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In this paper, a class of scalar quartic polynomial delay systems is investigated. We found rich dynamics in this system through numerical simulation, including chaotic attractors, chaotic saddles, and intermittent chaos. Moreover, this chaotic quartic system may serve as an approximation, through Taylor expansion, for a wide class of scalar delay differential equations. Thus, these nonlinear systems may exhibit chaotic behaviors, and the studies in our paper may provide an insight into the emergence of chaos in other time-delay nonlinear systems. We also conduct a detailed theoretical analysis of the system, including the stability of equilibria and Hopf bifurcation analysis based on the theory of normal form and center manifold. Additionally, a numerical analysis is provided to give numerical evidence for the existence of chaos. [ABSTRACT FROM AUTHOR]

Birhythmicity is evident in many nonlinear systems, which include physical and biological systems. In some living systems, birhythmicity is necessary for response to the varying environment while unnecessary in some physical systems as it limits their efficiency. Therefore, its control is an important area of research. This paper proposes a space-dependent intermittent control scheme capable of controlling birhythmicity in various dynamical systems. We apply the proposed control scheme in five nonlinear systems from diverse branches of natural science and demonstrate that the scheme is efficient enough to control the birhythmic oscillations in all the systems. We derive the analytical condition for controlling birhythmicity by applying harmonic decomposition and energy balance methods in a birhythmic van der Pol oscillator. Further, the efficacy of the control scheme is investigated through numerical and bifurcation analyses in a wide parameter space. Since the proposed control scheme is general and efficient, it may be employed to control birhythmicity in several dynamical systems. [ABSTRACT FROM AUTHOR]

Aging phenomena have been observed in numerous physical systems. Many statistical quantities depend on the aging time t a for aging anomalous diffusion processes. This paper pays more attention to how an external force field affects the aging Lévy walk. Based on the Langevin picture of the Lévy walk and the generalized Green–Kubo formula, we investigate the quantities that include the ensemble- and time-averaged mean-squared displacements in both weak aging t a ≪ t and strong aging t a ≫ t cases and compare them to the ones in the absence of any force field. Two typical force fields, constant force F and time-dependent periodic force F (t) = f 0 sin ⁡ (ω t) , are considered for comparison. The generalized Einstein relation is also discussed in the case with the constant force. We find that the constant force is the key to causing the aging phenomena and enhancing the diffusion behavior of the aging Lévy walk, while the time-dependent periodic force is not. The different effects of the two kinds of forces on the aging Lévy walk are verified by both theoretical analyses and numerical simulations. [ABSTRACT FROM AUTHOR]

This paper presents review and comparison of alternative methodologies for control performance assessment. The approach uses nonlinear time series analysis, such as non-Gaussian statistics, fractal, crossover analysis, or entropy-based approaches. There is a presented practical rationale for the analysis. Evaluation is based on the real data gathered from industrial systems. Non-Gaussian analysis starts with statistical methods using different probabilistic distribution functions. As another potential measure, the Hurst exponent is calculated using different approaches. Finally, R/S plot analysis together with crossover point phenomenon discussion is presented. The paper ends with conclusions and presentation of open issues attractive for further development. [ABSTRACT FROM AUTHOR]

This paper reports a method to derive a novel class of 3D generalized thermostatted oscillators from a simple damped harmonic oscillator. Its detailed procedure is obtained through a mathematical derivation. Then, we propose an example system to show the effectiveness of the method. Furthermore, the numerical analysis is performed to investigate its rich conservative dynamics, including chaotic sea, hierarchical invariant tori, and the coexistence of chaotic seas and invariant tori with islands-around-islands hierarchy, even if there are three dissipative terms in the example system. To verify the existence of conservative chaos at the physical level, an analog circuit is presented to observe the existing conservative chaotic flows in National Instruments multisim. We finally provide two systems, which can produce conservative chaotic flows with more complicated topologies, and draw our conclusions. [ABSTRACT FROM AUTHOR]

This paper investigates biological models that represent the transition equation from a system in the past to a system in the future. It is shown that finite-time Lyapunov exponents calculated along a locally pullback attractive solution are efficient indicators (early-warning signals) of the presence of a tipping point. Precise time-dependent transitions with concave or d-concave variation in the state variable giving rise to scenarios of rate-induced tracking are shown. They are classified depending on the internal dynamics of the set of bounded solutions. Based on this classification, some representative features of these models are investigated by means of a careful numerical analysis. [ABSTRACT FROM AUTHOR]

Nonlinear parametric systems have been widely used in modeling nonlinear dynamics in science and engineering. Bifurcation analysis of these nonlinear systems on the parameter space is usually used to study the solution structure, such as the number of solutions and the stability. In this paper, we develop a new machine learning approach to compute the bifurcations via so-called equation-driven neural networks (EDNNs). The EDNNs consist of a two-step optimization: the first step is to approximate the solution function of the parameter by training empirical solution data; the second step is to compute bifurcations using the approximated neural network obtained in the first step. Both theoretical convergence analysis and numerical implementation on several examples have been performed to demonstrate the feasibility of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we investigate transport of an inertial particle in a spatially symmetric potential and subjected to two harmonic signals with different frequencies in both deterministic and stochastic cases. Numerical results indicate that: (i) In the deterministic case, the two harmonic signals can induce many (up to six) segments of negative slopes in the curve of mean velocity vs. external constant force, i.e., a multiple absolute negative mobilities (ANMs) effect. But the occurrence of the effect depends on their frequencies and amplitudes. (ii) For the stochastic case, the multiple ANMs relay on stable index and symmetry parameter of the Levy noise. In the case of symmetric noise, appropriate stable index makes the multiple ANMs be the strongest. Our further investigations indicate that an indispensable condition for the multiple ANMs to occur in the system is the temporal symmetry breaking by one multiplicative periodic signal and one additive periodic signal. [ABSTRACT FROM AUTHOR]

We theoretically study noise-induced phase switch phenomena in an inertial majority-vote (IMV) model introduced in a recent paper [Chen et al., Phys. Rev. E 95, 042304 (2017)]. The IMV model generates a strong hysteresis behavior as the noise intensity f goes forward and backward, a main characteristic of a first-order phase transition, in contrast to a second-order phase transition in the original MV model. Using the Wentzel-Kramers-Brillouin approximation for the master equation, we reduce the problem to finding the zero-energy trajectories in an effective Hamiltonian system, and the mean switching time depends exponentially on the associated action and the number of particles N. Within the hysteresis region, we find that the actions, along the optimal forward switching path from the ordered phase (OP) to disordered phase (DP) and its backward path show distinct variation trends with f, and intersect at f=fc that determines the coexisting line of the OP and DP. This results in a nonmonotonic dependence of the mean switching time between two symmetric OPs on f, with a minimum at fc for sufficiently large N. Finally, the theoretical results are validated by Monte Carlo simulations. [ABSTRACT FROM AUTHOR]

This paper presents a chaotic circuit based on a nonvolatile locally active memristor model, with non-volatility and local activity verified by the power-off plot and the DC V–I plot, respectively. It is shown that the memristor-based circuit has no equilibrium with appropriate parameter values and can exhibit three hidden coexisting heterogeneous attractors including point attractors, periodic attractors, and chaotic attractors. As is well known, for a hidden attractor, its attraction basin does not intersect with any small neighborhood of any unstable equilibrium. However, it is found that some attractors of this circuit can be excited from an unstable equilibrium in the locally active region of the memristor, meaning that its basin of attraction intersects with neighborhoods of an unstable equilibrium of the locally active memristor. Furthermore, with another set of parameter values, the circuit possesses three equilibria and can generate self-excited chaotic attractors. Theoretical and simulated analyses both demonstrate that the local activity and an unstable equilibrium of the memristor are two reasons for generating hidden attractors by the circuit. This chaotic circuit is implemented in a digital signal processing circuit experiment to verify the theoretical analysis and numerical simulations. [ABSTRACT FROM AUTHOR]

The aim of this paper is to model trading decisions of financial investors based on a sentiment index. For this purpose, we analyze a dynamical model, which includes the sentiment index in the agents' trading behavior. We consider the setup of a discrete dynamical system, assuming that in financial markets, transactions take place between two groups of fundamentalists that differ in their perception of fundamental value. This assumption is motivated by a degree of uncertainty about the true fundamental value. The proportion of fundamentalists in the two groups is assumed to depend on the sentiment index. The sentiment index used is related to the risk asymmetry index, enabling us to consider both the variance and the asymmetry of the prediction error between the two groups of fundamentalists. We identify the equilibria of the model and conduct a numerical analysis in order to capture stylized facts documented empirically in the financial literature. [ABSTRACT FROM AUTHOR]

Spatially extended oscillatory systems can be entrained by pacemakers, regions that oscillate with a higher frequency than the rest of the medium. Entrainment happens through waves originating at a pacemaker. Typically, biological and chemical media can contain multiple pacemaker regions, which compete with each other. In this paper, we perform a detailed numerical analysis of how wave propagation and synchronization of the medium depend on the properties of these pacemakers. We discuss the influence of the size and intrinsic frequency of pacemakers on the synchronization properties. We also study a system in which the pacemakers are embedded in a medium without any local dynamics. In this case, synchronization occurs if the coupling determined by the distance and diffusion is strong enough. The transition to synchronization is reminiscent of systems of discrete coupled oscillators. [ABSTRACT FROM AUTHOR]

This paper proposes a simple no-equilibrium chaotic system with only one signum function as compared with the existing no-equilibrium chaotic ones with at least one quadratic or higher nonlinearity. The system has the offset boosting of three variables through adjusting the corresponding controlled constants. The resulting hidden attractors can be distributed in a 1D line, a 2D lattice, a 3D grid, and even in an arbitrary location of the phase space. Particularly, a hidden chaotic bursting oscillation is also observed in this system, which is an uncommon phenomenon. In addition, complex hidden dynamics is investigated via phase portraits, time series, Kaplan–Yorke dimensions, bifurcation diagrams, Lyapunov exponents, and two-parameter bifurcation diagrams. Then, a very simple hardware circuit without any multiplier is fabricated, and the experimental results are presented to demonstrate theoretical analyses and numerical simulations. Furthermore, the randomness test of the chaotic pseudo-random sequence generated by the system is tested by the National Institute of Standards and Technology test suite. The tested results show that the proposed system has good randomness, thus being suitable for chaos-based applications such as secure communication and image encryption. [ABSTRACT FROM AUTHOR]

In this paper, the harmonic balance method with the alternating frequency/time (HB-AFT) domain technique is extended to the dynamical systems with state-dependent delays and non-smooth right-hand side for the first time. Two types of network congestion control models [the modified transmission control protocol-random early detection (TCP-RED) model and the fluid-flow TCP-additive increase multiplicative decease (AIMD)/RED model] with state-dependent round-trip time delays and non-smooth right-hand side are considered in detail. First, their dynamics and bifurcation are analyzed by the numerical analysis method. Then, the analytical approximate expressions of the periodic solutions are obtained by employing the semi-analytical method named as HB-AFT. The results of the numerical simulation and HB-AFT agree with each other very well. It indicates that the HB-AFT technique is simple, valid, effective, and accurate for the non-smooth dynamical systems with state-dependent time delays. Besides, more complicated and rich dynamical phenomena, such as period- 2 solution, period- 3 solution, chaos and the period-doubling bifurcation, and the window of period- 3 solution, which lead to chaos, are explored in the TCP-AIMD/RED model. Moreover, two different kinds of multi-stabilities, such as the coexistence of a stable period- 1 solution, a stable period- 3 solution, and an asymptotically stable equilibrium, and that of a stable period- 1 solution, a stable period- 2 solution, chaos, and an asymptotically stable equilibrium, are also discovered in the TCP-AIMD/RED congestion control model. It discloses that the interaction of the state-dependent delay and non-smooth function can easily induce the richer and more complicated dynamics. It can explain the complicated dynamical phenomena in the Internet network. Thus, the control and parameters tuning based on the above results could be used to avoid the congestion and optimize the performance. It can give the theoretical guidance to the application and research in network congestion control, which is very important in practical application. [ABSTRACT FROM AUTHOR]

This paper presents a novel memristor-based dynamical system with circuit implementation, which has a 2 × 3 -wing, 2 × 2 -wing, and 2 × 1 -wing non-Shilnikov type of chaotic attractors. The system has two index-2 saddle-focus equilibria, symmetrical with respect to the x -axis. The system is analyzed with bifurcation diagrams and Lyapunov exponents, demonstrating its complex dynamical behaviors: the system reaches the chaotic state from the periodic state through alternating period-doubling bifurcations and then from the chaotic state back to the periodic state through inverse bifurcations, as one parameter changes. It shows two interesting phenomena: a jump-switching periodic state and jump-switching chaotic state. Also, the system can sustain chaos with a constant Lyapunov spectrum in some initial conditions and a parameter set. In addition, a class of symmetric periodic bursting phenomena is surprisingly observed under a particular set of parameters, and its generation mechanism is revealed through bifurcation analysis. Finally, the circuit implementation verifies the theoretical analysis and the jump-switching numerical simulation results. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the integrable Hirota equation, which describes the propagation of ultrashort light pulses in optical fibers. First, we numerically study spectral signatures of the spatial Lax pair with distinct potentials [e.g., solitons, Akhmediev-Kuznetsov-Ma (AKM) and Kuznetsov-Ma (KM) breathers, and rogue waves (RWs)] of the Hirota equation. Second, we discuss the RW generation by using the dam-break problem with a decaying initial condition and further analyze spectral signatures of periodized wavetrains. Third, we explore two kinds of noise-derived modulational instabilities: (i) the one case is based on the initial condition (one plus a random noise) such that the KM and AKM breathers, and RWs can be generated, and they agree well with analytical solutions; (ii) another case is to consider another initial condition (one plus a Gaussian wave with a random noise phase) such that some RWs with higher amplitudes can be found. Moreover, we also investigate the spectral signatures of corresponding periodic wavetrains. Finally, we find that the interactions of two waves can also generate the RW phenomena with higher amplitudes. These obtained results will be useful to understand the RW generation in the third-order nonlinear Schrödinger equation and other related models. [ABSTRACT FROM AUTHOR]

Previous results have shown that a large class of complex systems consisting of many interacting heterogeneous phase oscillators exhibit an attracting invariant manifold. This result has enabled reduced analytic system descriptions from which all the long term dynamics of these systems can be calculated. Although very useful, these previous results are limited by the restriction that the individual interacting system components have one-dimensional dynamics, with states described by a single, scalar, angle-like variable (e.g., the Kuramoto model). In this paper, we consider a generalization to an appropriate class of coupled agents with higher-dimensional dynamics. For this generalized class of model systems, we demonstrate that the dynamics again contain an invariant manifold, hence enabling previously inaccessible analysis and improved numerical study, allowing a similar simplified description of these systems. We also discuss examples illustrating the potential utility of our results for a wide range of interesting situations. [ABSTRACT FROM AUTHOR]

Recovering network topologies is of great significance in the study of complex networks. In this paper, a method for identifying structures of multilayer networks is proposed via compressive sensing and Taylor expansion. By using this method, the topologies of multilayer networks with unknown node dynamical functions can be identified from a relatively small number of observations. Numerical experiments are provided to show the effectiveness and efficiency of the method on different types of multilayer networks, where the intralayer topology and the interlayer topology of a multilayer network can be identified simultaneously. In particular, the topology of one layer can be identified even when nodes of the other layer are unobservable. [ABSTRACT FROM AUTHOR]

The paper describes the effects of mutual and external synchronization of spiral wave structures in two coupled two-dimensional lattices of coupled discrete-time oscillators. Each lattice is given by a 2D N × N network of nonlocally coupled Nekorkin maps which model neuronal activity. We show numerically that spiral wave structures, including spiral wave chimeras, can be synchronized and establish the mechanism of the synchronization scenario. Our numerical studies indicate that when the coupling strength between the lattices is sufficiently weak, only a certain part of oscillators of the interacting networks is imperfectly synchronized, while the other part demonstrates a partially synchronous behavior. If the spatiotemporal patterns in the lattices do not include incoherent cores, imperfect synchronization is realized for most oscillators above a certain value of the coupling strength. In the regime of spiral wave chimeras, the imperfect synchronization of all oscillators cannot be achieved even for sufficiently large values of the coupling strength. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to investigate an accurate numerical method for solving a biological fractional model via Atangana-Baleanu fractional derivative. We focused our attention on linear and nonlinear Fisher's equations. We use the spectral collocation method based on the Chebyshev approximations. This method reduced the nonlinear equations to a system of ordinary differential equations by using the properties of Chebyshev polynomials and then solved them by using the finite difference method. This is the first time that this method is used to solve nonlinear equations in Atangana-Baleanu sense. We present the effectiveness and accuracy of the proposed method by computing the absolute error and the residual error functions. The results show that the given procedure is an easy and efficient tool to investigate the solution of nonlinear equations with local and non-local singular kernels. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the problem of controlling the source function for an optimal control problem involving the fractional wave equation. We show that an optimal solution exists and it is unique for the considered fractional optimal control problem. We calculate the Frechet derivative of the cost functional by means of an adjoint problem and derive necessary optimality conditions. Also, we introduce an efficient numerical approximation for the fractional wave equation with the Atangana-Baleanu derivative. [ABSTRACT FROM AUTHOR]

In this paper, we propose a method to calculate asymptotically period two sinks and define the range of stability of fixed points for a variety of discrete fractional systems of the order 0 < α < 2. The method is tested on various forms of fractional generalizations of the standard and logistic maps. Based on our analysis, we make a conjecture that chaos is impossible in the corresponding continuous fractional systems. [ABSTRACT FROM AUTHOR]

In this paper, we propose a Parameter Switching (PS) algorithm as a new chaos control method for the Hastings-Powell (HP) system. The PS algorithm is a convergent scheme that switches the control parameter within a set of values while the controlled system is numerically integrated. The attractor obtained with the PS algorithm matches the attractor obtained by integrating the system with the parameter replaced by the averaged value of the switched parameter values. The switching rule can be applied periodically or randomly over a set of given values. In this way, every stable cycle of the HP system can be approximated if its underlying parameter value equalizes the average value of the switching values. Moreover, the PS algorithm can be viewed as a generalization of Parrondo's game, which is applied for the first time to the HP system, by showing that losing strategy can win: "losing+losing=winning." If "loosing" is replaced with "chaos" and, "winning" with "order" (as the opposite to "chaos"), then by switching the parameter value in the HP system within two values, which generate chaotic motions, the PS algorithm can approximate a stable cycle so that symbolically one can write "chaos+chaos=regular." Also, by considering a different parameter control, new complex dynamics of the HP model are revealed. [ABSTRACT FROM AUTHOR]

There has been growing interest in exploring the interplay between epidemic spreading with human response, since it is natural for people to take various measures when they become aware of epidemics. As a proper way to describe the multiple connections among people in reality, multiplex network, a set of nodes interacting through multiple sets of edges, has attracted much attention. In this paper, to explore the coupled dynamical processes, a multiplex network with two layers is built. Specifically, the information spreading layer is a time varying network generated by the activity driven model, while the contagion layer is a static network. We extend the microscopic Markov chain approach to derive the epidemic threshold of the model. Compared with extensive Monte Carlo simulations, the method shows high accuracy for the prediction of the epidemic threshold. Besides, taking different spreading models of awareness into consideration, we explored the interplay between epidemic spreading with awareness spreading. The results show that the awareness spreading can not only enhance the epidemic threshold but also reduce the prevalence of epidemics. When the spreading of awareness is defined as susceptible-infected-susceptible model, there exists a critical value where the dynamical process on the awareness layer can control the onset of epidemics; while if it is a threshold model, the epidemic threshold emerges an abrupt transition with the local awareness ratio a approximating 0.5. Moreover, we also find that temporal changes in the topology hinder the spread of awareness which directly affect the epidemic threshold, especially when the awareness layer is threshold model. Given that the threshold model is a widely used model for social contagion, this is an important and meaningful result. Our results could also lead to interesting future research about the different time-scales of structural changes in multiplex networks. [ABSTRACT FROM AUTHOR]

This paper investigates the properties of unstable single-spiral wave solutions arising in the Karma model of two-dimensional cardiac tissue. In particular, we discuss how such solutions can be computed numerically on domains of arbitrary shape and study how their stability, rotational frequency, and spatial drift depend on the size of the domain as well as the position of the spiral core with respect to the boundaries. We also discuss how the breaking of local Euclidean symmetry due to finite size effects as well as the spatial discretization of the model is reflected in the structure and dynamics of spiral waves. This analysis allows identification of a self-sustaining process responsible for maintaining the state of spiral chaos featuring multiple interacting spirals. [ABSTRACT FROM AUTHOR]

This paper focuses on the resonance dynamics of a modular neuronal network consisting of several small-world subnetworks. The considered network is composed of delay-coupled FitzHugh-Nagumo (FHN) neurons, whose characteristic parameters present diversity in the form of quenched noise. Our numerical results indicate that when such a network is subjected to an external subthreshold periodic signal, its collective response is optimized for an intermediate level of diversity, namely, a resonant behavior can be induced by an appropriate level of diversity. How the probabilities of intramodule and intermodule connections, as well as the number of subnetworks influence the diversity-induced resonance are also discussed. Further, conclusive evidences demonstrate the nontrivial role of time-delayed coupling on the diversity-induced resonance properties. Especially, multiple resonance is obviously detected when time delays are located at integer multiples of the oscillation period of the signal. Moreover, the phenomenon of fine-tuned delays in inducing multiple resonance remains when diversity is within an intermediate range. Our findings have implications that neural systems may profit from their generic diversity and delayed coupling to optimize the response to external stimulus. [ABSTRACT FROM AUTHOR]

This article introduces an adaptive control scheme with a feedback delay, specifically designed for controlling partial networks, to achieve desynchronization in a coupled network with two or multiple clusters. The proposed scheme's effectiveness is validated through several representative examples of coupled neuronal networks with two interconnected clusters. The efficacy of this scheme is attributed to the rigorous and numerical analyses on the corresponding transcendental characteristic equation, which includes time delay and other network parameters. In addition to investigating the impact of time delay and inter-connectivity on the stability of an incoherent state, we also rigorously find that controlling only one cluster cannot realize the desynchronization in the coupled oscillators within three or more clusters. All these, we believe, can deepen the understanding of the deep brain stimulation techniques presently used in the clinical treatment of neurodegenerative diseases and suggest future avenues for enhancing these clinical techniques through adaptive feedback settings. [ABSTRACT FROM AUTHOR]

Chimera states in spatiotemporal dynamical systems have been investigated in physical, chemical, and biological systems, while how the system is steering toward different final destinies upon spatially localized perturbation is still unknown. Through a systematic numerical analysis of the evolution of the spatiotemporal patterns of multi-chimera states, we uncover a critical behavior of the system in transient time toward either chimera or synchronization as the final stable state. We measure the critical values and the transient time of chimeras with different numbers of clusters. Then, based on an adequate verification, we fit and analyze the distribution of the transient time, which obeys power-law variation process with the increase in perturbation strengths. Moreover, the comparison between different clusters exhibits an interesting phenomenon, thus we find that the critical value of odd and even clusters will alternatively converge into a certain value from two sides, respectively, implying that this critical behavior can be modeled and enabling the articulation of a phenomenological model. [ABSTRACT FROM AUTHOR]

We identify effective stochastic differential equations (SDEs) for coarse observables of fine-grained particle- or agent-based simulations; these SDEs then provide useful coarse surrogate models of the fine scale dynamics. We approximate the drift and diffusivity functions in these effective SDEs through neural networks, which can be thought of as effective stochastic ResNets. The loss function is inspired by, and embodies, the structure of established stochastic numerical integrators (here, Euler–Maruyama and Milstein); our approximations can thus benefit from backward error analysis of these underlying numerical schemes. They also lend themselves naturally to "physics-informed" gray-box identification when approximate coarse models, such as mean field equations, are available. Existing numerical integration schemes for Langevin-type equations and for stochastic partial differential equations can also be used for training; we demonstrate this on a stochastically forced oscillator and the stochastic wave equation. Our approach does not require long trajectories, works on scattered snapshot data, and is designed to naturally handle different time steps per snapshot. We consider both the case where the coarse collective observables are known in advance, as well as the case where they must be found in a data-driven manner. [ABSTRACT FROM AUTHOR]

A heterogeneous structure of social networks induces various intriguing phenomena. One of them is the friendship paradox, which states that on average, your friends have more friends than you do. Its generalization, called the generalized friendship paradox (GFP), states that on average, your friends have higher attributes than yours. Despite successful demonstrations of the GFP by empirical analyses and numerical simulations, analytical, rigorous understanding of the GFP has been largely unexplored. Recently, an analytical solution for the probability that the GFP holds for an individual in a network with correlated attributes was obtained using the copula method but by assuming a locally tree structure of the underlying network [Jo et al., Phys. Rev. E 104, 054301 (2021)]. Considering the abundant triangles in most social networks, we employ a vine copula method to incorporate the attribute correlation structure between neighbors of a focal individual in addition to the correlation between the focal individual and its neighbors. Our analytical approach helps us rigorously understand the GFP in more general networks, such as clustered networks and other related interesting phenomena in social networks. [ABSTRACT FROM AUTHOR]

A dynamical system is said to undergo rate-induced tipping when it fails to track its quasi-equilibrium state due to an above-critical-rate change of system parameters. We study a prototypical model for rate-induced tipping, the saddle-node normal form subject to time-varying equilibrium drift and noise. We find that both most commonly used early-warning indicators, increase in variance and increase in autocorrelation, occur not when the equilibrium drift is fastest but with a delay. We explain this delay by demonstrating that the most likely trajectory for tipping also crosses the tipping threshold with a delay, and therefore, the tipping itself is delayed. We find solutions of the variational problem determining the most likely tipping path using numerical continuation techniques. The result is a systematic study of the most likely tipping time in the plane of two parameters, distance from tipping threshold and noise intensity. [ABSTRACT FROM AUTHOR]

The present study considers the nonlinear dynamics of a Duffing oscillator under a symmetric potential subjected to a pulse-type excitation with a deformable shape. Our attention is focused on the effects of the external excitation shape parameter r and its period. The frequency response of the system is derived by using a semi-analytical approach. Interestingly, the frequency–response curve displays a large number of resonance peaks and anti-resonance peaks as well. Surprisingly, a resonance phenomenon termed here as shape-induced-resonance is noticed as it occurs solely due to the change in the shape parameter of the external periodic force. The system exhibits amplitude jumps and hysteresis depending on r. The critical driving magnitude for the chaos occurrence is investigated through Melnikov's method. Numerical analysis based on bifurcation diagrams and Lyapunov exponent is used to show how chaos occurs in the system. It is shown that the threshold amplitude of the excitation to observe chaotic dynamics decreases/increases for small/large values of r. In general, the theoretical estimates match with numerical simulations and electronic simulations as well. [ABSTRACT FROM AUTHOR]

Most classical chaotic systems, such as the Lorenz system and the Chua circuit, have chaotic attractors in bounded regions. This article constructs and analyzes a different kind of non-smooth impulsive systems, which have growing numbers of attractors in the sense that the number of attractors or the scrolls of an attractor is growing as time increases, and these attractors or scrolls are not located in bounded regions. It is found that infinitely many chaotic attractors can be generated in some of such systems. As an application, both theoretical and numerical analyses of an impulsive Lorenz-like system with infinitely many attractors are demonstrated. [ABSTRACT FROM AUTHOR]

We study the dynamic control of birhythmicity under an impulsive feedback control scheme where the feedback is made ON for a certain rather small period of time and for the rest of the time, it is kept OFF. We show that, depending on the height and width of the feedback pulse, the system can be brought to any of the desired limit cycles of the original birhythmic oscillation. We derive a rigorous analytical condition of controlling birhythmicity using the harmonic decomposition and energy balance methods. The efficacy of the control scheme is investigated through numerical analysis in the parameter space. We demonstrate the robustness of the control scheme in a birhythmic electronic circuit where the presence of noise and parameter fluctuations are inevitable. Finally, we demonstrate the applicability of the control scheme in controlling birhythmicity in diverse engineering and biochemical systems and processes, such as an energy harvesting system, a glycolysis process, and a p53-mdm2 network. [ABSTRACT FROM AUTHOR]

In this paper, we study resonances of the ABC-flow in the near integrable case ( C ≪ 1 ). This is an interesting example of a Hamiltonian system with 3/2 degrees of freedom in which simultaneous existence of two resonances of the same order is possible. Analytical conditions of the resonance existence are received. It is shown numerically that the largest n : 1 (n = 1, 2, 3) resonances exist, and their energies are equal to theoretical energies in the near integrable case. We provide analytical and numerical evidences for existence of two branches of the two largest n : 1 (n = 1, 2) resonances in the region of finite motion. [ABSTRACT FROM AUTHOR]

This work investigates experimentally and numerically frontal polymerization in a thermally anisotropic system with parallel copper strips embedded in 1,6-hexanediol diacrylate resin. Both experiments and multiphysics finite element analyses reveal that the front propagation in the thermally anisotropic system is orientation-dependent, leading to variations in the front shape and the front velocity due to the different front–metal strip interaction mechanisms along and across the metal strips. The parameters entering the cure kinetics model used in this work are chosen to capture the key characteristics of the polymerization front, i.e., the front temperature and velocity. Numerical parametric analyses demonstrate that the front velocity in the directions parallel and perpendicular to the metal strips increases as the system size decreases and approaches the analytical prediction for homogenized systems. A two-dimensional homogenized model for anisotropic frontal polymerization in the metal–resin system is proposed. [ABSTRACT FROM AUTHOR]

Heterogeneity in the load capacity of nodes is a common characteristic of many real-world networks that can dramatically affect their robustness to cascading overloads. However, most studies seeking to model cascading failures have ignored variations in nodal load capacity and functionality. The present study addresses this issue by extending the local load redistribution model to include heterogeneity in nodal load capacity and heterogeneity in the types of nodes employed in the network configuration and exploring how these variations affect network robustness. Theoretical and numerical analyses demonstrate that the extent of cascading failure is influenced by heterogeneity in nodal load capacity, while it is relatively insensitive to heterogeneity in nodal configuration. Moreover, the probability of cascading failure initiation at the critical state increases as the range of nodal load capacities increases. However, for large-scale networks with degree heterogeneity, a wide range of nodal load capacities can also suppress the spread of failure after its initiation. In addition, the analysis demonstrates that heterogeneity in nodal load capacity increases and decreases the extent of cascading failures in networks with sublinear and superlinear load distributions, respectively. These findings may provide some practical implications for controlling the spread of cascading failure. [ABSTRACT FROM AUTHOR]

To better explore asymmetrical interaction between epidemic spreading and awareness diffusion in multiplex networks, we distinguish susceptibility and infectivity between aware and unaware individuals, relax the degree of immunization, and take into account three types of generation mechanisms of individual awareness. We use the probability trees to depict the transitions between distinct states for nodes and then write the evolution equation of each state by means of the microscopic Markovian chain approach (MMCA). Based on the MMCA, we theoretically analyze the possible steady states and calculate the critical threshold of epidemics, related to the structure of epidemic networks, the awareness diffusion, and their coupling configuration. The achieved analytical results of the mean-field approach are consistent with those of the numerical Monte Carlo simulations. Through the theoretical analysis and numerical simulations, we find that global awareness can reduce the final scale of infection when the regulatory factor of the global awareness ratio is less than the average degree of the epidemic network but it cannot alter the onset of epidemics. Furthermore, the introduction of self-awareness originating from infected individuals not only reduces the epidemic prevalence but also raises the epidemic threshold, which tells us that it is crucial to enhance the early warning of symptomatic individuals during pandemic outbreaks. These results give us a more comprehensive and deep understanding of the complicated interaction between epidemic transmission and awareness diffusion and also provide some practical and effective recommendations for the prevention and control of epidemics. [ABSTRACT FROM AUTHOR]

Master stability functions (MSFs) are significant tools to identify the synchronizability of nonlinear dynamical systems. For a network of coupled oscillators to be synchronized, the corresponding MSF should be negative. The study of MSF will normally be discussed considering the coupling factor as a control variable. In our study, we considered various neuron models with electromagnetic flux induction and investigated the MSF's zero-crossing points for various values of the flux coupling coefficient. Our numerical analysis has shown that in all the neuron models we considered, flux coupling has increased the synchronization of the coupled neuron by increasing the number of zero-crossing points of MSFs or by achieving a zero-crossing point for a lesser value of a coupling parameter. [ABSTRACT FROM AUTHOR]

We study the interplay of global attractive coupling and individual noise in a system of identical active rotators in the excitable regime. Performing a numerical bifurcation analysis of the nonlocal nonlinear Fokker–Planck equation for the thermodynamic limit, we identify a complex bifurcation scenario with regions of different dynamical regimes, including collective oscillations and coexistence of states with different levels of activity. In systems of finite size, this leads to additional dynamical features, such as collective excitability of different types and noise-induced switching and bursting. Moreover, we show how characteristic quantities such as macroscopic and microscopic variability of interspike intervals can depend in a non-monotonous way on the noise level. [ABSTRACT FROM AUTHOR]

To explore the complexity of the locally active memristor and its application circuits, a tristable locally active memristor is proposed and applied in periodic, chaotic, and hyperchaotic circuits. The quantitative numerical analysis illustrated the steady-state switching mechanism of the memristor using the power-off plot and dynamic route map. For any pulse amplitude that can achieve a successful switching, there must be a minimum pulse width that enables the state variable to move beyond the attractive region of the equilibrium point. As local activity is the origin of complexity, the locally active memristor can oscillate periodically around a locally active operating point when connected in series with a linear inductor. A chaotic oscillation evolves from periodic oscillation by adding a capacitor in the periodic oscillation circuit, and a hyperchaotic oscillation occurs by further putting an extra inductor into the chaotic circuit. Finally, the dynamic behaviors and complexity mechanism are analyzed by utilizing coexisting attractors, dynamic route map, bifurcation diagram, Lyapunov exponent spectrum, and the basin of attraction. [ABSTRACT FROM AUTHOR]

An understanding of the underlying mechanism of side-branching is paramount in controlling and/or therapeutically treating mammalian organs, such as lungs, kidneys, and glands. Motivated by an activator-inhibitor-substrate approach that is conjectured to dominate the initiation of side-branching in a pulmonary vascular pattern, I demonstrate a distinct transverse front instability in which new fingers grow out of an oscillatory breakup dynamics at the front line without any typical length scale. These two features are attributed to unstable peak solutions in 1D that subcritically emanate from Turing bifurcation and that exhibit repulsive interactions. The results are based on a bifurcation analysis and numerical simulations and provide a potential strategy toward also developing a framework of side-branching for other biological systems, such as plant roots and cellular protrusions. [ABSTRACT FROM AUTHOR]

The effect of reaction delay, temporal sampling, sensory quantization, and control torque saturation is investigated numerically for a single-degree-of-freedom model of postural sway with respect to stability, stabilizability, and control effort. It is known that reaction delay has a destabilizing effect on the balancing process: the later one reacts to a perturbation, the larger the possibility of falling. If the delay is larger than a critical value, then stabilization is not even possible. In contrast, numerical analysis showed that quantization and control torque saturation have a stabilizing effect: the region of stabilizing control gains is greater than that of the linear model. Control torque saturation allows the application of larger control gains without overcontrol while sensory quantization plays a role of a kind of filter when sensory noise is present. These beneficial effects are reflected in the energy demand of the control process. On the other hand, neither control torque saturation nor sensory quantization improves stabilizability properties. In particular, the critical delay cannot be increased by adding saturation and/or sensory quantization. [ABSTRACT FROM AUTHOR]

We study the steady-state patterns of population of the coupled oscillators that sync and swarm, where the interaction distances among the oscillators have a finite-cutoff in the interaction distance. We examine how the static patterns known in the infinite-cutoff are reproduced or deformed and explore a new static pattern that does not appear until a finite-cutoff is considered. All steady-state patterns of the infinite-cutoff, static sync, static async, and static phase wave are repeated in space for proper finite-cutoff ranges. Their deformation in shape and density takes place for the other finite-cutoff ranges. Bar-like phase wave states are observed, which has not been the case for the infinite-cutoff. All the patterns are investigated via numerical and theoretical analyses. [ABSTRACT FROM AUTHOR]

In this study, we focus on the fractal property of recurrence networks constructed from the two-dimensional fractional Brownian motion (2D fBm), i.e., the inter-system recurrence network, the joint recurrence network, the cross-joint recurrence network, and the multidimen- sional recurrence network, which are the variants of classic recurrence networks extended for multiple time series. Generally, the fractal dimension of these recurrence networks can only be estimated numerically. The numerical analysis identifies the existence of fractality in these constructed recurrence networks. Furthermore, it is found that the numerically estimated fractal dimension of these networks can be connected to the theoretical fractal dimension of the 2D fBm graphs, because both fractal dimensions are piecewisely associated with the Hurst exponent H in a highly similar pattern, i.e., a linear decrease (if H varies from 0 to 0.5) followed by an inversely proportional-like decay (if H changes from 0.5 to 1). Although their fractal dimensions are not exactly identical, their difference can actually be deciphered by one single parameter with the value around 1. Therefore, it can be concluded that these recurrence networks constructed from the 2D fBms must inherit some fractal properties of its associated 2D fBms with respect to the fBm graphs. [ABSTRACT FROM AUTHOR]

We formulate a tumor-immune interaction model with a constant delay to capture the behavior following application of a dendritic cell therapy. The model is validated using experimental data from melanoma-induced mice. Through theoretical and numerical analyses, the model is shown to produce rich dynamics, such as a Hopf bifurcation and bistability. We provide thresholds for tumor existence and, in a special case, tumor elimination. Our work indicates a sensitivity in model outcomes to the immune response time. We provide a stability analysis for the high tumor equilibrium. For small delays in response, the tumor and immune system coexist at a low level. Large delays give rise to fatally high tumor levels. Our computational and theoretical work reveals that there exists an intermediate region of delay that generates complex oscillatory, even chaotic, behavior. The model then reflects uncertainty in treatment outcomes for varying initial tumor burdens, as well as tumor dormancy followed by uncontrolled growth to a lethal size, a phenomenon seen in vivo. Theoretical and computational analyses suggest efficacious treatments to use in conjunction with the dendritic cell vaccine. Additional analysis of a highly aggressive tumor additionally confirms the importance of representation with a time delay, as periodic solutions are strictly able to be generated when a delay is present. [ABSTRACT FROM AUTHOR]

In this study, an extended spatiotemporal model of a human immunodeficiency virus (HIV) CD4+ T cell with a drug therapy effect is proposed for the numerical investigation. The stability analysis of equilibrium points is carried out for temporal and spatiotemporal cases where stability regions in the space of parameters for each case are acquired. Three numerical techniques are used for the numerical simulations of the proposed HIV reaction–diffusion system. These techniques are the backward Euler, Crank–Nicolson, and a proposed structure preserving an implicit technique. The proposed numerical method sustains all the important characteristics of the proposed HIV model such as positivity of the solution and stability of equilibria, whereas the other two methods have failed to do so. We also prove that the proposed technique is positive, consistent, and Von Neumann stable. The effect of different values for the parameters is investigated through numerical simulations by using the proposed method. The stability of the proposed model of the HIV CD4+ T cell with the drug therapy effect is also analyzed. [ABSTRACT FROM AUTHOR]

We study an excitable active rotator with slowly adapting nonlinear feedback and noise. Depending on the adaptation and the noise level, this system may display noise-induced spiking, noise-perturbed oscillations, or stochastic bursting. We show how the system exhibits transitions between these dynamical regimes, as well as how one can enhance or suppress the coherence resonance or effectively control the features of the stochastic bursting. The setup can be considered a paradigmatic model for a neuron with a slow recovery variable or, more generally, as an excitable system under the influence of a nonlinear control mechanism. We employ a multiple timescale approach that combines the classical adiabatic elimination with averaging of rapid oscillations and stochastic averaging of noise-induced fluctuations by a corresponding stationary Fokker–Planck equation. This allows us to perform a numerical bifurcation analysis of a reduced slow system and to determine the parameter regions associated with different types of dynamics. In particular, we demonstrate the existence of a region of bistability, where the noise-induced switching between a stationary and an oscillatory regime gives rise to stochastic bursting. [ABSTRACT FROM AUTHOR]

We introduce a representation space to contrast chaotic with stochastic dynamics. Following the complex network representation of a time series through ordinal pattern transitions, we propose to assign each system a position in a two-dimensional plane defined by the permutation entropy of the network (global network quantifier) and the minimum value of the permutation entropy of the nodes (local network quantifier). The numerical analysis of representative chaotic maps and stochastic systems shows that the proposed approach is able to distinguish linear from non-linear dynamical systems by different planar locations. Additionally, we show that this characterization is robust when observational noise is considered. Experimental applications allow us to validate the numerical findings and to conclude that this approach is useful in practical contexts. [ABSTRACT FROM AUTHOR]

Bursting phenomena are found in a wide variety of fast–slow systems. In this article, we consider the Hindmarsh–Rose neuron model, where, as it is known in the literature, there are homoclinic bifurcations involved in the bursting dynamics. However, the global homoclinic structure is far from being fully understood. Working in a three-parameter space, the results of our numerical analysis show a complex atlas of bifurcations, which extends from the singular limit to regions where a fast–slow perspective no longer applies. Based on this information, we propose a global theoretical description. Surfaces of codimension-one homoclinic bifurcations are exponentially close to each other in the fast–slow regime. Remarkably, explained by the specific properties of these surfaces, we show how the Hindmarsh–Rose model exhibits isolas of homoclinic bifurcations when appropriate two-dimensional slices are considered in the three-parameter space. On the other hand, these homoclinic bifurcation surfaces contain curves corresponding to parameter values where additional degeneracies are exhibited. These codimension-two bifurcation curves organize the bifurcations associated with the spike-adding process and they behave like the "spines-of-a-book," gathering "pages" of bifurcations of periodic orbits. Depending on how the parameter space is explored, homoclinic phenomena may be absent or far away, but their organizing role in the bursting dynamics is beyond doubt, since the involved bifurcations are generated in them. This is shown in the global analysis and in the proposed theoretical scheme. [ABSTRACT FROM AUTHOR]