Dynamical Bifurcation for a Class of Large-Scale Fractional Delayed Neural Networks With Complex Ring-Hub Structure and Hybrid Coupling

Coexistence of Cyclic Sequential Pattern Recognition and Associative Memory in Neural Networks by Attractor Mechanisms

Neural networks are developed to model the behavior of the brain. One crucial question in this field pertains to when and how a neural network can memorize a given set of patterns. There are two mechanisms to store information: associative memory and sequential pattern recognition. In the case of associative memory, the neural network operates with dynamical attractors that are point attractors, each corresponding to one of the patterns to be stored within the network. In contrast, sequential pattern recognition involves the network memorizing a set of patterns and subsequently retrieving them in a specific order over time. From a dynamical perspective, this corresponds to the presence of a continuous attractor or a cyclic attractor composed of the sequence of patterns stored within the network in a given order. Evidence suggests that the brain is capable of simultaneously performing both associative memory and sequential pattern recognition. Therefore, these types of attractors coexist within the neural network, signifying that some patterns are stored as point attractors, while others are stored as continuous or cyclic attractors. This article investigates the coexistence of cyclic attractors and continuous or point attractors in certain nonlinear neural networks, enabling the simultaneous emergence of various memory mechanisms. By selectively grouping neurons, conditions are established for the existence of cyclic attractors, continuous attractors, and point attractors, respectively. Furthermore, each attractor is explicitly represented, and a competitive dynamic emerges among these coexisting attractors, primarily regulated by adjustments to external inputs.

Tipping prediction of a class of large-scale radial-ring neural networks

Understanding the emergence and evolution of collective dynamics in large-scale neural networks remains a complex challenge. This paper seeks to address this gap by applying dynamical systems theory, with a particular focus on tipping mechanisms. First, we introduce a novel (n+mn)-scale radial-ring neural network and employ Coates' flow graph topological approach to derive the characteristic equation of the linearized network. Second, through deriving stability conditions and predicting the tipping point using an algebraic approach based on the integral element concept, we identify critical factors such as the synaptic transmission delay, the self-feedback coefficient, and the network topology. Finally, we validate the methodology's effectiveness in predicting the tipping point. The findings reveal that increased synaptic transmission delay can induce and amplify periodic oscillations. Additionally, the self-feedback coefficient and the network topology influence the onset of tipping points. Moreover, the selection of activation function impacts both the number of equilibrium solutions and the convergence speed of the neural network. Lastly, we demonstrate that the proposed large-scale radial-ring neural network exhibits stronger robustness compared to lower-scale networks with a single topology. The results provide a comprehensive depiction of the dynamics observed in large-scale neural networks under the influence of various factor combinations.

Stability and Bifurcation Exploration of Delayed Neural Networks With Radial-Ring Configuration and Bidirectional Coupling

For decades, studying the dynamic performances of artificial neural networks (ANNs) is widely considered to be a good way to gain a deeper insight into actual neural networks. However, most models of ANNs are focused on a finite number of neurons and a single topology. These studies are inconsistent with actual neural networks composed of thousands of neurons and sophisticated topologies. There is still a discrepancy between theory and practice. In this article, not only a novel construction of a class of delayed neural networks with radial-ring configuration and bidirectional coupling is proposed, but also an effective analytical approach to dynamic performances of large-scale neural networks with a cluster of topologies is developed. First, Coates' flow diagram is applied to acquire the characteristic equation of the system, which contains multiple exponential terms. Second, by means of the idea of the holistic element, the sum of the neuron synapse transmission delays is regarded as the bifurcation argument to investigate the stability of the zero equilibrium point and the beingness of Hopf bifurcation. Finally, multiple sets of computerized simulations are utilized to confirm the conclusions. The simulation results expound that the increase in transmission delay may cause a leading impact on the generation of Hopf bifurcation. Meanwhile, the number and the self-feedback coefficient of neurons are also playing significant roles in the appearance of periodic oscillations.

Stability and Hopf bifurcation analysis of a fractional-order ring-hub structure neural network with delays under parameters delay feedback control

In this paper, a fractional-order two delays neural network with ring-hub structure is investigated. Firstly, the stability and the existence of Hopf bifurcation of proposed system are obtained by taking the sum of two delays as the bifurcation parameter. Furthermore, a parameters delay feedback controller is introduced to control successfully Hopf bifurcation. The novelty of this paper is that the characteristic equation corresponding to system has two time delays and the parameters depend on one of them. Selecting two time delays as the bifurcation parameters simultaneously, stability switching curves in $ (\tau_{1}, \tau_{2}) $ plane and crossing direction are obtained. Sufficient criteria for the stability and the existence of Hopf bifurcation of controlled system are given. Ultimately, numerical simulation shows that parameters delay feedback controller can effectively control Hopf bifurcation of system.

Bifurcations of a delayed fractional-order BAM neural network via new parameter perturbations

This paper makes a new breakthrough in deliberating the bifurcations of fractional-order bidirectional associative memory neural network (FOBAMNN). In the beginning, the corresponding bifurcation results are established according to self-regulating parameter, which is different from bifurcation outcomes available by using time delay as the bifurcation parameter, and greatly enriches the bifurcation results of continuous neural networks(NNs). The deived results manifest that a larger self-regulating parameter is more conducive to the stability of the system, which is consistent with the actual meaning of the self-regulating parameter representing the decay rate of activity. In addition to the innovation in the research object, this paper also has innovation in the procedure of calculating the bifurcation critical point. In the face of the quartic equation about the bifurcation parameters, this paper utilizes the methodology of implicit array to calculate the bifurcation critical point succinctly and effectively, which eschews the disadvantages of the conventional Ferrari approach, such as cumbersome formula and huge computational efforts. Our developed technique can be employed as a general method to solve the bifurcation point including the problem of dealing with the bifurcation critical point of delay. Ultimately, numerical experiments test the key theoretical fruits of this paper.

Understanding Dynamics and Bifurcation Control Mechanism for a Fractional-Order Delayed Duopoly Game Model in Insurance Market

Recently, the insurance industry in China has been greatly developed. The number of domestic insurance companies and foreign investment insurance companies has greatly increased. Competition between different insurance companies is becoming increasingly fierce. Grasping the internal competition law of different insurance companies is a very meaningful work. In this present work, we set up a novel fractional-order delayed duopoly game model in insurance market and discuss the dynamics including existence and uniqueness, non-negativeness, and boundedness of solution for the established fractional-order delayed duopoly game model in insurance market. By selecting the delay as a bifurcation parameter, we build a new delay-independent condition ensuring the stability and creation of Hopf bifurcation of the built fractional-order delayed duopoly game model. Making use of a suitable definite function, we explore the globally asymptotic stability of the involved fractional-order delayed duopoly game model. By virtue of hybrid controller which includes state feedback and parameter perturbation, we can effectively control the stability and the time of creation of Hopf bifurcation for the involved fractional-order delayed duopoly game model. The research indicates that time delay plays an all-important role in stabilizing the system and controlling the time of onset of Hopf bifurcation of the involved fractional-order delayed duopoly game model. To check the rationality of derived primary conclusions, Matlab simulation plots are explicitly presented. The established results in this manuscript are wholly novel and own immense theoretical guiding significance in managing and operating insurance companies.