Stability and bifurcation analysis on a Logistic model with discrete and distributed delays

In this paper, a class of three-neuron network with discrete and distributed delays is introduced. We first give a detailed Hopf bifurcation analysis for the proposed network. Choosing the discrete time delay as a bifurcation parameter, the existence of Hopf bifurcation is studied. Moreover, by using the normal form theory and center manifold theorem, the formulae determining the direction of the bifurcations and the stability of the bifurcating periodic solutions are derived. Finally, numerical simulations are presented to demonstrate the effectiveness of our theoretical results.

In this paper, the dynamical behavior of an eco-epidemiological model with discrete and distributed delay is studied. Sufficient conditions for the local asymptotical stability of the nonnegative equilibria are obtained. We prove that there exists a threshold value of the feedback time delay τ beyond which the positive equilibrium bifurcates towards a periodic solution. Using the normal form theory and center manifold argument, the explicit formulae which determine the stability, the direction and the periodic of bifurcating period solutions are derived. Numerical simulations are carried out to explain the mathematical conclusions.

Classical reaction–diffusion systems exhibit diverse dynamical behaviors, which become more complex particularly in the context of bifurcation analysis and normal form theory when chemotactic terms are incorporated. This paper investigates the stability of a positive constant equilibrium and the Turing–Hopf (TH) bifurcation in a diffusive predator–prey model with searching, handling predators, and chemotaxis. A computational method is developed for deriving the normal form of the TH bifurcation on the center manifold in general three-dimensional chemotactic reaction–diffusion systems with Neumann boundary conditions. The method enables an explicit dynamical classification near the bifurcation point. Combined theoretical and numerical analyses reveal that TH interactions significantly enrich spatiotemporal dynamics.

This paper focuses on the effect of leakage delays on the dynamics of nonlinear delay systems based on a neuron system with discrete and distributed delay. By introducing a virtual neuron, we can eliminate the impact of distributed delay and transform the original neuron system with mixed delays to an equivalent new system involving only discrete delay. This paper extends the existing works on neural networks to the case of leakage delays. We choose the leakage delay as the bifurcation parameter and analyze the role of leakage delays on the stability and Hopf bifurcation. Sufficient conditions for ensuring the neuron system to be stable, and undergoing the Hopf bifurcation are derived. Finally, by using the software package DDE-BIFTOOL, we provide the simulation results to substantiate our theoretical analysis, and give the relationship between the onset of Hopf bifurcation and the system parameters by drawing the bifurcation curves.

Analysis of stability and Hopf bifurcation for a viral infectious model with delay

A remark on stabilization of nonlinear systems with discrete and distributed delays

Since the ratio-dependent theory reflects the fact that predators must share and compete for food, it is suitable for describing the relationship between predators and their preys and has recently become a very important theory put forward by biologists. In order to investigate the dynamical relationship between predators and their preys, a so-called Michaelis-Menten ratio-dependent predator-prey model is studied in this paper with gestation time delays of predators and preys taken into consideration. The stability of the positive equilibrium is investigated by the Nyquist criteria, and the existence of the local Hopf bifurcation is analyzed by employing the theory of Hopf bifurcation. By means of the center manifold and the normal form theories, explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. The above theoretical results are validated by numerical simulations with the help of dynamical software WinPP. The results show that if both the gestation delays are small enough, their sizes will keep stable in the long run, but if the gestation delays of predators are big enough, their sizes will periodically fluctuate in the long term. In order to reveal the effects of time delays on the ratio-dependent predator-prey model, a ratio-dependent predator-prey model without time delays is considered. By Hurwitz criteria, the local stability of positive equilibrium of this model is investigated. The conditions under which the positive equilibrium is locally asymptotically stable are obtained. By comparing the results with those of the model with time delays, it shows that the dynamical behaviors of ratio-dependent predator-prey model with time delays are more complicated. Under the same conditions, namely, with the same parameters, the stability of positive equilibrium of ratio-dependent predator-prey model would change due to the introduction of gestation time delays for predators and preys. Moreover, with the variation of time delays, the positive equilibrium of the ratio-dependent predator-prey model subjects to Hopf bifurcation.

A three-neuron network model with mixed delays involving multiple discrete and distributed delays is considered in this paper. Taking the discrete and distribute delays as bifurcation parameters respectively, we investigate the stability of the system structure and the conditions for the generation of Hopf bifurcation from the perspective of the distribution of the root of the characteristic equation of the linearized system at the equilibrium state of the nonlinear system. The intervals of parameters that make the system stable and unstable are also given. In addition, when the conditions of Hopf bifurcation theorem are satisfied, the calculation formulas for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solution are presented by means of the central manifold theorem and normal form method. Finally, numerical experiments are carried out to support the correctness of the theoretical results in different cases. It can be concluded that the oscillation and instability caused by delays obviously affect the stability of the network.

Generalized norm for existence, uniqueness and stability of Hopfield neural networks with discrete and distributed delays

Passivity criteria for continuous-time neural networks with mixed time-varying delays

This paper studies the stability and Hopf bifurcation in a class of high-dimension neural network involving the discrete and distributed delays under a new framework. By introducing some virtual neurons to the original system, the impact of distributed delay can be described in a simplified way via an equivalent new model. This paper extends the existing works on neural networks to high-dimension cases, which is much closer to complex and real neural networks. Here, we first analyze the Hopf bifurcation in this special class of high dimensional model with weak delay kernel from two aspects: one is induced by the time delay, the other is induced by a rate parameter, to reveal the roles of discrete and distributed delays on stability and bifurcation. Sufficient conditions for keeping the original system to be stable, and undergoing the Hopf bifurcation are obtained. Besides, this new framework can also apply to deal with the case of the strong delay kernel and corresponding analysis for different dynamical behaviors is provided. Finally, the simulation results are presented to justify the validity of our theoretical analysis.

In this paper, a two-neuron reaction–diffusion neural network with discrete and distributed delays is proposed, and the state feedback control strategy is adopted to achieve control of its spatiotemporal dynamical behaviours. Adding two virtual neurons, the original system is transformed into a neural network only containing the discrete delay. The conditions under which Hopf bifurcation and Turing instability arise are determined through analysis of the characteristic equation. Additionally, the amplitude equations are derived with the aid of weakly nonlinear analysis, and the selection of the Turing patterns is determined. The simulation results demonstrate that the state feedback controller can delay the onset of Hopf bifurcation and suppress the generation of Turing patterns.

This paper considered the state estimation for stochastic neural networks of neutral type with discrete and distributed delays. By using available output measurements, the state estimator can approximate the neuron states, and the asymptotic property of the state error is mean square exponential stable and also almost surely exponential stable in the presence of discrete and distributed delays. Under the Lipschitz assumptions for the activation functions and the measurement nonlinearity, a delay-dependent linear matrix inequality (LMI) criterion is proposed to guarantee the existence of the desired estimators by constructing an appropriate Lyapunov-Krasovskii function. It is shown that the existence conditions and the explicit expression of the state estimator can be parameterised in terms of the solution to a LMI. Finally, two numerical examples are presented to demonstrate the validity of the theoretical results and show that the theorem can provide less conservative conditions.

Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Peer Review Share Icon Share Twitter Facebook Reddit LinkedIn Tools Icon Tools Reprints and Permissions Cite Icon Cite Search Site Citation A. Suryanto, I. Yanti, W. M. Kusumawinahyu; Stability analysis of a discrete Hutchinson equation with discrete and distributed delay. AIP Conference Proceedings 21 February 2014; 1587 (1): 115–118. https://doi.org/10.1063/1.4866545 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentAIP Publishing PortfolioAIP Conference Proceedings Search Advanced Search |Citation Search

In this paper, we consider a modified predator–prey model with Michaelis–Menten-type predator harvesting and diffusion term. We give sufficient conditions to ensure that the coexisting equilibrium is asymptotically stable by analyzing the distribution of characteristic roots. We also study the Turing instability of the coexisting equilibrium. In addition, we use the natural growth rate r_{1} of the prey as a parameter and carry on Hopf bifurcation analysis including the existence of Hopf bifurcation, bifurcation direction, and the stability of the bifurcating periodic solution by the theory of normal form and center manifold method. Our results suggest that the diffusion term is important for the study of the predator–prey model, since it can induce Turing instability and spatially inhomogeneous periodic solutions. The natural growth rate r_{1} of the prey can also affect the stability of positive equilibrium and induce Hopf bifurcation.