Dynamical behaviors of fuzzy reaction–diffusion periodic cellular neural networks with variable coefficients and delays

When modeling neural networks in a real world, not only diffusion effect and fuzziness cannot be avoided, but also self-inhibitions, interconnection weights, and inputs should vary as time varies. In this paper, we discuss the dynamical behaviors of delayed reaction–diffusion fuzzy cellular neural networks with varying periodic self-inhibitions, interconnection weights as well as inputs. By using Halanay’s delay differential inequality, M$M$-matrix theory and analytic methods, some new sufficient conditions are obtained to ensure the existence, uniqueness, and global exponential stability of the periodic solution, and the exponentially convergent rate index is also estimated. In particular, the traditional assumption on the differentiability of the time-varying delays is no longer needed. The methodology developed in this paper is shown to be simple and effective for the exponential periodicity and stability analysis of neural networks with time-varying delays. Two examples are given to show the usefulness of the obtained results that are less restrictive than recently known criteria.     

Synchronization analysis for complex networks with coupling delay based on T–S fuzzy theory

In this paper, synchronization problems for a general complex networks are investigated by Takagi–Sugeno (T–S) fuzzy theory. A novel concept named linear approximation method is firstly proposed to solve the synchronization problems for T–S fuzzy complex networks. This novel method can linearize the system into some time-delay subsystems, which can effectively simplify the complicated system and then be easy to acquire the synchronization results. Since the expression based on linear matrix inequality (LMI) is used, the synchronization criteria can be easily checked in practice. Numerical simulation examples are provided to verify the effectiveness and the applicability of the proposed method.     

Synchronization of Takagi–Sugeno fuzzy stochastic discrete-time complex networks with mixed time-varying delays

In this paper, we propose and investigate a new general model of fuzzy stochastic discrete-time complex networks (SDCNs) described by Takagi–Sugeno (T–S) fuzzy model with discrete and distributed time-varying delays. The proposed model takes some well-studied models as special cases. By employing a new Lyapunov functional candidate, we utilize some stochastic analysis techniques and Kronecker product to deduce delay-dependent synchronization criteria that ensure the mean-square synchronization of the proposed T–S fuzzy SDCNs with mixed time-varying delays. These sufficient conditions are computationally efficient as it can be solved numerically by the LMI toolbox in Matlab. A numerical simulation example is provided to verify the effectiveness and the applicability of the proposed approach.     

Convergence analysis for second-order interval Cohen–Grossberg neural networks

This paper presents new theoretical results on global stability of a class of second-order interval Cohen–Grossberg neural networks. The new criteria is derived to ensure the existence, uniqueness and global stability of the equilibrium point of neural networks under uncertainties. And we make some comparisons between our results with the existed corresponding results. Some examples are provided to show the effectiveness of the obtained results.     

Synchronization of drive–response singular Boolean networks

Network synchronization, explicating typical collective behaviors of coupled systems, plays a crucial role in social production and life. This paper addresses the synchronization problem of drive–response singular Boolean networks (SBNs). The solvability of drive–response SBNs is investigated based on the matrix representation. In view of the existence and uniqueness of the solutions to drive–response SBNs, three types of concepts, synchronization, strong synchronization and weak synchronization, are put forward for the first time. By two new systems, a restricted BN and a switched restricted BN, which are constructed from the considered systems, several synchronization conditions are provided to deal with the circumstances of unique solutions and multiple solutions, respectively. Besides, the synchronous ratio is defined to characterize the synchronization capability of drive–response SBNs for the case of multiple solutions. Finally, several examples are given to illustrate the effectiveness of the obtained results.     

Exponential stability of fuzzy Cohen–Grossberg neural networks with time delays and impulsive effects

In this paper, we investigate a class of fuzzy Cohen-Grossberg neural networks with time delays and impulsive effects. By employing an inequality technique, we find sufficient conditions for the existence, uniqueness, global exponential stability of the equilibrium without using the M-matrix theory. An example is given to illustrate the effectiveness of the obtained results.     

Synchronization of boundary coupled Hindmarsh–Rose neuron network

In this work, we present a new mathematical model of a boundary coupled neuron network described by the partly diffusive Hindmarsh–Rose equations. We prove the global absorbing property of the solution semiflow and then the main result on the asymptotic synchronization of this neuron network at a uniform exponential rate provided that the boundary coupling strength and the stimulating signal exceed a quantified threshold in terms of the parameters.     

Stability analysis of fuzzy Markovian jumping Cohen–Grossberg BAM neural networks with mixed time-varying delays

In this paper, we investigate the robust stability of uncertain fuzzy Markovian jumping Cohen–Grossberg BAM neural networks with discrete and distributed time-varying delays. A new delay-dependent stability condition is derived under uncertain switching probabilities by Takagi–Sugeno fuzzy model. Based on the linear matrix inequality (LMI) technique, upper bounds for the discrete and distributed delays are calculated using the LMI toolbox in MATLAB. Numerical examples are provided to illustrate the effectiveness of the proposed method.     

Dynamical behaviors of reaction–diffusion fuzzy neural networks with mixed delays and general boundary conditions

In this paper, we study delayed reaction–diffusion fuzzy neural networks with general boundary conditions. By using topology degree theory and constructing suitable Lyapunov functional, some sufficient conditions are given to ensure the existence, uniqueness and globally exponential stability of the equilibrium point. Finally, an example is given to verify the theoretical analysis.     

Global stability analysis for stochastic coupled reaction–diffusion systems on networks

Coupled systems on networks (CSNs) can be used to model many real systems, such as food webs, ecosystems, metabolic pathways, the Internet, World Wide Web, social networks, and global economic markets. This paper is devoted to investigation of the stability problem for some stochastic coupled reaction–diffusion systems on networks (SCRDSNs). A systematic method for constructing global Lyapunov function for these SCRDSNs is provided by using graph theory. The stochastic stability, asymptotically stochastic stability and globally asymptotically stochastic stability of the systems are investigated. The derived results are less conservative than the results recently presented in Luo and Zhang [Q. Luo, Y. Zhang, Almost sure exponential stability of stochastic reaction diffusion systems. Non-linear Analysis: Theory, Methods & Applications 71(12) (2009) e487–e493]. In fact, the system discussed in Q. Luo and Y. Zhang [Q. Luo, Y. Zhang, Almost sure exponential stability of stochastic reaction diffusion systems. Non-linear Analysis: Theory, Methods & Applications 71(12) (2009) e487–e493] is a special case of ours. Moreover, our novel stability principles have a close relation to the topological property of the networks. Our new method which constructs a relation between the stability criteria of a CSN and some topology property of the network, can help analyzing the stability of the complex networks by using the Lyapunov functional method.     

Soliton interactions and Yang–Baxter maps for the complex coupled short-pulse equation

The complex coupled short-pulse equation (ccSPE) describes the propagation of ultrashort optical pulses in nonlinear birefringent fibers. The system admits a variety of vector soliton solutions: fundamental solitons, fundamental breathers, composite breathers (generic or nongeneric), as well as so-called self-symmetric composite solitons. In this work, we use the dressing method and the Darboux matrices corresponding to the various types of solitons to investigate soliton interactions in the focusing ccSPE. The study combines refactorization problems on generators of certain rational loop groups, and long-time asymptotics of these generators, as well as the main refactorization theorem for the dressing factors that leads to the Yang–Baxter property for the refactorization map and the vector soliton interactions. Among the results obtained in this paper, we derive explicit formulas for the polarization shift of fundamental solitons that are the analog of the well-known formulas for the interaction of vector solitons in the Manakov system. Our study also reveals that upon interacting with a fundamental breather, a fundamental soliton becomes a fundamental breather and, conversely, that the interaction of two fundamental breathers generically yields two fundamental breathers with a polarization shifts, but may also result into a fundamental soliton and a fundamental breather. Explicit formulas for the coefficients that characterize the fundamental breathers, as well as for their polarization vectors are obtained. The interactions of other types of solitons are also derived and discussed in detail and illustrated with plots. New Yang–Baxter maps are obtained in the process.     

Stability analysis of Takagi–Sugeno fuzzy Cohen–Grossberg BAM neural networks with discrete and distributed time-varying delays

In this paper, the global asymptotic stability problem of Takagi–Sugeno (TS) fuzzy Cohen–Grossberg Bidirectional Associative Memory neural networks (FCGBAMNNs) with discrete and distributed time-varying delays is considered. A novel LMI-based stability criterion is obtained by using Lyapunov functional theory to guarantee the asymptotic stability of FCGBAMNNs which are represented by TS fuzzy models. Our results can be easily verified and are also less restrictive than previously known criteria and can be applied to Cohen–Grossberg neural networks, recurrent neural networks and cellular neural networks. Finally, the proposed stability conditions are demonstrated with a numerical example.     

Exponential synchronization of memristive Hindmarsh–Rose neural networks

A new model of neural networks in terms of the memristive Hindmarsh–Rose equations is proposed. Globally dissipative dynamics is shown with absorbing sets in the state spaces. Through sharp and uniform grouping estimates and by leverage of integral and interpolation inequalities tackling the linear network coupling against the memristive nonlinearity, it is proved that exponential synchronization at a uniform convergence rate occurs when the coupling strengths satisfy the threshold conditions which are quantitatively expressed by the parameters.     

Global stability for discrete Cohen–Grossberg neural networks with finite and infinite delays

In this paper, the global stability problem for a general discrete Cohen–Grossberg neural network with finite and infinite delays is investigated. A simple criterion ensuring the global asymptotical stability is established, by applying the Lyapunov method and graph theory. Finally, an example showing the effectiveness of the provided criterion is given.     

Adaptive synchronization in an array of linearly coupled neural networks with reaction–diffusion terms and time delays

In this paper, the adaptive synchronization in an array of linearly coupled neural networks with reaction–diffusion terms and time delays is discussed. Based on the LaSalle invariant principle of functional differential equations and the adaptive feedback control technique, some sufficient conditions for adaptive synchronization of such a system are obtained. Finally, a numerical example is given to show the effectiveness of the proposed synchronization method.     

Almost periodicity for a class of delayed Cohen–Grossberg neural networks with discontinuous activations

The objective of this paper is to investigate the dynamics of a class of delayed Cohen–Grossberg neural networks with discontinuous neuron activations. By means of retarded differential inclusions, we obtain a result on the local existence of solutions, which improves the previous related results for delayed neural networks. It is shown that an M-matrix condition satisfied by the neuron interconnections, can guarantee not only the existence and uniqueness of an almost periodic solution, but also its global exponential stability. It is also shown that the M-matrix condition ensures that all solutions of the system display a common asymptotic behavior. In this paper, we prove that the existence interval of the almost periodic solution is (?∞, +∞), whereas the existence interval is only proved to be [0, +∞) in most of the literature. As special cases, we derive the results of existence, uniqueness and global exponential stability of a periodic solution for delayed neural networks with periodic coefficients, as well as the similar results of an equilibrium for the systems with constant coefficients. To the author’s knowledge, the results in this paper are the only available results on almost periodicity for Cohen–Grossberg neural networks with discontinuous activations and delays.     

LMI conditions for stability of impulsive stochastic Cohen–Grossberg neural networks with mixed delays

In this paper, the global asymptotic stability of impulsive stochastic Cohen–Grossberg neural networks with mixed delays is investigated by using Lyapunov–Krasovskii functional method and the linear matrix inequality (LMI) technique. The mixed time delays comprise both the multiple time-varying and continuously distributed delays. Some new sufficient conditions are obtained to guarantee the global asymptotic stability of the addressed model in the stochastic sense using the powerful MATLAB LMI toolbox. The results extend and improve the earlier publications. Two numerical examples are given to illustrate the effectiveness of our results.