Nonlinear Coupling Function Research Articles

Synchronization of spatiotemporal chaos in large scale rich-club network

The Plankton spatiotemporal chaos system is taken as network node and constructed as a rich-club network through nonlinear coupling. The synchronization of spatiotemporal chaos for the above network is investigated. The general selection rule of nonlinear coupling function connecting nodes in the rich-club network is presented. Furthermore, the condition to realize the network synchronization is analyzed theoretically based on Lyapunov stability theory. Finally, the synchronization effect of spatiotemporal chaos for the rich-club network is checked through artificial simulation. The results show that complete synchronization can be realized for all rich nodes in the rich-club network and all nodes in every subnetwork constructed in star-shape.

Robust adaptive synchronization of dynamical networks via scalar controllers

Based on high gain feedback control theory, robust adaptive synchronization of dynamical network is investigated in this paper. When the non-linear coupling functions are unknown but with unknown bounded, some fairly simple robust adaptive scalar feedback controllers are derived. The key idea is that a time-varying gain parameter is introduced in designing controllers which can guarantee that the states of uncertain coupled dynamical networks robust adaptive asymptotically synchronize with each other. Numerical simulation is given to validate the proposed theoretical result.

Synchronization analysis for nonlinearly-coupled complex networks with an asymmetrical coupling matrix

In this paper, the global synchronization for an array of nonlinearly coupled identical chaotic systems is investigated. A distinctive feature of this work is to address synchronization issues for nonlinearly coupled complex networks with an asymmetrical coupling matrix. By projecting the nonlinear coupling function onto a linear one and assuming the difference between them as a disturbing function, we give some criteria for the global synchronization in virtual of the left eigenvector corresponding to the zero eigenvalue of the coupling matrix. Numerical examples are also provided to demonstrate the effectiveness of the theory.

Decentralized tracking-type games for multi-agent systems with coupled ARX models: Asymptotic Nash equilibria

A class of decentralized tracking-type games is considered for large population multi-agent systems (MAS). The agents are described by stochastic discrete-time auto-regressive models with exogenous inputs (ARX models), and coupled together through their individual dynamics and performance indexes by terms of the unknown population state average (PSA). The performance index of each agent to minimize is a stochastic long term averaged group-tracking-type functional, in which there is a nonlinear term of the unknown PSA. The control law is decentralized and implemented via the Nash certainty equivalence principle. By probability limit theory, under mild conditions it is shown that: (a) the estimate of the PSA is strongly consistent; (b) the closed-loop system is stable almost surely, and the stability is independent of the number N of agents; (c) the decentralized control law is an asymptotic Nash equilibrium almost surely or in probability according to the property of the nonlinear coupling function in the performance indexes.

Chaotic synchronization of nearest-neighbor diffusive coupling Hindmarsh–Rose neural networks in noisy environments

The chaotic synchronization of Hindmarsh–Rose neural networks linked by a nonlinear coupling function is discussed. The HR neural networks with nearest-neighbor diffusive coupling form are treated as numerical examples. By the construction of a special nonlinear-coupled term, the chaotic system is coupled symmetrically. For three and four neurons network, a certain region of coupling strength corresponding to full synchronization is given, and the effect of network structure and noise position are analyzed. For five and more neurons network, the full synchronization is very difficult to realize. All the results have been proved by the calculation of the maximum conditional Lyapunov exponent.

Exponential synchronization of nonlinear coupled dynamical networks with a delayed coupling

In this paper, we discuss exponential synchronization of nonlinear coupled dynamical networks with a delayed coupling. Sufficient conditions for both delay-independent and delay-dependent global exponential synchronization are obtained based on Lyapunov functional method. It is shown that, for some coupled chaotic systems and nonlinear coupling functions, if the coupling delay is less than a positive threshold, then the coupled network will be synchronized. On the other hand, with the increase of coupling delay, the synchronizability of the coupled network will be restrained, even eventually de-synchronized. Finally, an example is provided to demonstrate the effectiveness of the proposed results.

Chaotic synchronization and control in nonlinear-coupled Hindmarsh–Rose neural systems

Abstract A new approach for chaotic synchronization of Hindmarsh–Rose (HR) neural networks linked by special nonlinear coupling function is proposed. The method expands SC method in investigation of chaotic synchronization based on the stability criterion. We provide the error evolutional equation to determine the stability of synchronized states, which has very simple forms corresponding to matrix of star coupling coefficients. The synchronization can be achieved without the requirement to calculate the maximum Lyapunov exponents when the coupling strengths are taken as reference values, and there is a region of stability around them. Besides, the stability criterion control method is applied to control chaotic behaviors of individual Hindmarsh–Rose neuron model. The chaotic orbit is stabilized on 5spike/burst orbit embedded in the chaotic attractor by an input of the nonlinear time-continuous feedback perturbation to membrane potential.

An Early Prediction of Maximum Sunspot Number in Solar Cycle 24

A non-linear coupling function between sunspot maxima and aa minima modulations has been found as a result of a wavelet analysis of geomagnetic index aa and Wolf sunspot number yearly means since 1844. It has been demonstrated that the increase of these modulations for the past 158 years has not been steady, instead, it has occurred in less than 30 years starting around 1923. Otherwise sunspot maxima have oscillated about a constant level of 90 and 141, prior to 1923 and after 1949, respectively. The relevance of these findings regarding the forecasting of solar activity is analyzed here. It is found that if sunspot cycle maxima were still oscillating around the 141 constant value, then the Gnevyshev–Ohl rule would be violated for two consecutive even–odd sunspot pairs (22–23 and 24–25) for the first time in 1700 years. Instead, we present evidence that solar activity is in a declining episode that started about 1993. A value for maximum sunspot number in solar cycle 24 (87.5±23.5) is estimated from our results.

Factors influencing the intensity of magnetospheric substorms

A magnetospheric substorm is the sequence of processes which occur throughout the magnetosphere following a southward turning of the interplanetary magnetic field and the onset of dayside reconnection. An isolated substorm consists of three phases, growth, expansion and recovery. Two primary processes make up a substorm. The directly driven component visible in the global, two cell current system is powered by the direct projection of the solar wind electric field onto the conducting ionosphere. The unloading component seen as the one cell intensification of the westward electrojet near midnight is powered by processes internal to the plasma sheet. The intensity of substorms can be measured by a variety of activity indices, the most common of which are magnetic indices. The solar wind controls the intensity of substorms through the process of magnetic reconnection. Time series analysis has established that 40 min after a southward turning of the interplanetary magnetic field the strength of the electrojets is correlated with solar wind parameters through a nonlinear coupling function similar to V 2B Sin 4( θ 2 ) ( where θ = IMF clock angle about Sun vector). Linear filter analysis of the magnetospheric system response shows that the average impulse response for the AL index is a Rayleigh function peaking at a lag time of 60 min and diminishing to zero after 3 h. The average response function for a sequence of moderate substorms is bimodal with peaks at 25 and 70 min, while more intense activity is uni-modal with only the 25 min peak. Response functions for individual substorms have peak amplitudes and delays that can be substantially different from the average. The linear analysis predicts less than half the variance in the AL index when high resolution indices are used. These results imply that reconnection on the day and night side are delayed relative to the solar wind by differing amounts which vary with conditions in the solar wind, magnetosphere, and ionosphere. Currents are driven through the ionosphere that are proportional to the reconnection rates which are in turn proportional to the solar wind electric field, but their strength depends on ionospheric conductivity. Conductivity depends on season, prior activity, and the way in which a particular substorm develops. Computer simulation of substorms with a simple ‘leaky tap’ model suggests that one reason for lack of short term predictabilty is likely to be that the magnetosphere is nonlinear with behavior depending both on previous conditions and the strength and wave form of the solar wind input. New techniques of analysis of magnetic indices support this conjecture. As with the Earth's surface weather, accurate space weather predictions will probably require extensive real-time observatory networks both in the solar wind and magnetosphere.

Method of solution of a special class of two-point boundary-value problems

A numerical method is presented for the solution of a special class of two-point boundary value problems characterised by state equations of the form x = f(U, t), a nonlinear coupling function and fixed end points.