Cohen-Grossberg Neural Networks Research Articles

Existence and exponential stability of a periodic solution of an infinite delay differential system with applications to Cohen–Grossberg neural networks

In the present paper, we investigate both the global exponential stability and the existence of a periodic solution of a general differential equation with unbounded distributed delays. The main stability criterion depends on the dominance of the non-delay terms over the delay terms. The criterion for the existence of a periodic solution is obtained with the application of the coincidence degree theorem. We use the main results to get criteria for the existence and global exponential stability of periodic solutions of a generalized higher-order periodic Cohen–Grossberg neural network model with discrete-time varying delays and infinite distributed delays. Additionally, we provide a comparison with the results in the literature and two numerical examples to illustrate the effectiveness of some of our results.

Coexistence and locally exponential stability of multiple equilibrium points for fractional-order impulsive control Cohen–Grossberg neural networks

Different from the existing multiple Mittag-Leffler stability or multiple asymptotic stability, the multiple exponential stability, which has explicit and faster convergence rate, is investigated in this paper for fractional-order impulsive control Cohen–Grossberg neural networks. First, by using the definition of Dirac delta function, the fractional-order control Cohen–Grossberg neural networks are translated into fractional-order impulsive neural networks, in which pulse effect relies on the fractional order of the addressed system. Then, based on maximum norm, 1−norm and general q−norm (q=2n), a series of novel criteria are obtained respectively to ensure that such n−neuron neural networks can have ∏i=1n(2Li+1) total equilibrium points and ∏i=1n(Li+1) locally exponentially stable equilibrium points, by utilizing the known fixed point theorem, the method of average impulsive interval, the theory of fractional-order differential equations, and the method of Lyapunov function. This paper’s investigation reveals the effects of impulsive function, impulsive interval, and fractional order on the dynamic behaviors. Finally, theoretical results are shown to be effective by four illustrative examples.

An ingenious scheme to bifurcations in a fractional-order Cohen–Grossberg neural network with different delays

An ingenious scheme to bifurcations in a fractional-order Cohen–Grossberg neural network with different delays

Almost automorphic behaviours of nonlocal stochastic fuzzy Cohen–Grossberg lattice neural networks

Discrete-time Cohen–Grossberg neural networks (CGNNs) play important role in feedback neural networks. The existing literatures of CGNNs only regarded integral order discrete-time models without other variables. This paper builds the lattice model for nonlocal stochastic fuzzy CGNNs with reaction diffusions by employing the finite difference and Mittag–Leffler time Euler difference techniques. Further, the existence of a unique bounded almost automorphic sequence solution in distribution and global exponential convergence in the mean-square sense to the achieved difference model are investigated. At last, by using the tool of Matlab and time Euler difference for the Brownian motions, an illustrative example with simulations is used to show the feasible of the works of the current paper. We think that this work can contribute to achieve some effective real tasks, e.g. content-addressable memory, auto-association, dynamic reconstruction of a chaotic process, etc.

Novel passivity and dissipativity criteria for discrete-time fractional generalized delayed Cohen–Grossberg neural networks

This paper pays attention to the passivity and dissipativity for discrete-time fractional generalized delayed Cohen-Grossberg neural networks. A new fractional passive lemma is firstly proposed for discrete-time system by means of the Lyapunov functional. This facilitates the discussion of system stabilization in terms of input and output energy. Some passive and dissipative conditions are established under comparison principle, reduction to absurdity, inequality techniques and the proposed passive lemma. Finally, two examples are performed to demonstrate the validity of proposed methods.

Fixed-time synchronization of Inertial Cohen-Grossberg Neural Networks with state dependent delayed impulse control and its application to multi-image encryption

In this paper, we discussed about fixed-time synchronization (FXTS) of Inertial Cohen-Grossberg Neural Networks (ICGNNs) with state-dependent delayed impulses. The Lyapunov stability theory and several useful criteria are utilized to make sure that the control parameters are selected in sync with the intended settling time. Two types of the controller are developed in order to guarantee that error-delayed ICGNNs can be synchronized. A sufficient condition for ensuring FXTS for delayed ICGNNs with desynchronization impulses is investigated. In FXTS, the settling time of ICGNNs will have the smallest upper bound and the settling time of desynchronization will have the largest upper bound. We subsequently conducted numerical simulations to substantiate the validity of the proposed discoveries. Finally, we proposed a multi-image encryption algorithm with the help of ICGNNs and presented the statistical analysis to test its efficacy.

Periodic Self-Triggered Impulsive Synchronization of Hybrid Stochastic Complex-Valued Delayed Networks

This paper studies the synchronization problem for hybrid stochastic delay complex-valued networks with time-varying multi-links (HSDCNT) via periodic self-triggered impulsive control (PSIC). Therein, PSIC combines with the benefits of impulsive control and periodic self-triggered strategy. Moreover, time delays, stochastic disturbances, and time-varying multi-links are considered simultaneously. Particularly, the synchronization for HSDCNT is studied on the complex space and no split real and imaginary parts. Actually, different from the previous PSIC-based results, this paper extends the real-valued neural networks to HSSs, delayed systems, and complex-valued networks, which is significant and challenging. Then, as a practical application, we consider the synchronization of a kind of Cohen-Grossberg neural network in the complex space. Finally, a numerical example is utilized for demonstration.

Besicovitch Almost Anti-periodic Solution of Octonion-Valued Cohen–Grossberg Neural Networks with Delays on Time Scales

In this work, we put forward a concept of Besicovitch almost anti-periodic functions on time scales, which is new even when time scale T=R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {T}=\\mathbb {R}$$\\end{document} or Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}$$\\end{document}. Based on this, we use the fixed point theorem and analytical techniques to obtain the existence, uniqueness and global exponential stability of Besicovitch almost anti-periodic solutions for a class of octonion-valued Cohen–Grossberg neural networks with time delays on time scales. Finally, the validity of the results is verified by a numerical example.

Exponential synchronization of quaternion-valued memristor-based Cohen–Grossberg neural networks with time-varying delays: norm method

Exponential synchronization of quaternion-valued memristor-based Cohen–Grossberg neural networks with time-varying delays: norm method

Stability Analysis of Anti-Periodic Solutions for Cohen–Grossberg Neural Networks with Inertial Term and Time Delays

This work is dedicated to exploring the globally exponential stability of anti-periodic solutions in inertial CGNNs that incorporate time delays. This is based on a strategic variable substitution to transform the complex system into a first-order differential equation. By leveraging the Lyapunov functional and demonstrating uniformly converging properties, we establish sufficient conditions that guarantee the existence and global exponential stability of anti-periodic solutions for the system. Finally, examples are presented to illustrate the effectiveness of the obtained theoretical results. This work contributes significantly to enhancing our understanding of the stability dynamics in neural networks with time delays and provides valuable insights for applications across various fields.

Cohen-Grossberg Neural Networks with Time-Varying Delays Generalized Synchronization

Cohen-Grossberg Neural Networks with Time-Varying Delays Generalized Synchronization

Robustness analysis of Cohen-Grossberg neural network with piecewise constant argument and stochastic disturbances

<abstract><p>Robustness of neural networks has been a hot topic in recent years. This paper mainly studies the robustness of the global exponential stability of Cohen-Grossberg neural networks with a piecewise constant argument and stochastic disturbances, and discusses the problem of whether the Cohen-Grossberg neural networks can still maintain global exponential stability under the perturbation of the piecewise constant argument and stochastic disturbances. By using stochastic analysis theory and inequality techniques, the interval length of the piecewise constant argument and the upper bound of the noise intensity are derived by solving transcendental equations. In the end, we offer several examples to illustrate the efficacy of the findings.</p></abstract>

Multimode function multistability of Cohen-Grossberg neural networks with Gaussian activation functions and mixed time delays

<abstract><p>This paper explores multimode function multistability of Cohen-Grossberg neural networks (CGNNs) with Gaussian activation functions and mixed time delays. We start by using the geometrical properties of Gaussian functions. The state space is partitioned into $ 3^\mu $ subspaces, where $ 0\le \mu\le n $. Moreover, through the utilization of Brouwer's fixed point theorem and contraction mapping, some sufficient conditions are acquired to ensure the existence of precisely $ 3^\mu $ equilibria for $ n $-dimensional CGNNs. Meanwhile, there are $ 2^\mu $ and $ 3^\mu-2^\mu $ multimode function stable and unstable equilibrium points, respectively. Ultimately, two illustrative examples are provided to confirm the efficacy of theoretical results.</p></abstract>

Generalized-Type Multistability of Almost Periodic Solutions for Memristive Cohen-Grossberg Neural Networks.

This article investigates a generalized type of multistability about almost periodic solutions for memristive Cohen-Grossberg neural networks (MCGNNs). As the inevitable disturbances in biological neurons, almost periodic solutions are more common in nature than equilibrium points (EPs). They are also generalizations of EPs in mathematics. According to the concepts of almost periodic solutions and Ψ -type stability, this article presents a generalized-type multistability definition of almost periodic solutions. The results show that (K+1)n generalized stable almost periodic solutions can coexist in a MCGNN with n neurons, where K is a parameter of the activation functions. The enlarged attraction basins are also estimated based on the original state space partition method. Some comparisons and convincing simulations are given to verify the theoretical results at the end of this article.

Bifurcation analysis of a fractional-order Cohen–Grossberg neural network with three delays

This paper pays more attention to the dynamic bifurcations of a fractional-order Cohen–Grossberg neural network in the sense of the Caputo fractional derivative with three delays. In the beginning, the dynamical behaviors of the system in several different cases are discussed when time delay can be viewed as zero, and the corresponding Hopf bifurcation critical values are derived. Subsequently, the system’s dynamics with three delays are investigated for three cases where none are zero. The reduction method of transcendental terms is applied to solve the third transcendental term in the characteristic equation. The exact bifurcation conditions are obtained by choosing one of the delays as the bifurcation parameter, respectively. It is worth noting that previous studies on multi-delay systems only discuss the case where the multi-delay can be transformed into a single delay or two delays based on certain transformations. This paper explores how to study the problem of a system with three delays without transforming the delays. In addition, the effects of time delays on the critical points of the system are examined. Finally, to verify the correctness of the proposed results, numerical simulations are addressed.

General stability for a Cohen–Grossberg neural network system

Of concern is a Cohen–Grossberg neural network (CGNNs) system taking into account distributed and discrete delays. The class of delay kernels ensuring exponential stability existing in the previous papers is enlarged to an extended class of functions guaranteeing more general types of stability. The exponential and polynomial (or power type) type stabilities becomes particular cases of our result. This is achieved using appropriate Lyapunov-type functionals and the characteristics of the considered class.

Fixed-time synchronization of delayed inertial Cohen–Grossberg neural networks with desynchronizing impulses

This article focuses on achieving fixed-time (FXT) synchronization of inertial Cohen–Grossberg neural networks (ICGNNs) in the presence of time-varying delays and desynchronizing impulsive effects. Firstly, a new lemma is presented to achieve FXT stability of the impulsive systems with destabilizing impulses. The settling-time function is shown to depend on both the parameters of impulsive sequence and continuous-time subsystems. Furthermore, based on the proposed lemma, sufficient conditions are established to achieve FXT synchronization of ICGNNS with desynchronizing impulses by designing a unified controller. Two numerical examples are taken into consideration to verify the efficacy of the proposed theoretical results.

Cohen-Grossberg neural networks with unpredictable and Poisson stable dynamics

In this paper, we provide theoretical as well as numerical results concerning recurrent oscillations in Cohen-Grossberg neural networks with variable inputs and strengths of connectivity for cells, which are unpredictable or Poisson stable functions. A special case of the compartmental coefficients with periodic and unpredictable ingredients is also carefully researched. By numerical and graphical analysis, it is shown how a constructive technical characteristic, the degree of periodicity, reflects contributions of the ingredients to final outputs of the neural networks. Sufficient conditions are obtained to guarantee the existence of exponentially stable unpredictable outputs of the models. They are specified for Poisson stability by utilizing the original method of included intervals. Examples with numerical simulations that support the theoretical results are provided.

Fixed/Preassigned-Time Synchronization of Fully Quaternion-Valued Cohen–Grossberg Neural Networks with Generalized Time Delay

This article is concerned with fixed-time synchronization and preassigned-time synchronization of Cohen–Grossberg quaternion-valued neural networks with discontinuous activation functions and generalized time-varying delays. Firstly, a dynamic model of Cohen–Grossberg neural networks is introduced in the quaternion field, where the time delay successfully integrates discrete-time delay and proportional delay. Secondly, two types of discontinuous controllers employing the quaternion-valued signum function are designed. Without utilizing the conventional separation technique, by developing a direct analytical approach and using the theory of non-smooth analysis, several adequate criteria are derived to achieve fixed-time synchronization of Cohen–Grossberg neural networks and some more precise convergence times are estimated. To cater to practical requirements, preassigned-time synchronization is also addressed, which shows that the drive-slave networks reach synchronization within a specified time. Finally, two numerical simulations are presented to validate the effectiveness of the designed controllers and criteria.

Hyperbolic function-based fixed/preassigned-time stability of nonlinear systems and synchronization of delayed fuzzy Cohen–Grossberg neural networks

In this article, the fixed-time (FIT) and preassigned-time (PRT) stability of nonlinear models and the FIT/PRT synchronization of discontinuous delayed fuzzy Cohen–Grossberg neural networks (DFCGNNs) are explored, respectively. Firstly, by means of hyperbolic-cosine function and Gudermannian function, a novel FIT stability theorem is established by utilizing Lyapunov method. Then, based on the developed stability theorem, several new discontinuous control schemes with hyperbolic-cosine function are developed to discuss the FIT synchronization of DFCGNNs. Furthermore, the PRT synchronization is also explored by slightly modifying FIT control schemes, where the synchronization time is predetermined. Eventually, two numerical examples are provided to validate the theoretical research.