Nonlinear Coupling Function Research Articles

Stability analysis for memristor-based stochastic multi-layer neural networks with coupling disturbance

This paper discusses the asymptotical synchronization and the input-to-state exponential stability for memrist or-based multi-layers networks with delays under the coupling disturbance and stochastic noise. First, in order to solve the nonlinear coupling function of mismatched parameter and discontinuous activation in the system, methods of differential inclusion and Laplace transform are used. Then, based on the Lyapunov–Krasovskii functional, technique of inequality and linear matrix inequality, new sufficient conditions are also derived, in order to ensure the asymptotic synchronization and the input-to-state exponential stability of the considered model. Finally, two examples and simulations are given to illustrate the validity and correctness of our conclusions.

A 2D hybrid NMM-UPM method for waterflooding processes modelling considering reservoir fracturing

This study presents a methodology to integrate two simulation codes, i.e., cohesive element-based numerical manifold method (Co-NMM) and unified pipe network method (UPM), for 2D coupled two-phase seepage-stress analysis of waterflooding processes in fractured rock masses. The Co-NMM is an extension of the numerical manifold method (NMM) and designed for efficiently addressing continuity-discontinuity problems of rock especially complex multiple-fracture propagation, while the UPM is a relatively novel approach for underground seepage analysis with single−/multiphase fluid flow in both fractured and fractured porous reservoirs. To link the Co-NMM and UPM properly, a two-phase seepage-stress coupling model is proposed for considering interactions between the fractured rock mass and two-phase fluids. The mechanical behaviour of the fractured rock mass responds to the variation in skeletal effective stress induced by the average two-phase fluid pressure in both the rock matrix and fractures, including deformation of the rock matrix, change in fracture aperture and the initiation and propagation of fractures. Accordingly, the induced changes in hydraulic behaviours such as the porosity, permeability and capillary pressure are captured by nonlinear stress-dependent coupling functions. Furthermore, an implicit sequential solution is adopted to execute the two solvers, consisting of two iteration loops, i.e., the two-phase flow solving loop and seepage-stress coupling solving loop. The relative performance of the developed 2D hybrid NMM-UPM method is illustrated for four cases related to the waterflooding process in porous and fractured porous mediums.

IFOSMONDI Co-simulation Algorithm with Jacobian-Free Methods in PETSc

Co-simulation is a widely used solution to enable global simulation of a modular system via the composition of black-boxed simulators. Among co-simulation methods, the IFOSMONDI implicit iterative algorithm, previously introduced by the authors, enables us to solve the non-linear coupling function while keeping the smoothness of interfaces without introducing a delay. Moreover, it automatically adapts the size of the steps between data exchanges among the subsystems according to the difficulty of solving the coupling constraint. The latter was solved by a fixed-point algorithm, whereas this paper introduces the Jacobian-Free Methods version. Most implementations of Newton-like methods require a jacobian matrix which, except in the Zero-Order-Hold case, can be difficult to compute in the co-simulation context. As IFOSMONDI coupling algorithm uses Hermite interpolation for smoothness enhancement, we propose hereafter a new formulation of the non-linear coupling function including both the values and the time-derivatives of the coupling variables. This formulation is well designed for solving the coupling through jacobian-free Newton-type methods. Consequently, successive function evaluations consist in multiple simulations of the systems on a co-simulation time-step using rollback. The orchestrator-workers structure of the algorithm enables us to combine the PETSc framework on the orchestrator side for the non-linear Newton-type solvers with the parallel integrations of the systems on the workers’ side thanks to MPI processes. Different non-linear methods will be compared to one another and to the original fixed-point implementation on a newly proposed 2-system academic test case with direct feedthrough on both sides. An industrial model will also be considered to investigate the performance of the method.

A Synchronization Criterion for Two Hindmarsh-Rose Neurons with Linear and Nonlinear Coupling Functions Based on the Laplace Transform Method.

In this paper, an analytical criterion is proposed to investigate the synchronization between two Hindmarsh-Rose neurons with linear and nonlinear coupling functions based on the Laplace transform method. Different from previous works, the synchronization error system is expressed in its integral form, which is more convenient to analyze. The synchronization problem of two HR coupled neurons is ultimately converted into the stability problem of roots to a nonlinear algebraic equation. Then, an analytical criterion for synchronization between the two HR neurons can be given by using the Routh-Hurwitz criterion. Numerical simulations show that the synchronization criterion derived in this paper is valid, regardless of the periodic spikes or burst-spike chaotic behavior of the two HR neurons. Furthermore, the analytical results have almost the same accuracy as the conditional Lyapunov method. In addition, the calculation quantities always are small no matter the linear and nonlinear coupling functions, which show that the approach presented in this paper is easy to be developed to study synchronization between a large number of HR neurons.

Comment on "The Winfree model with non-infinitesimal phase-response curve: Ott-Antonsen theory" [Chaos 30, 073139 (2020)

In a recent paper [Chaos 30, 073139 (2020)], we analyzed an extension of the Winfree model with nonlinear interactions. The nonlinear coupling function Q was mistakenly identified with the non-infinitesimal phase-response curve (PRC). Here, we assess to what extent Q and the actual PRC differ in practice. By means of numerical simulations, we compute the PRCs corresponding to the Q functions previously considered. The results confirm a qualitative similarity between the PRC and the coupling function Q in all cases.

Finite-time synchronization of stochastic complex networks with random coupling delay via quantized aperiodically intermittent control

This paper studies the finite-time synchronization issue of stochastic complex networks (SCNs) with random coupling delay and nonlinear coupling function. Internal delay, perturbation delay are also considered into this model. Meanwhile, a quantized aperiodically intermittent control strategy is proposed to realize finite-time synchronization of SCNs. By means of Lyapunov stability theory and graph theory, two types of sufficient conditions are derived to ensure finite-time synchronization of SCNs. The obtained results are closely related to the maximum proportion of rest width and the topological structure of considered networks. Moreover, the theoretical results is applied to study finite-time synchronization of stochastic coupled oscillators. Finally, a numerical example is presented to demonstrate the effectiveness and feasibility of the obtained results.

Scalar modes, spontaneous scalarization and circular null-geodesics of black holes in scalar-Gauss–Bonnet gravity

In general relativity, astrophysical black holes (BHs) are simple objects, described by just their mass and spin. These simple solutions are not exclusive to general relativity, as they also appear in theories that allow for an extra scalar degree of freedom. Recently, it was shown that some theories which couple a scalar field with the Gauss–Bonnet invariant can have the same classic black hole solutions from general relativity as well as hairy BHs. These scalarized solutions can be stable, having an additional “charge” term that has an impact on the gravitational-wave emission by black hole binaries. In this paper, we overview black hole solutions in scalar-Gauss–Bonnet gravity, considering self-interacting terms for the scalar field. We present the mode analysis for the monopolar and dipolar perturbations around the Schwarzschild black hole in scalar-Gauss–Bonnet, showing the transition between stable and unstable solutions. We also present the time-evolution of scalar Gaussian wave packets, analyzing the impact of the scalar-Gauss–Bonnet term in their evolution. We then present some scalarized solutions, showing that nonlinear coupling functions and self-interacting terms can stabilize them. Finally, we compute the light-ring frequency and the Lyapunov exponent, which possibly estimate the black hole quasinormal modes in the eikonal limit.

Identification of the coupling functions between the process and the degradation dynamics by means of the variational Bayesian inference: an application to the solid-oxide fuel cells.

Understanding the way in which the degradation of a system's component is coupled to the system's dynamics is highly relevant for the monitoring and control of modern engineering systems. This paper focuses on the identification of coupling functions that describe the relationship between the system's dynamics and the degradation rate in solid-oxide fuel cell (SOFC) stacks. Based on the degradation dynamics estimated from data acquired online, we can design timely mitigation actions as well as optimal maintenance interventions. We introduce a computationally tractable identification approach that takes into account prior knowledge of the form of the coupling function that is found experimentally for a certain degradation mechanism. The nonlinear coupling function is estimated using variational Bayesian inference. The approach is tested on a 1600 h recording from a SOFC system. It is shown that the use of the prior form of the coupling function results in a superior prediction of the degradation, when compared with that obtained using purely data-driven black-box models. The reliable convergence of the variational Bayesian method and the simplicity of its implementation make it a promising tool for the in-field performance monitoring of SOFC systems. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.

Nonlinear phase coupling functions: a numerical study.

Phase reduction is a general tool widely used to describe forced and interacting self-sustained oscillators. Here, we explore the phase coupling functions beyond the usual first-order approximation in the strength of the force. Taking the periodically forced Stuart-Landau oscillator as the paradigmatic model, we determine and numerically analyse the coupling functions up to the fourth order in the force strength. We show that the found nonlinear phase coupling functions can be used for predicting synchronization regions of the forced oscillator. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.

Convergence Analysis of Signed Nonlinear Networks

This paper analyzes the convergence properties of signed networks with nonlinear edge functions. We consider diffusively coupled networks comprised of maximal equilibrium-independent passive (MEIP) dynamics on the nodes, and a general class of nonlinear coupling functions on the edges. The first contribution of this paper is to generalize the classical notion of signed networks for graphs with scalar weights to graphs with nonlinear edge functions using notions from the passivity theory. We show that the output of the network can finally form one or several steady-state clusters if all edges are positive and, in particular, all nodes can reach an output agreement if there is a connected subnetwork spanning all nodes and strictly positive edges. When there are nonpositive edges added to the network, we show that the tension of the network still converges to the equilibria of the edge functions if the relative outputs of the nodes connected by nonpositive edges converge to their equilibria. Furthermore, we establish the equivalent circuit models for signed nonlinear networks, and define the concept of equivalent edge functions, which is a generalization of the notion of the effective resistance. We finally characterize the relationship between the convergence property and the equivalent edge function, when a nonpositive edge is added to a strictly positive network comprised of nonlinear integrators. We show that the convergence of the network is always guaranteed, if the sum of the equivalent edge function of the previous network and the new edge function is passive.

Improved electro‐optic chaotic system with nonlinear electrical coupling

Concealing the time delay signature and enhancing the complexity of electro-optic (EO) chaotic system is important for practical applications. Therefore, in this study, the authors propose a new coupled EO chaotic system based on the classical one. In the new model, the electrical signals are mixed by using nonlinear coupling functions. Some simulation experiments are used to illustrate its chaotic performances and complexity. All results indicate that the new coupled model has a high complexity level, and can be competitive to other proposed EO systems. Furthermore, the time-delay signature can be effectively concealed with a smaller gain coefficient than other systems. All these results demonstrate that this new system is more potential to be used in random number generation and chaotic secure communication.

The Impact of Coupling Function on Finite‐Time Synchronization Dynamics of Multi‐Weighted Complex Networks with Switching Topology

This paper is not only concerned with the problem of finite‐time synchronization control for a class of nonlinear coupling multi‐weighted complex networks (NCMWCNs) with switching topology but also an attempt at using the derived results and Lyapunov stability theory to study the impact of nonlinear coupling function on finite‐time synchronization dynamics of the raised network model. Firstly, different from the existing related results, based on the existing and new finite‐time theories, two finite‐time synchronization controllers are, respectively, designed to make the considered network achieve finite‐time synchronization. Secondly, according to the obtained results, several finite‐time synchronization dynamics criteria are established to show that nonlinear coupled function and the switching of outer‐coupling matrix are how to impact finite‐time synchronization dynamics. Finally, two illustrated examples are provided to verify the effectiveness of theoretical results proposed in this paper.

Synchronization of Interconnected Discontinuous Neural Networks With Nonlinear Coupling Functions

Some criteria on exponential synchronization of multiple identical discontinuous neural networks with time delay are given in this paper. Different from the available results, the nodes of complex networks are coupled via nonlinear discontinuous functions. First, we study the existence of Filippov solution of the nonlinear coupled discontinuous neural networks (NCDNNs) based on differential inclusion theory. In addition, we derive some exponential synchronization criteria of the NCDNNs. Finally, we design a pinning impulsive controller to realize the exponential synchronization of NCDNNs. Numerical examples are presented to illustrate the obtained results.

Finite-time Synchronization Control Relationship Analysis of Two Classes of Markovian Switched Complex Networks

In this paper, finite-time global synchronization control problem for a class of nonlinear coupling Markovian switched complex networks (NCMSCNs) is investigated. Furthermore, according to differentiability of nonlinear coupling function g(x,y), g(x,y) how to affect synchronization dynamics of the class of NCMSCNs is analyzed by two viewpoints. The first is that if g(x,y) satisfies the Lipschitz condition and is derivable, the above question is discussed by taking g(x,y) = L1x+L2y, g(x,y) =–L1x+L2y, g(x,y) = L1x–L2y and g(x,y) =–L1x–L2y, where L1 > 0, L2 > 0. The second is that if nonlinear coupling function g(x,y) only satisfies the Lipschitz condition, by analyzing the differences of synchronization control rules for the class of NCMSCNs and a class of linear coupling Markovian switched complex networks (LCMSCNs), the problem is explored. Comparing the previous works [12,21,22,26,33,34], the main improvement of this paper is that the works of this paper extend the existed analyzing ideas of the finite-time global synchronization for nonlinear coupling complex networks, including NCMSCNs.

Experimental Realization of the Coupling Function Secure Communications Protocol and Analysis of Its Noise Robustness

There is an increasing need for everyday communications to be both secure and resistant to external perturbations. We have therefore created an experimental implementation of the coupling-function-based secure communication protocol, in order to assess its robustness to channel noise. The transmitter and receiver are implemented on single-board computers, thereby facilitating communication of the analog electronic signals. The information signals are encrypted at the transmitter as the timevariability of nonlinear coupling functions between dynamical systems. This results in a complicated nonlinear mixing and scrambling of the information. To replicate the channel noise, analog white noise is added to the encrypted signals. After digitization at the receiver, the decryption is performed with dynamical Bayesian inference to take account of time-varying dynamics in the presence of noise. The dynamical Bayesian approach effectively separates the deterministic information signals from the perturbations of dynamical channel noise. The experimental realization has demonstrated the feasibility, and established the performance, of the protocol for secure, reliable, communication even with high levels of channel noise.

Thermal and load rate-dependent interaction between embedded rail system and bridge

Recent developments in the technology of a modern embedded rail system grant many new possibilities for improving the railway infrastructure. An embedded rail system significantly reduces noise and the dynamical impact affecting both the infrastructure and the rolling stock itself. An embedded rail system may be used for constructing any type of railway infrastructure including high-speed railways. Its attributes are also suitable for modernization of the current steel railway bridges or constructing new structures, where slender bridge decks are required. However, in a railway track equipped with an embedded rail system, stress increments rise due to its restricted expansion movement. This effect is naturally higher when the embedded rail system is placed on a bridge, because of their different deformation possibilities. For a wider use of embedded rail system, examining the interaction behavior between the embedded rail system and the substructure is a matter of essential importance. Furthermore, the nonlinear character of embedded rail system polymer-based components needs to be considered when examining their interaction behavior. Therefore, this paper aims at investigating the nonlinear coupling functions of an embedded rail system under the effects of temperature and load rate. For this purpose, a comprehensive analysis consisting of laboratory experiments, material tests of the embedded rail system components and subsequent numerical validation was performed. Results are concluded in the paper.

Reconstruction of ensembles of generalized Van der Pol oscillators from vector time series

The method for reconstruction of nonlinear potential, dissipation and coupling functions for all elements of ensembles of coupled generalized Van der Pol oscillators (including Rayleigh and Bonhoeffer–Van der Pol oscillators) from multivariate time series is proposed.It is based on the idea of using the smoothness of reconstructed potential function as a criterion for quality of reconstruction. Therefore, no parametrization for potential is required and an arbitrary potential function unique for each oscillator in the ensemble can be considered. Different approaches to reconstruction of dissipation are studied. Linear and nonlinear coupling between oscillators is considered, including coupling by derivatives. Effects of measurement noise, contaminating the series, are taken into account.

Local [formula omitted] synchronization of uncertain complex networks via non-fragile state feedback control

This paper is concerned with local H∞ synchronization control of uncertain complex networks, in which there are nonlinear coupling functions between network nodes, and the coupling terms as well as network nodes are subject to time-varying delays. In order to make a complex network achieve H∞ synchronization, a non-fragile state feedback control scheme is used. By employing the Lyapunov functional method, a sufficient condition on the existence of H∞ non-fragile controllers is derived. This condition can not only guarantee the asymptotic stability of each network node, but also ensure a prescribed robust H∞ performance level for the synchronization error system under study. The effectiveness of the proposed method is finally verified through a numerical example.

Switch of synchronization states of aeroacoustical coupled organ pipes

The separation distance of two equally tuned, aeroacoustically coupled organ pipes is varied stepwise in a synchronization experiment. At distances of multiples of about half the wavelength of the fundamentals, abrupt changes of SPL and fundamental frequency of the coupled and synchronized system can be observed. Frequency jumps also occur at higher harmonics. The nonlinear coupling function of the system implicitly takes into account the time delay of the system caused by the separation distance and therefore plays an important role for the observed phenomenon. The effect can be explained as a switch of synchronization states from in-phase to anti-phase and vice versa.

Chimera states in two-dimensional networks of locally coupled oscillators

Chimera state is defined as a mixed type of collective state in which synchronized and desynchronized subpopulations of a network of coupled oscillators coexist and the appearance of such anomalous behavior has strong connection to diverse neuronal developments. Most of the previous studies on chimera states are not extensively done in two-dimensional ensembles of coupled oscillators by taking neuronal systems with nonlinear coupling function into account while such ensembles of oscillators are more realistic from a neurobiological point of view. In this paper, we report the emergence and existence of chimera states by considering locally coupled two-dimensional networks of identical oscillators where each node is interacting through nonlinear coupling function. This is in contrast with the existence of chimera states in two-dimensional nonlocally coupled oscillators with rectangular kernel in the coupling function. We find that the presence of nonlinearity in the coupling function plays a key role to produce chimera states in two-dimensional locally coupled oscillators. We analytically verify explicitly in the case of a network of coupled Stuart-Landau oscillators in two dimensions that the obtained results using Ott-Antonsen approach and our analytical finding very well matches with the numerical results. Next, we consider another type of important nonlinear coupling function which exists in neuronal systems, namely chemical synaptic function, through which the nearest-neighbor (locally coupled) neurons interact with each other. It is shown that such synaptic interacting function promotes the emergence of chimera states in two-dimensional lattices of locally coupled neuronal oscillators. In numerical simulations, we consider two paradigmatic neuronal oscillators, namely Hindmarsh-Rose neuron model and Rulkov map for each node which exhibit bursting dynamics. By associating various spatiotemporal behaviors and snapshots at particular times, we study the chimera states in detail over a large range of coupling parameter. The existence of chimera states is confirmed by instantaneous angular frequency, order parameter and strength of incoherence.