Stability Switches, Hopf Bifurcations, and Spatio-temporal Patterns in a Delayed Neural Model with Bidirectional Coupling

The dynamical behavior of a delayed neural network with bi-directional coupling is investigated by taking the delay as the bifurcating parameter. Some parameter regions are given for conditional/absolute stability and Hopf bifurcations by using the theory of functional differential equations. As the propagation time delay in the coupling varies, stability switches for the trivial solution are found. Conditions ensuring the stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. We also discuss the spatio-temporal patterns of bifurcating periodic oscillations by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. In particular, we obtain that the spatio-temporal patterns of bifurcating periodic oscillations will alternate according to the change of the propagation time delay in the coupling, i.e., different ranges of delays correspond to different patterns of neural activities. Numerical simulations are given to illustrate the obtained results and show the existence of bursts in some interval of the time for large enough delay.     

Multiple periodic solutions in a delay-coupled system of neural oscillators

In this paper, effects of the synaptic delay of signal transmissions on the pattern formation of nonlinear waves in a bidirectional ring of neural oscillators is studied. Firstly, the linear stability of the model is investigated by analyzing the associated characteristic transcendental equation. Meanwhile, using the symmetric bifurcation theory of delay differential equations coupled with the representation theory of Lie groups, we discuss the spontaneous bifurcation of multiple branches of periodic solutions and their spatio-temporal patterns. Finally, Hopf bifurcation directions and corresponding stabilities of bifurcating periodic orbits are derived by using the normal form approach and the center manifold theory. These theoretical results are significant to complement experimental and numerical observations made in living neuronal systems and artificial neural networks, in order to better understand the mechanisms underlying the system’s dynamics.     

Dynamics of a delayed two-coupled oscillator with excitatory-to-excitatory connection

In this paper, we consider a neural network model consisting of two coupled oscillators with delayed feedback and excitatory-to-excitatory connection. We study how the strength of the connections between the oscillators affects the dynamics of the neural network. We give a full classification of all equilibria in the parameter space and obtain its linear stability by analyzing the characteristic equation of the linearized system. We also investigate the spatio-temporal patterns of bifurcated periodic oscillations by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. Moreover, the stability and bifurcation direction of the bifurcated periodic solutions are obtained by employing center manifold reduction and normal form theory. Some numerical simulations are provided to illustrate the theoretical results.     

Synchronous Dynamics and Bifurcation Analysis in Two Delay Coupled Oscillators with Recurrent Inhibitory Loops

In this study, the dynamics and low-codimension bifurcation of the two delay coupled oscillators with recurrent inhibitory loops are investigated. We discuss the absolute synchronization character of the coupled oscillators. Then the characteristic equation of the linear system is examined, and the possible low-codimension bifurcations of the coupled oscillator system are studied by regarding eigenvalues of the connection matrix as bifurcation parameter, and the bifurcation diagram in the γ–ρ plane is obtained. Applying normal form theory and the center manifold theorem, the stability and direction of the codimension bifurcations are determined. Moreover, the symmetric bifurcation theory and representation theory of Lie groups are used to investigate the spatio-temporal patterns of the periodic oscillations. Finally, numerical results are applied to illustrate the results obtained.     

Influence of discrete delay on pattern formation in a ratio-dependent prey–predator model

In this paper we explore how the two mechanisms, Turing instability and Hopf bifurcation, interact to determine the formation of spatial patterns in a ratio-dependent prey–predator model with discrete time delay. We conduct both rigorous analysis and extensive numerical simulations. Results show that four types of patterns, cold spot, labyrinthine, chaotic as well as mixture of spots and labyrinthine can be observed with and without time delay. However, in the absence of time delay, the two aforementioned mechanisms have a significant impact on the emergence of spatial patterns, whereas only Hopf bifurcation threshold is derived by considering the discrete time delay as the bifurcation parameter. Moreover, time delay promotes the emergence of spatial patterns via spatio-temporal Hopf bifurcation compared to the non-delayed counterpart, implying the destabilizing role of time delay. In addition, the destabilizing role is prominent when the magnitude of time delay and the ratio of diffusivity are comparatively large.     

Bifurcation and synchronization of synaptically coupled FHN models with time delay

This paper presents an investigation of dynamics of the coupled nonidentical FHN models with synaptic connection, which can exhibit rich bifurcation behavior with variation of the coupling strength. With the time delay being introduced, the coupled neurons may display a transition from the original chaotic motions to periodic ones, which is accompanied by complex bifurcation scenario. At the same time, synchronization of the coupled neurons is studied in terms of their mean frequencies. We also find that the small time delay can induce new period windows with the coupling strength increasing. Moreover, it is found that synchronization of the coupled neurons can be achieved in some parameter ranges and related to their bifurcation transition. Bifurcation diagrams are obtained numerically or analytically from the mathematical model and the parameter regions of different behavior are clarified.     

Linear stability and Hopf bifurcation in a delayed two-coupled oscillator with excitatory-to-inhibitory connection

We consider the dynamical behavior of a delayed two-coupled oscillator with excitatory-to-inhibitory connection. Some parameter regions are given for linear stability, absolute synchronization, and Hopf bifurcations by using the theory of functional differential equations. Conditions ensuring the stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. We also investigate the spatio-temporal patterns of bifurcating periodic oscillations by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. Finally, numerical simulations are given to illustrate the results obtained.     

Synchronous dynamics of two coupled oscillators with inhibitory-to-inhibitory connection

We investigate the behaviour of a neural network model consisting of two coupled oscillators with delays and inhibitory-to-inhibitory connections. We consider the absolute synchronization and show that the connection topology of the network plays a fundamental role in classifying the rich dynamics and bifurcation phenomena. Regarding eigenvalues of the connection matrix as bifurcation parameters, we obtain codimension one bifurcations (including fold bifurcation and Hopf bifurcation) and codimension two bifurcation (including fold-Hopf bifurcations and Hopf–Hopf bifurcations). Based on the normal form theory and center manifold reduction, we obtain detailed information about the bifurcation direction and stability of various bifurcated equilibria as well as periodic solutions with some kinds of spatio-temporal patterns. Numerical simulation is also given to support the obtained results.     

时滞对磁通耦合及化学耦合神经元分岔及同步的影响

以化学突触耦合神经元模型为基础,讨论了抑制性及兴奋性条件下达到同步的区别及同步的类型。并根据磁通耦合对神经元放电的影响,讨论了具有时滞、磁通耦合和化学耦合Morris-Lecar (ML)神经元模型的放电状态、分岔类型及其同步情况。发现具有磁通耦合和化学耦合ML神经元系统在不同参数下会产生丰富的逆倍周期分岔或加周期分岔行为。而时滞的引入,虽然可以增加系统的周期性,但同时也会破环系统同步。相反,适当的耦合强度能够增加同步。     

Hopf bifurcation and multiple periodic solutions in Lotka–Volterra systems with symmetries

The purpose of this paper is to study Hopf bifurcations in a delayed Lotka–Volterra system with dihedral symmetry. By treating the response delay as bifurcation parameter and employing equivariant degree method, we obtain the existence of multiple branches of nonconstant periodic solutions through a local Hopf bifurcation around an equilibrium. We find that competing coefficients and the response delay in the system can affect the spatio-temporal patterns of bifurcating periodic solutions. According to their symmetric properties, a topological classification is given for these periodic solutions. Furthermore, an estimation is presented on minimal number of bifurcating branches. These theoretical results are helpful to better understand the complex dynamics induced by response delays and symmetries in Lotka–Volterra systems.     

Driven response of time delay coupled limit cycle oscillators

We study the periodic forced response of a system of two limit cycle oscillators that interact with each other via a time delayed coupling. Detailed bifurcation diagrams in the parameter space of the forcing amplitude and forcing frequency are obtained for various interesting limits using numerical and analytical means. In particular, the effects of the coupling strength, the natural frequency spread of the two oscillators and the time delay parameter on the size and nature of the entrainment domain are delineated. For an appropriate choice of time delay, synchronization can occur with infinitesimal forcing amplitudes even at off-resonant driving. The system is also found to display a nonlinear response on certain critical contours in the space of the coupling strength and time delay. Numerical simulations with a large number of coupled driven oscillators display similar behavior. Time delay offers a novel tuning knob for controlling the system response over a wide range of frequencies and this may have important practical applications.     

Stability analysis in a diffusional immunosuppressive infection model with delayed antiviral immune response

In this paper, the diffusion is introduced to an immunosuppressive infection model with delayed antiviral immune response. The direction and stability of Hopf bifurcation are effected by time delay, in the absence of which the positive equilibrium is locally asymptotically stable by means of analyzing eigenvalue spectrum; however, when the time delay increases beyond a threshold, the positive equilibrium loses its stability via the Hopf bifurcation. The stability and direction of the Hopf bifurcation is investigated with the norm form and the center manifold theory. The stability of the Hopf bifurcation leads to the emergence of spatial spiral patterns. Numerical calculations are performed to illustrate our theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.     

Delay Induced Hopf Bifurcation of Small-World Networks

In this paper,the stability and the Hopf bifurcation of small-world networks with time delay are studied.By analyzing the change of delay,we obtain several sufficient conditions on stable and unstable properties.When the delay passes a critical value,a Hopf bifurcation may appear.Furthermore,the direction and the stability of bifurcating periodic solutions are investigated by the normal form theory and the center manifold reduction.At last,by numerical simulations,we further illustrate the effectiveness of theorems in this paper.     

一类带无选择性捕获和时滞的捕食食饵系统的Hopf分支分析

本文主要研究了一个改进的带时滞和无选择捕获函数的捕食-食饵生态经济系统的稳定性和Hopf分支.利用微分代数系统的稳定性理论和分支理论,得到了系统正平衡点稳定性的条件,以及当时滞τ作为分支参数时系统产生Hopf分支的条件.对Leslie-Gower捕食-食饵模型进行了一定程度的完善,使得建立的模型更符合实际情况,因此得到的结论也更加科学.     

HOPF BIFURCATION OF NONLINEAR ADVERTISING CAPITAL MODEL WITH DISCRETE AND CONTINUOUS TIME DELAYED FEEDBACK

In this paper,stability and Hopf bifurcation of a nonlinear advertising ca- pital model with time delayed are studied.By analyzing the change of delay, we obtain several sufficient conditions on stable and unstable properties.When delay passes a critical value,Hopf bifurcation may appear.Furthermore,the di- rection and stability of bifurcating periodic solutions are investigated by normal form and center manifold theory.Additionally,we also have some discussion about the model with continuous time delay.     

Bifurcation analysis and control of chaos for a hybrid ratio-dependent three species food chain

In this paper, a hybrid ratio-dependent three species food chain model with time delay is studied by using the theory of functional differential equation and Hopf bifurcation, the condition on which positive equilibrium exists and the quality of Hopf bifurcation are given. Chaotic solutions are observed and are controlled by delay parameter. Finally, we indicate that when the delay passes through certain critical values, chaotic oscillation is converted into a stable state or a stable periodic orbit.     

Bursting mechanism in a time-delayed oscillator with slowly varying external forcing

This paper investigates the generation of complex bursting patterns in the Duffing oscillator with time-delayed feedback. We present the bursting patterns, including symmetric fold–fold bursting and symmetric Hopf–Hopf bursting when periodic forcing changes slowly. We make an analysis of the system bifurcations and dynamics as a function of the delayed feedback and the periodic forcing. We calculate the conditions of fold bifurcation and Hopf bifurcation as well as its stability related to external forcing and delay. We also identify two regimes of bursting depending on the magnitude of the delay itself and the strength of time delayed coupling in the model. Our results show that the dynamics of bursters in delayed system are quite different from those in systems without any delay. In particular, delay can be used as a tuning parameter to modulate dynamics of bursting corresponding to the different type. Furthermore, we use transformed phase space analysis to explore the evolution details of the delayed bursting behavior. Also some numerical simulations are included to illustrate the validity of our study.     

Bifurcation analysis of a modified Holling-Tanner predator-prey model with time delay

In this paper, a modified Holling-Tanner predator-prey model with time delay is considered. By regarding the delay as the bifurcation parameter, the local asymptotic stability of the positive equilibrium is investigated. Meanwhile, we find that the system can also undergo a Hopf bifurcation of nonconstant periodic solution at the positive equilibrium when the delay crosses through a sequence of critical values. In particular, we study the direction of Hopf bifurcation and the stability of bifurcated periodic solutions, an explicit algorithm is given by applying the normal form theory and the center manifold reduction for functional differential equations. Finally, numerical simulations supporting the theoretical analysis are also included.     

Existence of the backward bifurcation of a non-markovian SIS-network model with saturation treatment function

Facing an emerging infectious disease, the occurrence of the medical runs has an inevitable effect on curtailing the disease prevalence. The heterogeneity of individuals’ contacts significantly affects the patterns of the disease transmission. In this paper, we propose a SIS mean-field model coupling a non-markovian recovery process and a delay factor caused by the limitation of medical resources. The positivity and boundedness of the solution to the model have been established by the Volterra integral equation theory. Furthermore, if the lag effect of the treatment is terrible, the system exhibits a bistable phenomenon, whose stability of every feasible equilibrium is established by the Lyapunov–Schmidt approach and bifurcation analysis. Finally, numerical simulations have shown that the system may bifurcate multiple endemic steady states.     

Spatio-temporal symmetries and bifurcations via bi-orthogonal decompositions

Summary A tool for analyzing spatio-temporal complex physical phenomena was recently proposed by the authors, Aubry et al. [5]. This tool consists in decomposing a spatially and temporally evolving signal into orthogonal temporal modes (temporal “structures”) and orthogonal spatial modes (spatial “structures”) which are coupled. This allows the introduction of a temporal configuration space and a spatial one which are related to each other by an isomorphism. In this paper, we show how such a tool can be used to analyze space-time bifurcations, that is, qualitative changes in both space and time as a parameter varies. The Hopf bifurcation and various spatio-temporal symmetry related bifurcations, such as bifurcations to traveling waves, are studied in detail. In particular, it is shown that symmetry-breaking bifurcations can be detected by analyzing the temporal and spatial eigenspaces of the decomposition which then lose their degeneracy, namely their invariance under the symmetry. Furthermore, we show how an extension of the theory to “quasi-symmetries” permits the treatment of nondegenerate signals and leads to an exponentially decreasing law of the energy spectrum. Examples extracted from numerically obtained solutions of the Kuramoto-Sivashinsky equation, a coupled map lattice, and fully developed turbulence are given to illustrate the theory.