Hopf-pitchfork bifurcation of coupled van der Pol oscillator with delay

In this paper, the Hopf-pitchfork bifurcation of coupled van der Pol with delay is studied. The interaction coefficient and time delay are taken as two bifurcation parameters. Firstly, the normal form is gotten by performing a center manifold reduction and using the normal form theory developed by Faria and Magalhães. Secondly, bifurcation diagrams and phase portraits are given through analyzing the unfolding structure. Finally, numerical simulations are used to support theoretical analysis

Hopf-pitchfork bifurcation in van der Pol's oscillator with nonlinear delayed feedback

AbstractFirst, we identify the critical values for Hopf-pitchfork bifurcation. Second, we derive the normal forms up to third order and their unfolding with original parameters in the system near the bifurcation point, by the normal form method and center manifold theory. Then we give a complete bifurcation diagram for original parameters of the system and obtain complete classifications of dynamics for the system. Furthermore, we find some interesting phenomena, such as the coexistence of two asymptotically stable states, two stable periodic orbits, and two attractive quasi-periodic motions, which are verified both theoretically and numerically

Dynamics in a Delayed Neural Network Model of Two Neurons with Inertial Coupling

A delayed neural network model of two neurons with inertial coupling is dealt with in this paper. The stability is investigated and Hopf bifurcation is demonstrated. Applying the normal form theory and the center manifold argument, we derive the explicit formulas for determining the properties of the bifurcating periodic solutions. An illustrative example is given to demonstrate the effectiveness of the obtained results

Bifurcation Analysis and Spatiotemporal Patterns of Nonlinear Oscillations in a Ring Lattice of Identical Neurons with Delayed Coupling

We investigate the dynamics of a delayed neural network model consisting of n identical neurons. We first analyze stability of the zero solution and then study the effect of time delay on the dynamics of the system. We also investigate the steady state bifurcations and their stability. The direction and stability of the Hopf bifurcation and the pitchfork bifurcation are analyzed by using the derived normal forms on center manifolds. Then, the spatiotemporal patterns of bifurcating periodic solutions are investigated by using the symmetric bifurcation theory, Lie group theory and S1-equivariant degree theory. Finally, two neural network models with four or seven neurons are used to verify our theoretical results

Dynamic properties of the coupled Oregonator model with delay

This work explores a coupled Oregonator model. By analyzing the associated characteristic equation, linear stability is investigated and Hopf bifurcations are demonstrated, as well as the stability and direction of the Hopf bifurcation are determined by employing the normal form method and the center manifold reduction. We also discussed the Z2 equivariant property and the existence of multiple periodic solutions. Numerical simulations are presented to illustrate the results in Section 5

Nonlinear dynamics of full-range CNNs with time-varying delays and variable coefficients

In the article, the dynamical behaviours of the full-range cellular neural networks (FRCNNs) with variable coefficients and time-varying delays are considered. Firstly, the improved model of the FRCNNs is proposed, and the existence and uniqueness of the solution are studied by means of differential inclusions and set-valued analysis. Secondly, by using the Hardy inequality, the matrix analysis, and the Lyapunov functional method, we get some criteria for achieving the globally exponential stability (GES). Finally, some examples are provided to verify the correctness of the theoretical results

Dissipativity analysis of stochastic fuzzy neural networks with randomly occurring uncertainties using delay dividing approach

This paper focuses on the problem of delay-dependent robust dissipativity analysis for a class of stochastic fuzzy neural networks with time-varying delay. The randomly occurring uncertainties under consideration are assumed to follow certain mutually uncorrelated Bernoulli-distributed white noise sequences. Based on the Itô's differential formula, Lyapunov stability theory, and linear matrix inequalities techniques, several novel sufficient conditions are derived using delay partitioning approach to ensure the dissipativity of neural networks with or without time-varying parametric uncertainties. It is shown, by comparing with existing approaches, that the delay-partitioning projection approach can largely reduce the conservatism of the stability results. Numerical examples are constructed to show the effectiveness of the theoretical results

Synchronization analysis of coupled fractional-order neural networks with time-varying delays

In this paper, the complete synchronization and Mittag-Leffler synchronization problems of a kind of coupled fractional-order neural networks with time-varying delays are introduced and studied. First, the sufficient conditions for a controlled system to reach complete synchronization are established by using the Kronecker product technique and Lyapunov direct method under pinning control. Here the pinning controller only needs to control part of the nodes, which can save more resources. To make the system achieve complete synchronization, only the error system is stable. Next, a new adaptive feedback controller is designed, which combines the Razumikhin-type method and Mittag-Leffler stability theory to make the controlled system realize Mittag-Leffler synchronization. The controller has time delays, and the calculation can be simplified by constructing an appropriate auxiliary function. Finally, two numerical examples are given. The simulation process shows that the conditions of the main theorems are not difficult to obtain, and the simulation results confirm the feasibility of the theorems

Dynamics of neural systems with time delays

Complex networks are ubiquitous in nature. Numerous neurological diseases, such as Alzheimer's, Parkinson's, epilepsy are caused by the abnormal collective behaviour of neurons in the brain. In particular, there is a strong evidence that Parkinson's disease is caused by the synchronisation of neurons, and understanding how and why such synchronisation occurs will bring scientists closer to the design and implementation of appropriate control to support desynchronisation required for the normal functioning of the brain. In order to study the emergence of (de)synchronisation, it is necessary first to understand how the dynamical behaviour of the system under consideration depends on the changes in systems parameters. This can be done using a powerful mathematical method, called bifurcation analysis, which allows one to identify and classify different dynamical regimes, such as, for example, stable/unstable steady states, Hopf and fold bifurcations, and find periodic solutions by varying parameters of the nonlinear system. In real-world systems, interactions between elements do not happen instantaneously due to a finite time of signal propagation, reaction times of individual elements, etc. Moreover, time delays are normally non-constant and may vary with time. This means that it is vital to introduce time delays in any realistic model of neural networks. In this thesis, I consider four different models. First, in order to analyse the fundamental properties of neural networks with time-delayed connections, I consider a system of four coupled nonlinear delay differential equations. This model represents a neural network, where one subsystem receives a delayed input from another subsystem. The exciting feature of this model is the combination of both discrete and distributed time delays, where distributed time delays represent the neural feedback between the two sub-systems, and the discrete delays describe neural interactions within each of the two subsystems. Stability properties are investigated for different commonly used distribution kernels, and the results are compared to the corresponding stability results for networks with no distributed delays. It is shown how approximations to the boundary of stability region of an equilibrium point can be obtained analytically for the cases of delta, uniform, and gamma delay distributions. Numerical techniques are used to investigate stability properties of the fully nonlinear system and confirm our analytical findings. In the second part of this thesis, I consider a globally coupled network composed of active (oscillatory) and inactive (non-oscillatory) oscillators with distributed time delayed coupling. Analytical conditions for the amplitude death, where the oscillations are quenched, are obtained in terms of the coupling strength, the ratio of inactive oscillators, the width of the uniformly distributed delay and the mean time delay for gamma distribution. The results show that for uniform distribution, by increasing both the width of the delay distribution and the ratio of inactive oscillators, the amplitude death region increases in the mean time delay and the coupling strength parameter space. In the case of the gamma distribution kernel, we find the amplitude death region in the space of the ratio of inactive oscillators, the mean time delay for gamma distribution, and the coupling strength for both weak and strong gamma distribution kernels. Furthermore, I analyse a model of the subthalamic nucleus (STN)-globus palidus (GP) network with three different transmission delays. A time-shift transformation reduces the model to a system with two time delays, for which the existence of a unique steady state is established. Conditions for stability of the steady state are derived in terms of system parameters and the time delays. Numerical stability analysis is performed using traceDDE and DDE-BIFTOOL in Matlab to investigate different dynamical regimes in the STN-GP model, and to obtain critical stability boundaries separating stable (healthy) and oscillatory (Parkinsonian-like) neural ring. Direct numerical simulations of the fully nonlinear system are performed to confirm analytical findings, and to illustrate different dynamical behaviours of the system. Finally, I consider a ring of n neurons coupled through the discrete and distributed time delays. I show that the amplitude death occurs in the symmetric (asymmetric) region depending on the even (odd) number of neurons in the ring neural system. Analytical conditions for linear stability of the trivial steady state are represented in a parameter space of the synaptic weight of the self-feedback and the coupling strength between the connected neurons, as well as in the space of the delayed self-feedback and the coupling strength between the neurons. It is shown that both Hopf and steady-state bifurcations may occur when the steady state loses its stability. Stability properties are also investigated for different commonly used distribution kernels, such as delta function and weak gamma distributions. Moreover, the obtained analytical results are confirmed by the numerical simulations of the fully nonlinear system