditions for the existence of bistability phenomenon are also introduced.

Description: In this article, we show that if the nonlinear Schrödinger (NLS) equation is generalized by simultaneously taking into account higher-order dispersion, a quintic nonlinearity, and self-steepening terms, the resulting equation is interesting as it has exact soliton solutions which may be (depending on the values of the coefficients) stable or unstable, standard or “embedded,” fixed or “moving” (i.e., solitons which advance along the retarded-time axis). We investigate the stability of these solitons by means of a modified version of the Vakhitov-Kolokolov criterion, and numerical tests are carried out to corroborate that these solitons respond differently to perturbations. It is shown that this generalized NLS equation can be derived from a Lagrangian density which contains an auxiliary variable, and Noether's theorem is then used to show that the invariance of the action integral under infinitesimal gauge transformations generates a whole family of conserved quantities. Finally, we study if this equation has the Painlevé property.

Description: Switches in real systems take many forms, such as impacts, electronic relays, mitosis, and the implementation of decisions or control strategies. To understand what is lost, and what can be retained, when we model a switch as an instantaneous event, requires a consideration of so-called hidden terms. These are asymptotically vanishing outside the switch, but can be encoded in the form of nonlinear switching terms. A general expression for the switch can be developed in the form of a series of sigmoid functions. We review the key steps in extending Filippov's method of sliding modes to such systems. We show how even slight nonlinear effects can hugely alter the behaviour of an electronic control circuit, and lead to “hidden” attractors inside the switching surface.

Description: We present experimental results obtained under normal gravity on the dynamics of solid particles in periodic oscillatory thermocapillary-driven flows in a non-isothermal liquid bridge made of decane. Inertial particles of different densities and in the size range approximately 0.75 − 75 μ m are able to form stable coherent structures (particle accumulation structures, or PASs). Two image processing techniques were developed and successfully applied to compute time required for an ensemble of particles to form a structure. It is shown that the formation time grows with the decrease of the Stokes number. The observations indicate the probable irrelevance of the memory term for these experiments. Two types of PAS were observed—single (SL-I) and double-loop (SL-II)—which sometimes co-existed. Only large or very dense particles may form an SL-II type structure. A number of novel features of the system were perceived. In some cases, intermittently stable structures emerged (their dynamics is characterized by alternating time intervals during which a structure exists and is destroyed). Whereas in most experiments we observed a conventional symmetric and centered PAS, there were cases when a long-term stable asymmetric structure appeared. Experiments wherein two different types of PAS-forming particles were used simultaneously revealed the destructive role of collisions between the particles on formation of structures.

Description: In this paper, a new memristor-based multi-scroll hyper-chaotic system is designed. The proposed memristor-based system possesses multiple complex dynamic behaviors compared with other chaotic systems. Various coexisting attractors and hidden coexisting attractors are observed in this system, which means extreme multistability arises. Besides, by adjusting parameters of the system, this chaotic system can perform single-scroll attractors, double-scroll attractors, and four-scroll attractors. Basic dynamic characteristics of the system are investigated, including equilibrium points and stability, bifurcation diagrams, Lyapunov exponents, and so on. In addition, the presented system is also realized by an analog circuit to confirm the correction of the numerical simulations.

Description: This paper deals with the stability and bifurcation analysis of a general form of equation D α x ( t ) = g ( x ( t ) , x ( t − τ ) ) involving the derivative of order α ∈ (0, 1] and a constant delay τ  ≥ 0. The stability of equilibrium points is presented in terms of the stability regions and critical surfaces. We provide a necessary condition to exist chaos in the system also. A wide range of delay differential equations involving a constant delay can be analyzed using the results proposed in this paper. The illustrative examples are provided to explain the theory.

Description: In this paper, we propose a new consensus model in which the interactions among agents stochastically switch between attraction and repulsion. Such a positive-and-negative mechanism is described by the white-noise-based coupling. Analytic criteria for the consensus and non-consensus in terms of the eigenvalues of the noise intensity matrix are derived, which provide a better understanding of the constructive roles of random interactions. Specifically, we discover a positive role of noise coupling that noise can accelerate the emergence of consensus. We find that the converging speed of the multi-agent network depends on the square of the second smallest eigenvalue of its graph Laplacian. The influence of network topologies on the consensus time is also investigated.

Description: After having the issues of singularity and locality addressed recently in mathematical modelling, another question regarding the description of natural phenomena was raised: How influent is the second parameter β of the two-parameter Mittag-Leffler function E α , β ( z ) ,   z ∈ ℂ ? To answer this question, we generalize the newly introduced one-parameter derivative with non-singular and non-local kernel [A. Atangana and I. Koca, Chaos, Solitons Fractals 89 , 447 (2016); A. Atangana and D. Bealeanu (e-print)] by developing a similar two-parameter derivative with non-singular and non-local kernel based on E α , β ( z ). We exploit the Agarwal/Erdelyi higher transcendental functions together with their Laplace transforms to explicitly establish the Laplace transform's expressions of the two-parameter derivatives, necessary for solving related fractional differential equations. Explicit expression of the associated two-parameter fractional integral is also established. Concrete applications are done on atmospheric convection process by using Lorenz non-linear simple system. Existence result for the model is provided and a numerical scheme established. As expected, solutions exhibit chaotic behaviors for α less than 0.55, and this chaos is not interrupted by the impact of β . Rather, this second parameter seems to indirectly squeeze and rotate the solutions, giving an impression of twisting. The whole graphics seem to have completely changed its orientation to a particular direction. This is a great observation that clearly shows the substantial impact of the second parameter of E α , β ( z ), certainly opening new doors to modeling with two-parameter derivatives.

Description: The total electrostatic energy of systems of identical particles of equal charge is studied in configurations bounded in space, but divergent in the number of charges. This approach shall guide us to unveil a non-linear, functional form specifying the divergent nature of system energy. We consider fractals to be physical entities, with charges located in their vertices or nodes. This description is interesting since features, such as the corresponding fractal dimension, can characterize the total energy E N . Finally, at local length scales, we describe how energy diverges at charge accumulation points in the fractal, that is, almost everywhere by definition.

Description: The Kuramoto–Sakaguchi system of coupled phase oscillators, where interaction between oscillators is determined by a single harmonic of phase differences of pairs of oscillators, has very simple emergent dynamics in the case of identical oscillators that are globally coupled: there is a variational structure that means the only attractors are full synchrony (in-phase) or splay phase (rotating wave/full asynchrony) oscillations and the bifurcation between these states is highly degenerate. Here we show that nonpairwise coupling—including three and four-way interactions of the oscillator phases—that appears generically at the next order in normal-form based calculations can give rise to complex emergent dynamics in symmetric phase oscillator networks. In particular, we show that chaos can appear in the smallest possible dimension of four coupled phase oscillators for a range of parameter values.

Description: Basin of attraction of a stable equilibrium point is an effective concept for stability analysis in deterministic systems; however, it does not contain information on the external perturbations that may affect it. Here we introduce the concept of stochastic basin of attraction (SBA) by incorporating a suitable probabilistic notion of basin. We define criteria for the size of the SBA based on the escape probability, which is one of the deterministic quantities that carry dynamical information and can be used to quantify dynamical behavior of the corresponding stochastic basin of attraction. SBA is an efficient tool to describe the metastable phenomena complementing the known exit time, escape probability, or relaxation time. Moreover, the geometric structure of SBA gives additional insight into the system's dynamical behavior, which is important for theoretical and practical reasons. This concept can be used not only in models with small noise intensity but also with noise whose amplitude is proportional or in general is a function of an order parameter. As an application of our main results, we analyze a three potential well system perturbed by two types of noise: Brownian motion and non-Gaussian α -stable Lévy motion. Our main conclusions are that the thermal fluctuations stabilize the metastable system with an asymmetric three-well potential but have the opposite effect for a symmetric one. For Lévy noise with larger jumps and lower jump frequencies ( α = 0.5 ) metastability is enhanced for both symmetric and asymmetric potentials.

Description: We introduce mixing with piecewise isometries (PWIs) on a hemispherical shell, which mimics features of mixing by cutting and shuffling in spherical shells half-filled with granular media. For each PWI, there is an inherent structure on the hemispherical shell known as the exceptional set E , and a particular subset of E , E + , provides insight into how the structure affects mixing. Computer simulations of PWIs are used to visualize mixing and approximations of E + to demonstrate their connection. While initial conditions of unmixed materials add a layer of complexity, the inherent structure of E + defines fundamental aspects of mixing by cutting and shuffling.

Description: In stochastic systems, one is often interested in finding the optimal path that maximizes the probability of escape from a metastable state or of switching between metastable states. Even for simple systems, it may be impossible to find an analytic form of the optimal path, and in high-dimensional systems, this is almost always the case. In this article, we formulate a constructive methodology that is used to compute the optimal path numerically. The method utilizes finite-time Lyapunov exponents, statistical selection criteria, and a Newton-based iterative minimizing scheme. The method is applied to four examples. The first example is a two-dimensional system that describes a single population with internal noise. This model has an analytical solution for the optimal path. The numerical solution found using our computational method agrees well with the analytical result. The second example is a more complicated four-dimensional system where our numerical method must be used to find the optimal path. The third example, although a seemingly simple two-dimensional system, demonstrates the success of our method in finding the optimal path where other numerical methods are known to fail. In the fourth example, the optimal path lies in six-dimensional space and demonstrates the power of our method in computing paths in higher-dimensional spaces.

Description: The relation between the Fisher information and Rényi dimensions is established: the Fisher information can be expressed as a linear combination of the first and second derivatives of the Rényi dimensions with respect to the Rényi parameter β . The Rényi parameter β is the parameter of the Fisher information. A thermodynamical description based on the Fisher information with β being the inverse temperature is introduced for chaotic systems. The link between the Fisher information and the heat capacity is emphasized, and the Fisher heat capacity is introduced.

Description: A Riesz difference is defined by the use of the Riemann–Liouville differences on time scales. Then the definition is considered for discrete fractional modelling. A lattice fractional equation method is proposed among which the space variable is defined on discrete domains. Finite memory effects are introduced into the lattice system and the numerical formulae are given. Adomian decomposition method is adopted to solve the fractional partial difference equations numerically.

Description: We present a universal characterization scheme for chimera states applicable to both numerical and experimental data sets. The scheme is based on two correlation measures that enable a meaningful definition of chimera states as well as their classification into three categories: stationary , turbulent , and breathing . In addition, these categories can be further subdivided according to the time-stationarity of these two measures. We demonstrate that this approach is both consistent with previously recognized chimera states and enables us to classify states as chimeras which have not been categorized as such before. Furthermore, the scheme allows for a qualitative and quantitative comparison of experimental chimeras with chimeras obtained through numerical simulations.

Description: A large parameter mismatch can induce amplitude death in two instantaneously coupled oscillators. Alternatively, a time delay in the coupling can induce amplitude death in two identical oscillators. We unify the mechanism of quenching of oscillation in coupled oscillators, either by a large parameter mismatch or a delay coupling, by a common lag scenario that is, surprisingly, different from the conventional lag synchronization. We present numerical as well as experimental evidence of this unknown kind of lag scenario when the lag increases with coupling and at a critically large value at a critical coupling strength, amplitude death emerges in two largely mismatched oscillators. This is analogous to amplitude death in identical systems with increasingly large coupling delay. In support, we use examples of the Chua oscillator and the Bonhoeffer-van der Pol system. Furthermore, we confirm this lag scenario during the onset of amplitude death in identical Stuart-Landau system under various instantaneous coupling forms, repulsive, conjugate, and a type of nonlinear coupling.

Description: Nonlinear oscillations lie at the heart of numerous complex systems. Impulsive forcing arises naturally in many scenarios, and we endeavour to study nonlinear oscillators subject to such forcing. We model these kicked oscillatory systems as a piecewise smooth dynamical system, whereby their dynamics can be investigated. We investigate the problem of pattern formation in a turbulent combustion system and apply this formalism with the aim of explaining the observed dynamics. We identify that the transition of this system from low amplitude chaotic oscillations to large amplitude periodic oscillations is the result of a discontinuity induced bifurcation. Further, we provide an explanation for the occurrence of intermittent oscillations in the system.

Description: Following the long-lived qualitative-dynamics tradition of explaining behavior in complex systems via the architecture of their attractors and basins, we investigate the patterns of switching between distinct trajectories in a network of synchronized oscillators. Our system, consisting of nonlinear amplitude-phase oscillators arranged in a ring topology with reactive nearest-neighbor coupling, is simple and connects directly to experimental realizations. We seek to understand how the multiple stable synchronized states connect to each other in state space by applying Gaussian white noise to each of the oscillators' phases. To do this, we first analytically identify a set of locally stable limit cycles at any given coupling strength. For each of these attracting states, we analyze the effect of weak noise via the covariance matrix of deviations around those attractors. We then explore the noise-induced attractor switching behavior via numerical investigations. For a ring of three oscillators, we find that an attractor-switching event is always accompanied by the crossing of two adjacent oscillators' phases. For larger numbers of oscillators, we find that the distribution of times required to stochastically leave a given state falls off exponentially, and we build an attractor switching network out of the destination states as a coarse-grained description of the high-dimensional attractor-basin architecture.

Description: This paper addresses the need to deal with and control public opinion and rumors. Existing strategies to control public opinion include degree, random, and adaptive bridge control strategies. In this paper, we use the HK model to present a public opinion control strategy based on public authority (PA). This means utilizing the influence of expert or high authority individuals whose opinions we control to obtain the optimum effect in the shortest time possible and thus reach a consensus of public opinion. Public authority (PA) is only influenced by individuals' attributes (age, economic status, and education level) and not their degree distribution; hence, in this paper, we assume that PA complies with two types of public authority distribution (normal and power-law). According to the proposed control strategy, our experiment is based on random, degree, and public authority control strategies in three different social networks (small-world, scale-free, and random) and we compare and analyze the strategies in terms of convergence time ( T ), final number of controlled agents ( C ), and comprehensive efficiency ( E ). We find that different network topologies and the distribution of the PA in the network can influence the final controlling effect. While the effect of PA strategy differs in different network topology structures, all structures achieve comprehensive efficiency with any kind of public authority distribution in any network. Our findings are consistent with several current sociological phenomena and show that in the process of public opinion/rumor control, considerable attention should be paid to high authority individuals.

Description: Cells in the brain's Suprachiasmatic Nucleus (SCN) are known to regulate circadian rhythms in mammals. We model synchronization of SCN cells using the forced Kuramoto model, which consists of a large population of coupled phase oscillators (modeling individual SCN cells) with heterogeneous intrinsic frequencies and external periodic forcing. Here, the periodic forcing models diurnally varying external inputs such as sunrise, sunset, and alarm clocks. We reduce the dimensionality of the system using the ansatz of Ott and Antonsen and then study the effect of a sudden change of clock phase to simulate cross-time-zone travel. We estimate model parameters from previous biological experiments. By examining the phase space dynamics of the model, we study the mechanism leading to the difference typically experienced in the severity of jet-lag resulting from eastward and westward travel.

Description: The control of network-coupled nonlinear dynamical systems is an active area of research in the nonlinear science community. Coupled oscillator networks represent a particularly important family of nonlinear systems, with applications ranging from the power grid to cardiac excitation. Here, we study the control of network-coupled limit cycle oscillators, extending the previous work that focused on phase oscillators. Based on stabilizing a target fixed point, our method aims to attain complete frequency synchronization, i.e., consensus, by applying control to as few oscillators as possible. We develop two types of controls. The first type directs oscillators towards larger amplitudes, while the second does not. We present numerical examples of both control types and comment on the potential failures of the method.

Description: The Poisson white noise, as a typical non-Gaussian excitation, has attracted much attention recently. However, little work was referred to the study of stochastic systems with fractional derivative under Poisson white noise excitation. This paper investigates the stationary response of a class of quasi-linear systems with fractional derivative excited by Poisson white noise. The equivalent stochastic system of the original stochastic system is obtained. Then, approximate stationary solutions are obtained with the help of the perturbation method. Finally, two typical examples are discussed in detail to demonstrate the effectiveness of the proposed method. The analysis also shows that the fractional order and the fractional coefficient significantly affect the responses of the stochastic systems with fractional derivative.

Description: We construct a piecewise-linear (PWL) approximation of the Hindmarsh-Rose (HR) neuron model that is minimal, in the sense that the vector field has the least number of linearity zones, in order to reproduce all the dynamics present in the original HR model with classical parameter values. This includes square-wave bursting and also special trajectories called canards, which possess long repelling segments and organise the transitions between stable bursting patterns with n and n  + 1 spikes, also referred to as spike-adding canard explosions. We propose a first approximation of the smooth HR model, using a continuous PWL system, and show that its fast subsystem cannot possess a homoclinic bifurcation, which is necessary to obtain proper square-wave bursting. We then relax the assumption of continuity of the vector field across all zones, and we show that we can obtain a homoclinic bifurcation in the fast subsystem. We use the recently developed canard theory for PWL systems in order to reproduce the spike-adding canard explosion feature of the HR model as studied, e.g., in Desroches et al. , Chaos 23 (4), 046106 (2013).

Description: This paper proposes a novel approach for generating multi-scroll chaotic attractors in multi-directions for fractional-order (FO) systems. The stair nonlinear function series and the saturated nonlinear function are combined to extend equilibrium points with index 2 in a new FO linear system. With the help of stability theory of FO systems, stability of its equilibrium points is analyzed, and the chaotic behaviors are validated through phase portraits, Lyapunov exponents, and Poincaré section. Choosing the order 0.96 as an example, a circuit for generating 2-D grid multiscroll chaotic attractors is designed, and 2-D 9 × 9 grid FO attractors are observed at most. Numerical simulations and circuit experimental results show that the method is feasible and the designed circuit is correct.

Description: We develop a general classification of the infinite number of families of solitons and soliton complexes in the one-dimensional Gross-Pitaevskii/nonlinear Schrödinger equation with a nonlinear lattice pseudopotential , i.e., periodically modulated coefficient in front of the cubic term, which takes both positive and negative local values. This model finds direct implementations in atomic Bose-Einstein condensates and nonlinear optics. The most essential finding is the existence of two branches of dipole solitons (DSs), which feature an antisymmetric shape, being essentially squeezed into a single cell of the nonlinear lattice. This soliton species was not previously considered in nonlinear lattices. We demonstrate that one branch of the DS family (namely, which obeys the Vakhitov-Kolokolov criterion) is stable , while unstable DSs spontaneously transform into stable fundamental solitons (FSs). The results are obtained in numerical and approximate analytical forms, the latter based on the variational approximation. Some stable bound states of FSs are found too.

Description: Global bifurcations include sudden changes in chaotic sets due to crises. There are three types of crises defined by Grebogi et al. [Physica D 7 , 181 (1983)]: boundary crisis, interior crisis, and metamorphosis. In this paper, by means of the extended generalized cell mapping (EGCM), boundary and interior crises of a fractional-order Duffing system are studied as one of the system parameters or the fractional derivative order is varied. It is found that a crisis can be generally defined as a collision between a chaotic basic set and a basic set, either periodic or chaotic, to cause a sudden discontinuous change in chaotic sets. Here chaotic sets involve three different kinds: a chaotic attractor, a chaotic saddle on a fractal basin boundary, and a chaotic saddle in the interior of a basin and disjoint from the attractor. A boundary crisis results from the collision of a periodic (or chaotic) attractor with a chaotic (or regular) saddle in the fractal (or smooth) boundary. In such a case, the attractor, together with its basin of attraction, is suddenly destroyed as the control parameter passes through a critical value, leaving behind a chaotic saddle in the place of the original attractor and saddle after the crisis. An interior crisis happens when an unstable chaotic set in the basin of attraction collides with a periodic attractor, which causes the appearance of a new chaotic attractor, while the original attractor and the unstable chaotic set are converted to the part of the chaotic attractor after the crisis. These results further demonstrate that the EGCM is a powerful tool to reveal the mechanism of crises in fractional-order systems.

Description: This paper generalizes the stability test method via integral estimation for integer-order neutral time-delay systems to neutral fractional-delay systems. The key step in stability test is the calculation of the number of unstable characteristic roots that is described by a definite integral over an interval from zero to a sufficient large upper limit. Algorithms for correctly estimating the upper limits of the integral are given in two concise ways, parameter dependent or independent. A special feature of the proposed method is that it judges the stability of fractional-delay systems simply by using rough integral estimation. Meanwhile, the paper shows that for some neutral fractional-delay systems, the stability is extremely sensitive to the change of time delays. Examples are given for demonstrating the proposed method as well as the delay sensitivity.

Description: This paper focuses on impulsive synchronization of fractional Takagi-Sugeno (T-S) fuzzy complex networks. A novel comparison principle is built for the fractional impulsive system. Then a synchronization criterion is established for the fractional T-S fuzzy complex networks by utilizing the comparison principle. The method is also illustrated by applying the fractional T-S fuzzy Rössler's complex networks.

Description: In this paper, the computation schemes for periodic solutions of the forced fractional-order Mathieu-Duffing equation are derived based on incremental harmonic balance (IHB) method. The general forms of periodic solutions are founded by the IHB method, which could be useful to obtain the periodic solutions with higher precision. The comparisons of the approximate analytical solutions by the IHB method and numerical integration are fulfilled, and the results certify the correctness and higher precision of the solutions by the IHB method. The dynamical analysis of strongly nonlinear fractional-order Mathieu-Duffing equation is investigated by the IHB method. Then, the effects of the excitation frequency, fractional order, fractional coefficient, and nonlinear stiffness coefficient on the complex dynamical behaviors are analyzed. At last, the detailed results are summarized and the conclusions are made, which present some useful information to analyze and/or control the dynamical response of this kind of system.

Description: Tempered fractional processes offer a useful extension for turbulence to include low frequencies. In this paper, we investigate the stochastic phenomenological bifurcation, or stochastic P-bifurcation, of the Langevin equation perturbed by tempered fractional Brownian motion. However, most standard tools from the well-studied framework of random dynamical systems cannot be applied to systems driven by non-Markovian noise, so it is desirable to construct possible approaches in a non-Markovian framework. We first derive the spectral density function of the considered system based on the generalized Parseval's formula and the Wiener-Khinchin theorem. Then we show that it enjoys interesting and diverse bifurcation phenomena exchanging between or among explosive-like, unimodal, and bimodal kurtosis. Therefore, our procedures in this paper are not merely comparable in scope to the existing theory of Markovian systems but also provide a possible approach to discern P-bifurcation dynamics in the non-Markovian settings.

Description: Synchronization is the process of achieving identical dynamics among coupled identical units. If the units are different from each other, their dynamics cannot become identical; yet, after transients, there may emerge a functional relationship between them—a phenomenon termed “generalized synchronization.” Here, we show that the concept of transient uncoupling, recently introduced for synchronizing identical units, also supports generalized synchronization among nonidentical chaotic units. Generalized synchronization can be achieved by transient uncoupling even when it is impossible by regular coupling. We furthermore demonstrate that transient uncoupling stabilizes synchronization in the presence of common noise. Transient uncoupling works best if the units stay uncoupled whenever the driven orbit visits regions that are locally diverging in its phase space. Thus, to select a favorable uncoupling region, we propose an intuitive method that measures the local divergence at the phase points of the driven unit's trajectory by linearizing the flow and subsequently suppresses the divergence by uncoupling.

Description: The stable and unstable manifolds of an invariant set of a piecewise-smooth map are themselves piecewise-smooth. Consequently, as parameters of a piecewise-smooth map are varied, an invariant set can develop a homoclinic connection when its stable manifold intersects a non-differentiable point of its unstable manifold (or vice-versa). This is a codimension-one bifurcation analogous to a homoclinic tangency of a smooth map, referred to here as a homoclinic corner. This paper presents an unfolding of generic homoclinic corners for saddle fixed points of planar piecewise-smooth continuous maps. It is shown that a sequence of border-collision bifurcations limits to a homoclinic corner and that all nearby periodic solutions are unstable.

Description: In this study, we investigate the role of the mesoscopic structural properties of a scale-free social network on the contagion spreading. We focus on both the exponent of power-law community size distribution function ( β ) and the mixing parameter ( μ ). Findings show that increasing β reduces the rate of epidemic spreading. On the other hand, increasing μ increases the rate of epidemic spreading. Two innovating parameters, Temperature and cos   θ , are introduced here to analyze these effects.

Description: Spike-time correlations of neighbouring neurons depend on their intrinsic firing properties as well as on the inputs they share. Studies have shown that periodically firing neurons, when subjected to random shared input, exhibit asynchronicity. Here, we study the effect of random shared input on the synchronization of weakly coupled chaotic neurons. The cases of so-called electrical and chemical coupling are both considered, and we observe a wide range of synchronization behaviour. When subjected to identical shared random input, there is a decrease in the threshold coupling strength needed for chaotic neurons to synchronize in-phase. The system also supports lag–synchronous states, and for these, we find that shared input can cause desynchronization. We carry out a master stability function analysis for a network of such neurons and show agreement with the numerical simulations. The contrasting role of shared random input for complete and lag synchronized neurons is useful in understanding spike-time correlations observed in many areas of the brain.

Description: We consider the effects of several forms of delays on the existence and stability of travelling waves in non-locally coupled networks of Kuramoto-type phase oscillators and theta neurons. By passing to the continuum limit and using the Ott/Antonsen ansatz, we derive evolution equations for a spatially dependent order parameter. For phase oscillator networks, the travelling waves take the form of uniformly twisted waves, and these can often be characterised analytically. For networks of theta neurons, the waves are studied numerically.

Description: In this paper, transverse vibrations of an electrostatically actuated thin flexible cantilever perturbed by low-speed air flow are studied using both experiments and numerical modeling. In the experiments, the dynamic characteristics of the cantilever are studied by supplying a DC voltage with an AC component for electrostatic forcing and a constant uniform air flow around the cantilever system for aerodynamic forcing. A range of control parameters leading to stable vibrations are established using a dimensionless operating parameter that is the ratio of the induced and the free stream velocities. Numerical results are validated with experimental data. Assuming the amplitude of vibrations are small, then a non-linear dynamic Euler-Bernoulli beam equation with viscous damping and gravitational effects is used to model the equation of motion. Aerodynamic forcing is modelled as a temporally sinusoidal and uniform force acting perpendicular to the beam length. The forcing amplitude is found to be proportional to the square of the air flow velocity. Numerical results strongly agree with the experiments predicting accurate vibration amplitude, displacement frequency, and quasi-periodic displacement of the cantilever tip.

Description: We study optimal synchronization of networks of coupled phase oscillators. We extend previous theory for optimizing the synchronization properties of undirected networks to the important case of directed networks. We derive a generalized synchrony alignment function that encodes the interplay between the network structure and the oscillators' natural frequencies and serves as an objective measure for the network's degree of synchronization. Using the generalized synchrony alignment function, we show that a network's synchronization properties can be systematically optimized. This framework also allows us to study the properties of synchrony-optimized networks, and in particular, investigate the role of directed network properties such as nodal in- and out-degrees. For instance, we find that in optimally rewired networks, the heterogeneity of the in-degree distribution roughly matches the heterogeneity of the natural frequency distribution, but no such relationship emerges for out-degrees. We also observe that a network's synchronization properties are promoted by a strong correlation between the nodal in-degrees and the natural frequencies of oscillators, whereas the relationship between the nodal out-degrees and the natural frequencies has comparatively little effect. This result is supported by our theory, which indicates that synchronization is promoted by a strong alignment of the natural frequencies with the left singular vectors corresponding to the largest singular values of the Laplacian matrix.

Description: We investigate synchronization in complex networks of noisy phase oscillators. We find that, while too weak a coupling is not sufficient for the whole system to synchronize, too strong a coupling induces a nontrivial type of phase slip among oscillators, resulting in synchronization failure. Thus, an intermediate coupling range for synchronization exists, which becomes narrower when the network is more heterogeneous. Analyses of two noisy oscillators reveal that nontrivial phase slip is a generic phenomenon when noise is present and coupling is strong. Therefore, the low synchronizability of heterogeneous networks can be understood as a result of the difference in effective coupling strength among oscillators with different degrees; oscillators with high degrees tend to undergo phase slip while those with low degrees have weak coupling strengths that are insufficient for synchronization.

Description: We study the dynamics of coupled phase oscillators on a two-dimensional Kuramoto lattice with periodic boundary conditions. For coupling strengths just below the transition to global phase-locking, we find localized spatiotemporal patterns that we call “frequency spirals.” These patterns cannot be seen under time averaging; they become visible only when we examine the spatial variation of the oscillators' instantaneous frequencies, where they manifest themselves as two-armed rotating spirals. In the more familiar phase representation, they appear as wobbly periodic patterns surrounding a phase vortex. Unlike the stationary phase vortices seen in magnetic spin systems, or the rotating spiral waves seen in reaction-diffusion systems, frequency spirals librate: the phases of the oscillators surrounding the central vortex move forward and then backward, executing a periodic motion with zero winding number. We construct the simplest frequency spiral and characterize its properties using analytical and numerical methods. Simulations show that frequency spirals in large lattices behave much like this simple prototype.

Description: A “chimera state” is a dynamical pattern that occurs in a network of coupled identical oscillators when the symmetry of the oscillator population is broken into synchronous and asynchronous parts. We report the experimental observation of chimera and cluster states in a network of four globally coupled chaotic opto-electronic oscillators. This is the minimal network that can support chimera states, and our study provides new insight into the fundamental mechanisms underlying their formation. We use a unified approach to determine the stability of all the observed partially synchronous patterns, highlighting the close relationship between chimera and cluster states as belonging to the broader phenomenon of partial synchronization. Our approach is general in terms of network size and connectivity. We also find that chimera states often appear in regions of multistability between global, cluster, and desynchronized states.

Description: The existence of localized spin excitations and spin deviations along the site in a one-dimensional antiferromagnet with Dzyaloshinski-Moriya (D-M) interaction has been studied using quasiclassical approximation. By introducing the Holstein-Primakoff bosonic representation of spin operators, the coherent state ansatz, and the time dependent variational principle, a discrete set of coupled nonlinear partial differential equations governing the dynamics is derived. Employing the multiple-scale method, one, two and three solitary wave solutions are constructed and depicted graphically.

Description: We survey general results relating patterns of synchrony to network topology, applying the formalism of coupled cell systems. We also discuss patterns of phase-locking for periodic states, where cells have identical waveforms but regularly spaced phases. We focus on rigid patterns, which are not changed by small perturbations of the differential equation. Symmetry is one mechanism that creates patterns of synchrony and phase-locking. In general networks, there is another: balanced colorings of the cells. A symmetric network may have anomalous patterns of synchrony and phase-locking that are not consequences of symmetry. We introduce basic notions on coupled cell networks and their associated systems of admissible differential equations. Periodic states also possess spatio-temporal symmetries, leading to phase relations; these are classified by the H / K theorem and its analog for general networks. Systematic general methods for computing the stability of synchronous states exist for symmetric networks, but stability in general networks requires methods adapted to special classes of model equations.

Description: A hybrid multi-agent systems model integrating the advantages of both metric interaction and topological interaction rules, called the metric-topological model, is developed. This model describes planar motions of mobile agents, where each agent can interact with all the agents within a circle of a constant radius, and can furthermore interact with some distant agents to reach a pre-assigned number of neighbors, if needed. Some sufficient conditions imposed only on system parameters and agent initial states are presented, which ensure achieving synchronization of the whole group of agents. It reveals the intrinsic relationships among the interaction range, the speed, the initial heading, and the density of the group. Moreover, robustness against variations of interaction range, density, and speed are investigated by comparing the motion patterns and performances of the hybrid metric-topological interaction model with the conventional metric-only and topological-only interaction models. Practically in all cases, the hybrid metric-topological interaction model has the best performance in the sense of achieving highest frequency of synchronization, fastest convergent rate, and smallest heading difference.

Description: Exploring the dynamical behaviors of high water cut and low velocity oil-water flows remains a contemporary and challenging problem of significant importance. This challenge stimulates us to design a high-speed cycle motivation conductance sensor to capture spatial local flow information. We systematically carry out experiments and acquire the multi-channel measurements from different oil-water flow patterns. Then we develop a novel multivariate weighted recurrence network for uncovering the flow behaviors from multi-channel measurements. In particular, we exploit graph energy and weighted clustering coefficient in combination with multivariate time-frequency analysis to characterize the derived complex networks. The results indicate that the network measures are very sensitive to the flow transitions and allow uncovering local dynamical behaviors associated with water cut and flow velocity. These properties render our method particularly useful for quantitatively characterizing dynamical behaviors governing the transition and evolution of different oil-water flow patterns.

Description: We discuss the influence of small phase lags on the synchronization transitions in the Kuramoto model for a large inhomogeneous population of globally coupled phase oscillators. Without a phase lag, all unimodal distributions of the natural frequencies give rise to a classical synchronization scenario, where above the onset of synchrony at the Kuramoto threshold, there is an increasing synchrony for increasing coupling strength. We show that already for arbitrarily small phase lags, there are certain unimodal distributions of natural frequencies such that for increasing coupling strength synchrony may decrease and even complete incoherence may regain stability. Moreover, our example allows a qualitative understanding of the mechanism for such non-universal synchronization transitions.

Description: Some chaotic attractors produced by three-dimensional dynamical systems without any singular point have now been identified, but explaining how they are structured in the state space remains an open question. We here want to explain—in the particular case of the Wei system—such a structure, using one-dimensional sets obtained by vanishing two of the three derivatives of the flow. The neighborhoods of these sets are made of points which are characterized by the eigenvalues of a 2 × 2 matrix describing the stability of flow in a subspace transverse to it. We will show that the attractor is spiralling and twisted in the neighborhood of one-dimensional sets where points are characterized by a pair of complex conjugated eigenvalues. We then show that such one-dimensional sets are also useful in explaining the structure of attractors produced by systems with singular points, by considering the case of the Lorenz system.

Description: Topological approaches to mixing are important tools to understand chaotic fluid flows, ranging from oceanic transport to the design of micro-mixers. Typically, topological entropy, the exponential growth rate of material lines, is used to quantify topological mixing. Computing topological entropy from the direct stretching rate is computationally expensive and sheds little light on the source of the mixing. Earlier approaches emphasized that topological entropy could be viewed as generated by the braiding of virtual, or “ghost,” rods stirring the fluid in a periodic manner. Here, we demonstrate that topological entropy can also be viewed as generated by the braiding of ghost rods following heteroclinic orbits instead. We use the machinery of homotopic lobe dynamics, which extracts symbolic dynamics from finite-length pieces of stable and unstable manifolds attached to fixed points of the fluid flow. As an example, we focus on the topological entropy of a bounded, chaotic, two-dimensional, double-vortex cavity flow. Over a certain parameter range, the topological entropy is primarily due to the braiding of a period-three orbit. However, this orbit does not explain the topological entropy for parameter values where it does not exist, nor does it explain the excess of topological entropy for the entire range of its existence. We show that braiding by heteroclinic orbits provides an accurate computation of topological entropy when the period-three orbit does not exist, and that it provides an explanation for some of the excess topological entropy when the period-three orbit does exist. Furthermore, the computation of symbolic dynamics using heteroclinic orbits has been automated and can be used to compute topological entropy for a general 2D fluid flow.

Description: Dynamic properties of a nonlinear five-dimensional stoichiometric model of the hypothalamic-pituitary-adrenal (HPA) axis were systematically investigated. Conditions under which qualitative transitions between dynamic states occur are determined by independently varying the rate constants of all reactions that constitute the model. Bifurcation types were further characterized using continuation algorithms and scale factor methods. Regions of bistability and transitions through supercritical Andronov-Hopf and saddle loop bifurcations were identified. Dynamic state analysis predicts that the HPA axis operates under basal (healthy) physiological conditions close to an Andronov-Hopf bifurcation. Dynamic properties of the stress-control axis have not been characterized experimentally, but modelling suggests that the proximity to a supercritical Andronov-Hopf bifurcation can give the HPA axis both, flexibility to respond to external stimuli and adjust to new conditions and stability, i.e., the capacity to return to the original dynamic state afterwards, which is essential for maintaining homeostasis. The analysis presented here reflects the properties of a low-dimensional model that succinctly describes neurochemical transformations underlying the HPA axis. However, the model accounts correctly for a number of experimentally observed properties of the stress-response axis. We therefore regard that the presented analysis is meaningful, showing how in silico investigations can be used to guide the experimentalists in understanding how the HPA axis activity changes under chronic disease and/or specific pharmacological manipulations.

Description: In this article, we develop some approaches, which enable us to more accurately and analytically identify the essential patterns that guarantee the almost sure stability of discrete-time systems with random switches. We allow for the case that the elements in the switching connection matrix even obey some unbounded and continuous-valued distributions. In addition to the almost sure stability, we further investigate the almost sure synchronization in complex dynamical networks consisting of randomly connected nodes. Numerical examples illustrate that a chaotic dynamics in the synchronization manifold is preserved when statistical parameters enter some almost sure synchronization region established by the developed approach. Moreover, some delicate configurations are considered on probability space for ensuring synchronization in networks whose nodes are described by nonlinear maps. Both theoretical and numerical results on synchronization are presented by setting only a few random connections in each switch duration. More interestingly, we analytically find it possible to achieve almost sure synchronization in the randomly switching complex networks even with very large population sizes, which cannot be easily realized in non-switching but deterministically connected networks.

Description: This paper focuses on the study of the stochastic Van der Pol vibro-impact system with fractional derivative damping under Gaussian white noise excitation. The equations of the original system are simplified by non-smooth transformation. For the simplified equation, the stochastic averaging approach is applied to solve it. Then, the fractional derivative damping term is facilitated by a numerical scheme, therewith the fourth-order Runge-Kutta method is used to obtain the numerical results. And the numerical simulation results fit the analytical solutions. Therefore, the proposed analytical means to study this system are proved to be feasible. In this context, the effects on the response stationary probability density functions (PDFs) caused by noise excitation, restitution condition, and fractional derivative damping are considered, in addition the stochastic P-bifurcation is also explored in this paper through varying the value of the coefficient of fractional derivative damping and the restitution coefficient. These system parameters not only influence the response PDFs of this system but also can cause the stochastic P-bifurcation.

Description: Fractal analysis (FA) should be able to yield reliable and fast results for high-resolution digital images to be applicable in fields that require immediate outcomes. Triggered by an efficient implementation of FA for binary images, we present three new approaches for fractal dimension ( D ) estimation of images that utilize image pyramids, namely, the pyramid triangular prism, the pyramid gradient, and the pyramid differences method (PTPM, PGM, PDM). We evaluated the performance of the three new and five standard techniques when applied to images with sizes up to 8192 × 8192 pixels. By using artificial fractal images created by three different generator models as ground truth, we determined the scale ranges with minimum deviations between estimation and theory. All pyramidal methods (PM) resulted in reasonable D values for images of all generator models. Especially, for images with sizes ≥ 1024 × 1024 pixels, the PMs are superior to the investigated standard approaches in terms of accuracy and computation time. A measure for the possibility to differentiate images with different intrinsic D values did show not only that the PMs are well suited for all investigated image sizes, and preferable to standard methods especially for larger images, but also that results of standard D estimation techniques are strongly influenced by the image size. Fastest results were obtained with the PDM and PGM, followed by the PTPM. In terms of absolute D values best performing standard methods were magnitudes slower than the PMs. Concluding, the new PMs yield high quality results in short computation times and are therefore eligible methods for fast FA of high-resolution images.

Description: The use of reverse time chaos allows the realization of hardware chaotic systems that can operate at speeds equivalent to existing state of the art while requiring significantly less complex circuitry. Matched filter decoding is possible for the reverse time system since it exhibits a closed form solution formed partially by a linear basis pulse. Coefficients have been calculated and are used to realize the matched filter digitally as a finite impulse response filter. Numerical simulations confirm that this correctly implements a matched filter that can be used for detection of the chaotic signal. In addition, the direct form of the filter has been implemented in hardware description language and demonstrates performance in agreement with numerical results.

Description: The spatial distributions of system's frequencies have significant influences on the critical coupling strengths for amplitude death (AD) in coupled oscillators. We find that the left and right critical coupling strengths for AD have quite different relations to the increasing spatial period m of the frequency distribution in coupled oscillators. The left one has a negative linear relationship with m in log-log axis for small initial frequency mismatches while remains constant for large initial frequency mismatches. The right one is in quadratic function relation with spatial period m of the frequency distribution in log-log axis. There is an optimal spatial period m 0 of frequency distribution with which the coupled system has a minimal critical strength to transit from an AD regime to reviving oscillation. Moreover, the optimal spatial period m 0 of the frequency distribution is found to be related to the system size N . Numerical examples are explored to reveal the inner regimes of effects of the spatial frequency distribution on AD.

Description: We propose a systematic methodology for creating 2 N +  1-scroll chaotic attractors from a simple three-dimensional system, which is named as the translation chaotic system. It satisfies the condition a 12 a 21  =  0, while the Chua system satisfies a 12 a 21  〉  0. In this paper, we also propose a successful (an effective) design and an analytical approach for constructing 2 N +  1-scrolls, the translation transformation principle. Also, the dynamics properties of the system are studied in detail. MATLAB simulation results show very sophisticated dynamical behaviors and unique chaotic behaviors of the system. It provides a new approach for 2 N +  1-scroll attractors. Finally, to explore the potential use in technological applications, a novel block circuit diagram is also designed for the hardware implementation of 1 -, 3 - , 5 - , and 7 - scroll attractors via switching the switches. Translation chaotic system has the merit of convenience and high sensitivity to initial values, emerging potentials in future engineering chaos design.

Description: Electrocardiogram (ECG) data from patients with a variety of heart conditions are studied using ordinal pattern partition networks. The ordinal pattern partition networks are formed from the ECG time series by symbolizing the data into ordinal patterns. The ordinal patterns form the nodes of the network and edges are defined through the time ordering of the ordinal patterns in the symbolized time series. A network measure, called the mean degree, is computed from each time series-generated network. In addition, the entropy and number of non-occurring ordinal patterns (NFP) is computed for each series. The distribution of mean degrees, entropies, and NFPs for each heart condition studied is compared. A statistically significant difference between healthy patients and several groups of unhealthy patients with varying heart conditions is found for the distributions of the mean degrees, unlike for any of the distributions of the entropies or NFPs.

Description: This paper proposes an epilepsy detection and closed-loop control strategy based on Particle Swarm Optimization (PSO) algorithm. The proposed strategy can effectively suppress the epileptic spikes in neural mass models, where the epileptiform spikes are recognized as the biomarkers of transitions from the normal (interictal) activity to the seizure (ictal) activity. In addition, the PSO algorithm shows capabilities of accurate estimation for the time evolution of key model parameters and practical detection for all the epileptic spikes. The estimation effects of unmeasurable parameters are improved significantly compared with unscented Kalman filter. When the estimated excitatory-inhibitory ratio exceeds a threshold value, the epileptiform spikes can be inhibited immediately by adopting the proportion-integration controller. Besides, numerical simulations are carried out to illustrate the effectiveness of the proposed method as well as the potential value for the model-based early seizure detection and closed-loop control treatment design.

Description: We construct spatiotemporal localized envelope solutions of a (3 + 1)-dimensional nonlinear Schrödinger equation with varying coefficients such as dispersion, nonlinearity and gain parameters through similarity transformation technique. The obtained localized rational solutions can serve as prototypes of rogue waves in different branches of science. We investigate the characteristics of constructed localized solutions in detail when it propagates through six different dispersion profiles, namely, constant, linear, Gaussian, hyperbolic, logarithm, and exponential. We also obtain expressions for the hump and valleys of rogue wave intensity profiles for these six dispersion profiles and study the trajectory of it in each case. Further, we analyze how the intensity of another localized solution, namely, breather, changes when it propagates through the aforementioned six dispersion profiles. Our studies reveal that these localized solutions co-exist with the collapsing solutions which are already found in the (3 + 1)-dimensional nonlinear Schrödinger equation. The obtained results will help to understand the corresponding localized wave phenomena in related fields.

Description: The mechanical and electrical properties, and information processing capabilities of microtubules are the permanent subject of interest for carrying out experiments in vitro and in silico , as well as for theoretical attempts to elucidate the underlying processes. In this paper, we developed a new model of the mechano–electrical waves elicited in the rows of very flexible C–terminal tails which decorate the outer surface of each microtubule. The fact that C–terminal tails play very diverse roles in many cellular functions, such as recruitment of motor proteins and microtubule–associated proteins, motivated us to consider their collective dynamics as the source of localized waves aimed for communication between microtubule and associated proteins. Our approach is based on the ferroelectric liquid crystal model and it leads to the effective asymmetric double-well potential which brings about the conditions for the appearance of kink–waves conducted by intrinsic electric fields embedded in microtubules. These kinks can serve as the signals for control and regulation of intracellular traffic along microtubules performed by processive motions of motor proteins, primarly from kinesin and dynein families. On the other hand, they can be precursors for initiation of dynamical instability of microtubules by recruiting the proper proteins responsible for the depolymerization process.

Description: Vineeth Nair and R. I. Sujith In this paper, we show how the phenomenon of intermittency observed in systems with turbulent flow-sound interaction is related to the formation of homoclinic orbits in the phase space. Such orbits that emerge via the intersection of the stable and unstable manifold of an equilibrium configuration r ... [Chaos 23, 033136 (2013)] published Mon Sep 16, 2013.

Description: Bhagat Lal Dutta and Prashant M. Gade We study delayed circle map. A previously proposed analogy between delayed map and spatiotemporal system [F. T. Arecchi et al., Phys. Rev. A 45, R4225 (1992)] is employed to study this system. In the phase diagram, we observe laminar phase, travelling defect phase, and standing defect phase. We push ... [Chaos 23, 033138 (2013)] published Thu Sep 19, 2013.

Description: Stephan Bialonski and Klaus Lehnertz We investigate assortativity of functional brain networks before, during, and after one-hundred epileptic seizures with different anatomical onset locations. We construct binary functional networks from multi-channel electroencephalographic data recorded from 60 epilepsy patients; and from time-reso ... [Chaos 23, 033139 (2013)] published Fri Sep 20, 2013.

Description: Chenggui Yao, Ming Yi, and Jianwei Shuai Time delayed coupling plays a crucial role in determining the system's dynamics. We here report that the time delay induces transition from the asynchronous state to the complete synchronization (CS) state in the repulsively coupled chaotic oscillators. In particular, by changing the coupling streng ... [Chaos 23, 033140 (2013)] published Fri Sep 20, 2013.

Description: Lucia Valentina Gambuzza, Alessio Cardillo, Alessandro Fiasconaro, Luigi Fortuna, Jesus Gomez-Gardenes et al. A novel regime of synchronization, called remote synchronization, where the peripheral nodes form a phase synchronized cluster not including the hub, was recently observed in star motifs [Bergner et al., Phys. Rev. E 85, 026208 (2012)]. We show the existence of a more general dynamical state of remo ... [Chaos 23, 043103 (2013)] published Wed Oct 2, 2013.

Description: Nastaran Vasegh and Farhad Khellat In this paper a unified approach is presented for controlling chaos in nonlinear partial differential systems by a fuzzy control design. First almost all known chaotic partial differential equation systems are represented by Takagi-Sugeno fuzzy model. For investigating design procedure, Kuramoto-Siv ... [Chaos 23, 042101 (2013)] published Wed Oct 2, 2013.

Description: Satish Narayanan, Gemunu H. Gunaratne, and Fazle Hussain We present a strategy for control of chaos in open flows and provide its experimental validation in the near field of a transitional jet flow system. The low-dimensional chaotic dynamics studied here results from vortex ring formation and their pairings over a spatially extended region of the flow t ... [Chaos 23, 033133 (2013)] published Mon Sep 9, 2013.

Description: Aaron Luttman, Erik M. Bollt, Ranil Basnayake, Sean Kramer, and Nicholas B. Tufillaro Given image data of a fluid flow, the flow field, , governing the evolution of the system can be estimated using a variational approach to optical flow. Assuming that the flow field governing the advection is the symplectic gradient of a stream function or the gradient of a potential functionbo ... [Chaos 23, 033134 (2013)] published Wed Sep 11, 2013.

Description: Xiyun Zhang, Zhongyuan Ruan, and Zonghua Liu It has been found that contrarian oscillators usually take a negative role in the collective behaviors formed by conformist oscillators. However, experiments revealed that it is also possible to achieve a strong coherence even when there are contrarians in the system such as neuron networks with bot ... [Chaos 23, 033135 (2013)] published Mon Sep 16, 2013.

Description: Xing Lu A spectral problem, the x-derivative part of which is a simple generalization of the standard Ablowitz-Kaup-Newell-Segur and Kaup-Newell spectral problems, is presented with its associated generalized mixed nonlinear Schrodinger (GMNLS) model. The N-fold Darboux transformation with multi-parameters ... [Chaos 23, 033137 (2013)] published Wed Sep 18, 2013.

Description: Dane Taylor, Elana J. Fertig, and Juan G. Restrepo While considerable progress has been made in the analysis of large systems containing a single type of coupled dynamical component (e.g., coupled oscillators or coupled switches), systems containing diverse components (e.g., both oscillators and switches) have received much less attention. We analyz ... [Chaos 23, 033142 (2013)] published Wed Sep 25, 2013.

Description: Yang Ying and Chen Guopei This paper considers the problem of finite time control for a class of time-varying unified chaotic system. First, based on the finite-time stability theory, a novel adaptive control technique is presented to achieve finite-time stabilization for time-varying unified chaotic system. Comparing with t ... [Chaos 23, 033143 (2013)] published Mon Sep 30, 2013.

Description: Tanmoy Banerjee and Debabrata Biswas We explore and experimentally demonstrate the phenomena of amplitude death (AD) and the corresponding transitions through synchronized states that lead to AD in coupled intrinsic time-delayed hyperchaotic oscillators interacting through mean-field diffusion. We identify a novel synchronization trans ... [Chaos 23, 043101 (2013)] published Tue Oct 1, 2013.

Description: Nan Sheng, Youwei Jia, Zhao Xu, Siu-Lau Ho, and Chi Wai Kan Water distribution network (WDN) is a typical real-world complex network of major infrastructure that plays an important role in human's daily life. In this paper, we explore the formation of isolated communities in WDN based on complex network theory. A graph-algebraic model is proposed to effectiv ... [Chaos 23, 043102 (2013)] published Tue Oct 1, 2013.

Description: Eduardo Paucar Bravo, Kazuyuki Aihara, and Yoshito Hirata In this work, we introduce a model for predicting multivariate time series data. This model was obtained by partitioning the state space with joint permutations. We review the theoretical framework of the previous works, show a simple extension to multivariate data, and compare its performance to th ... [Chaos 23, 043104 (2013)] published Thu Oct 3, 2013.

Description: R. Gopal, A. Venkatesan, and M. Lakshmanan We show that the recently introduced 0-1 test can successfully distinguish between strange nonchaotic attractors (SNAs) and periodic/quasiperiodic/chaotic attractors, by suitably choosing the arbitrary parameter associated with the translation variables in terms of the golden mean number which avoid ... [Chaos 23, 023123 (2013)] published Wed Jun 5, 2013.

Description: Vlado A. Buljan, R. M. Damian Holsinger, D. Brown, J. J. Bohorquez-Florez, B. D. Hambly et al. We have studied a spontaneous self-organization dynamics in a closed, dissipative (in terms of guansine 5[prime]-triphosphate energy dissipation), reaction-diffusion system of acentrosomal microtubules (those nucleated and organized in the absence of a microtubule-organizing centre) multitude consti ... [Chaos 23, 023120 (2013)] published Wed Jun 5, 2013.

Description: A. Alias, R. H. J. Grimshaw, and K. R. Khusnutdinova In the weakly nonlinear limit, oceanic internal solitary waves for a single linear long wave mode are described by the KdV equation, extended to the Ostrovsky equation in the presence of background rotation. In this paper we consider the scenario when two different linear long wave modes have nearly ... [Chaos 23, 023121 (2013)] published Wed Jun 5, 2013.

Description: J. Levitan, A. Yahalom, L. Horwitz, and M. Lewkowicz We show here that a recently developed criterion for the stability of conservative Hamiltonian systems can be extended to Hamiltonians with weak time dependence. In this method, the geodesic equations contain the Hamilton equations of the original potential model through an inverse map in the tangen ... [Chaos 23, 023122 (2013)] published Wed Jun 5, 2013.

Description: Massimiliano Guzzo and Elena Lega The celebrated KAM and Nekhoroshev theorems provide essential informations about the long term dynamics of quasiintegrable Hamiltonian systems. In particular, longterm instability of the action variables can be observed only in the socalled Arnold web, which is the complement in the phasespace of al ... [Chaos 23, 023124 (2013)] published Thu Jun 6, 2013.

Description: Avi Robinson-Mosher, Tamar Shinar, Pamela A. Silver, and Jeffrey Way The predictable engineering of well-behaved transcriptional circuits is a central goal of synthetic biology. The artificial attachment of promoters to transcription factor genes usually results in noisy or chaotic behaviors, and such systems are unlikely to be useful in practical applications. Natur ... [Chaos 23, 025110 (2013)] published Wed Jun 12, 2013.