Bifurcations in a fractional-order neural network with multiple leakage delays

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.

A Comprehensive Review of Continuous-/Discontinuous-Time Fractional-Order Multidimensional Neural Networks

The dynamical study of continuous-/discontinuous-time fractional-order neural networks (FONNs) has been thoroughly explored, and several publications have been made available. This study is designed to give an exhaustive review of the dynamical studies of multidimensional FONNs in continuous/discontinuous time, including Hopfield NNs (HNNs), Cohen-Grossberg NNs, and bidirectional associative memory NNs, and similar models are considered in real ( [Formula: see text]), complex ( [Formula: see text]), quaternion ( [Formula: see text]), and octonion ( [Formula: see text]) fields. Since, in practice, delays are unavoidable, theoretical findings from multidimensional FONNs with various types of delays are thoroughly evaluated. Some required and adequate stability and synchronization requirements are also mentioned for fractional-order NNs without delays.

Stabilization of reaction-diffusion fractional-order memristive neural networks

This paper investigates the stabilization control of fractional-order memristive neural networks with reaction-diffusion terms. With regard to the reaction-diffusion model, a novel processing method based on Hardy-Poincarè inequality is introduced, as a result, the diffusion terms are estimated associated with the information of the reaction-diffusion coefficients and the regional feature, which may be beneficial to obtain conditions with less conservatism. Then, based on Kakutani's fixed point theorem of set-valued maps, new testable algebraic conclusion for ensuring the existence of the system's equilibrium point is obtained. Subsequently, by means of Lyapunov stability theory, it is concluded that the resulting stabilization error system is global asymptotic/Mittag-Leffler stable with a prescribed controller. Finally, an illustrative example about is provided to show the effectiveness of the established results.

Fixed-deviation stabilization and synchronization for delayed fractional-order complex-valued neural networks

In this paper, we study fixed-deviation stabilization and synchronization for fractional-order complex-valued neural networks with delays. By applying fractional calculus and fixed-deviation stability theory, sufficient conditions are given to ensure the fixed-deviation stabilization and synchronization for fractional-order complex-valued neural networks under the linear discontinuous controller. Finally, two simulation examples are presented to show the validity of theoretical results.

Fractional order-induced bifurcations in a delayed neural network with three neurons

This paper reports the novel results on fractional order-induced bifurcation of a tri-neuron fractional-order neural network (FONN) with delays and instantaneous self-connections by the intersection of implicit function curves to solve the bifurcation critical point. Firstly, it considers the distribution of the root of the characteristic equation in depth. Subsequently, it views fractional order as the bifurcation parameter and establishes the transversal condition and stability interval. The main novelties of this paper are to systematically analyze the order as a bifurcation parameter and concretely establish the order critical value through an implicit function array, which is a novel idea to solve the critical value. The derived results exhibit that once the value of the fractional order is greater than the bifurcation critical value, the stability of the system will be smashed and Hopf bifurcation will emerge. Ultimately, the validity of the developed key fruits is elucidated via two numerical experiments.

Pattern Formation in a Reaction-Diffusion BAM Neural Network With Time Delay: (k1, k2) Mode Hopf-Zero Bifurcation Case

This article investigates the joint effects of connection weight and time delay on pattern formation for a delayed reaction-diffusion BAM neural network (RDBAMNN) with Neumann boundary conditions by using the (k₁,k₂) mode Hopf-zero bifurcation. First, the conditions for k₁ mode zero bifurcation are obtained by choosing connection weight as the bifurcation parameter. It is found that the connection weight has a great impact on the properties of steady state. With connection weight increasing, the homogeneous steady state becomes inhomogeneous, which means that the connection weight can affect the spatial stability of steady state. Then, the specified conditions for the k₂ mode Hopf bifurcation and the (k₁,k₂) mode Hopf-zero bifurcation are established. By using the center manifold, the third-order normal form of the Hopf-zero bifurcation is obtained. Through the analysis of the normal form, the bifurcation diagrams on two parameters' planes (connection weight and time delay) are obtained, which contains six areas. Some interesting spatial patterns are found in these areas: a homogeneous periodic solution, a homogeneous steady state, two inhomogeneous steady state, and two inhomogeneous periodic solutions.

Stability and bifurcation control analysis of a delayed fractional-order eco-epidemiological system

Considering the factor of artificial intervention in biological control, a delayed fractional eco-epidemiological system with an extended feedback controller is proposed. By using the digestion delay as bifurcation parameter, the stability and Hopf bifurcation are investigated, and the branching conditions are given. The system undergoes Hopf bifurcation, when the parameter τ passes through the critical value. In addition, it can be pointed out that the negative feedback gain and the feedback delay could affect the bifurcation critical value of the system. Therefore, the Hopf bifurcation can also be induced by taking the feedback delay as a bifurcation parameter. Finally, by plotting the solution curve of the system, the significance of the controller to the stability of the eco-epidemiological system is verified.

State and Control Path-Dependent Stochastic Zero-Sum Differential Games: Viscosity Solutions of Path-Dependent Hamilton–Jacobi–Isaacs Equations

In this paper, we consider the two-player state and control path-dependent stochastic zero-sum differential game. In our problem setup, the state process, which is controlled by the players, is dependent on (current and past) paths of state and control processes of the players. Furthermore, the running cost of the objective functional depends on both state and control paths of the players. We use the notion of non-anticipative strategies to define lower and upper value functionals of the game, where unlike the existing literature, these value functions are dependent on the initial states and control paths of the players. In the first main result of this paper, we prove that the (lower and upper) value functionals satisfy the dynamic programming principle (DPP), for which unlike the existing literature, the Skorohod metric is necessary to maintain the separability of càdlàg (state and control) spaces. We introduce the lower and upper Hamilton–Jacobi–Isaacs (HJI) equations from the DPP, which correspond to the state and control path-dependent nonlinear second-order partial differential equations. In the second main result of this paper, we show that by using the functional Itô calculus, the lower and upper value functionals are viscosity solutions of (lower and upper) state and control path-dependent HJI equations, where the notion of viscosity solutions is defined on a compact κ-Hölder space to use several important estimates and to guarantee the existence of minimum and maximum points between the (lower and upper) value functionals and the test functions. Based on these two main results, we also show that the Isaacs condition and the uniqueness of viscosity solutions imply the existence of the game value. Finally, we prove the uniqueness of classical solutions for the (state path-dependent) HJI equations in the state path-dependent case, where its proof requires establishing an equivalent classical solution structure as well as an appropriate contradiction argument.

Stability Switching Curves and Hopf Bifurcation of a Fractional Predator–Prey System with Two Nonidentical Delays

In this paper, we propose and analyze a three-dimensional fractional predator–prey system with two nonidentical delays. By choosing two delays as the bifurcation parameter, we first calculate the stability switching curves in the delay plane. By judging the direction of the characteristic root across the imaginary axis in stability switching curves, we obtain that the stability of the system changes when two delays cross the stability switching curves, and then, the system appears to have bifurcating periodic solutions near the positive equilibrium, which implies that the trajectory of the system is the axial symmetry. Secondly, we obtain the conditions for the existence of Hopf bifurcation. Finally, we give one example to verify the correctness of the theoretical analysis. In particular, the geometric stability switch criteria are applied to the stability analysis of the fractional differential predator–prey system with two delays for the first time.

Chaos Control for a Fractional-Order Jerk System via Time Delay Feedback Controller and Mixed Controller

In this study, we propose a novel fractional-order Jerk system. Experiments show that, under some suitable parameters, the fractional-order Jerk system displays a chaotic phenomenon. In order to suppress the chaotic behavior of the fractional-order Jerk system, we design two control strategies. Firstly, we design an appropriate time delay feedback controller to suppress the chaos of the fractional-order Jerk system. The delay-independent stability and bifurcation conditions are established. Secondly, we design a suitable mixed controller, which includes a time delay feedback controller and a fractional-order PDσ controller, to eliminate the chaos of the fractional-order Jerk system. The sufficient condition ensuring the stability and the creation of Hopf bifurcation for the fractional-order controlled Jerk system is derived. Finally, computer simulations are executed to verify the feasibility of the designed controllers. The derived results of this study are absolutely new and possess potential application value in controlling chaos in physics. Moreover, the research approach also enriches the chaos control theory of fractional-order dynamical system.

Novel methods to global Mittag-Leffler stability of delayed fractional-order quaternion-valued neural networks

In this paper, a type of fractional-order quaternion-valued neural networks (FOQVNNs) with leakage and time-varying delays is established to simulate real-world situations, and the global Mittag-Leffler stability of the system is investigated by using the non-decomposition method. First, to avoid decomposing the system into two complex-valued systems or four real-valued systems, a new sign function for quaternion numbers is introduced based on the ones for real and complex numbers. And two novel lemmas for quaternion-valued sign function and Caputo fractional derivative are established in quaternion domain, which are used to investigate the stability of FOQVNNs. Second, a concise and flexible quaternion-valued state feedback controller is directly designed and a novel 1-norm Lyapunov function composed of the absolute values of real and imaginary parts is established. Then, based on the designed quaternion-valued state feedback controller and the proposed lemmas, some sufficient conditions are given to ensure the global Mittag-Leffler stability of the system. Finally, a numerical simulation is given to verify the theoretical results.

Bifurcations in a fractional-order BAM neural network with four different delays

This paper illuminates the issue of bifurcations for a fractional-order bidirectional associative memory neural network(FOBAMNN) with four different delays. On account of the affirmatory presumption, the developed FOBAMNN is firstly transformed into the one with two nonidentical delays. Then the critical values of Hopf bifurcations with respect to disparate delays are calculated quantitatively by establishing one delay and selecting remaining delay as a bifurcation parameter in the transformed model. It detects that the stability of the developed FOBAMNN with multiple delays can be fairly preserved if selecting lesser control delays, and Hopf bifurcation emerges once the control delays outnumber their critical values. The derived bifurcation results are numerically testified via the bifurcation graphs. The feasibility of theoretical analysis is ultimately corroborated in the light of simulation experiments. The analytic results available in this paper are beneficial to give impetus to resolve the issues of bifurcations of high-order FONNs with multiple delays.