Theory Of Functional Differential Equations Research Articles

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

Abstract 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.

The linear theory of functional differential equations: existence theorems and the problem of pointwise completeness of the solutions

A boundary value problem and an initial-boundary value problems are considered for a linear functional differential equation of point type. A suitable scale of functional spaces is introduced and existence theorems for solutions are stated in terms of this scale, in a form analogous to Noether's theorem. A key fact is established for the initial boundary value problem: the space of classical solutions of the adjoint equation must be extended to include impulsive solutions. A test for the pointwise completeness of solutions is obtained. The results presented are based on a formalism developed by the author for this type of equation.Bibliography: 7 titles.

Equivariant Hopf bifurcation for functional differential equations of mixed type

In this paper we employ the Lyapunov–Schmidt procedure to set up equivariant Hopf bifurcation theory of functional differential equations of mixed type. In the process we derive criteria for the existence and direction of branches of bifurcating periodic solutions in terms of the original system, avoiding the process of center manifold reduction.

Analysis of bifurcation in a system of n coupled oscillators with delays

A n-coupled BVP oscillators system with delays is considered. By choosing the delays as the bifurcating parameters, some results of the Hopf bifurcations occurring at the zero equilibrium as the delays increase are exhibited. Using the symmetric functional differential equation theories of Wu [Jianhong Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc. 350 (12) (1998) 4799–4838], the multiple Hopf bifurcations are obtained, and their spatio-temporal patterns: mirror-reflecting waves, standing waves, and discrete waves are demonstrated. Finally, computer simulations are performed to illustrate the analytical results found.

D3-Equivariant coupled advertising oscillators model

A ring of three coupled advertising oscillators with delay is considered. Using the symmetric functional differential equation theories, the multiple Hopf bifurcations of the equilibrium at the origin are demonstrated. The existence of multiple branches of bifurcating periodic solution is obtained. Numerical simulation supports our analysis results.

GLOBAL ROBUST STABILITY ANALYSIS FOR NEURAL NETWORKS WITH CONSTANT OR VARIABLE DELAYS

In this paper, some criteria are derived for global robust asymptotic stability of a class of interval neural networks with multiple constant or time-varying delays via the Lyapunov–Krasovskii stability theory for functional differential equations and the linear matrix inequality (LMI) approach. The main results are applied to the uncertain systems with single delay, and several new criteria to determine the asymptotical convergence of these systems are obtained. Numerical examples are also given to show the effectiveness of our results.

Synchronization of chaos for two-dimensional time-delayed chaotic systems with different structures

Generalized synchronization of chaos is studied for two-dimensional time-delayed chaotic systems with different structurs. The auxiliary system method and the stability theory of functional differential equations are used. Compared with the conventional generalized synchronization approaches, the proposed method is convenient to realize generalized synchronization of chaos. Simulation results illustrate the validity of the method.

Oscillation of partial population model with diffusion and delay

By using the upper- and lower-solution method of partial functional differential equations and the oscillation theory of functional differential equation, the oscillation of a population equation with diffusion and delay is studied and a sufficient condition for all positive solutions of the equation to oscillate about the positive equilibrium is obtained. Finally, a model arising from ecology is given to illustrate the obtained results.

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.

Rich dynamics in a non-local population model over three patches

We consider a system of nonlinear delay differential equations that describes the growth of the mature population of a species with age-structure living over three patches. We analyze existence of non-negative homogeneous equilibria and their stability and discuss possible Hopf bifurcation from these equilibria. More precisely, by employing both the standard Hopf bifurcation theory and the symmetric bifurcation theory for functional differential equations, we obtain very rich dynamics for the system, including bistable equilibria, transient oscillations, synchronous periodic solutions, phase-locked periodic solutions, mirror-reflecting waves and standing waves.

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

Abstract 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.

Hopf bifurcation in a partial dependent predator–prey system with delay

In this paper, a partial dependent predator–prey model with time delay is studied by using the theory of functional differential equation and Hassard’s method, the condition on which positive equilibrium exists and Hopf bifurcation occurs are given. Finally, numerical simulations are performed to support the analytical results, and the chaotic behaviors are observed.

Bifurcation analysis and spatio-temporal patterns of nonlinear oscillations in a delayed neural network with unidirectional coupling

In this paper, a delayed neural network with unidirectional coupling is considered which consists of two two-dimensional nonlinear differential equation systems with exponential decay where one system receives a delayed input from the other system. Some parameter regions are given for conditional/absolute stability 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 centre 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. Then the global continuation of phase-locked periodic solutions is investigated. Numerical simulations are given to illustrate the results obtained.

Solving linear variational inequalities by projection neural network with time-varying delays

A projection neural network with time-varying delays is proposed for solving linear variational inequalities. By the theory of functional differential equation and the linear matrix inequality technique, the proposed neural network is proved to be globally exponentially stable. The obtained results complement and extend the previously known work.

Equivariant Hopf Bifurcation in a Ring of Identical Cells with Delay

A kind of delay neural network with n elements is considered. By analyzing the distribution of the eigenvalues, a bifurcation set is given in an appropriate parameter space. Then by using the theory of equivariant Hopf bifurcations of ordinary differential equations due to Golubitsky et al. (1988) and delay differential equations due to Wu (1998), and combining the normal form theory of functional differential equations due to Faria and Magalhaes (1995), the equivariant Hopf bifurcation is completely analyzed.

Asymptotic stability of delayed neural networks: A descriptor system approach

In this paper some novel approaches to the analysis of asymptotic stability of artificial neural networks with time-varying delay are presented. These approaches are based on the Lyapunov–Krasovskii stability theory for functional differential equations and the linear matrix inequality (LMI) technique. Some corresponding Lyapunov–Krasovskii functionals are introduced for stability analysis of this system with use of the descriptor and “neutral-type” model transformation without producing any additional dynamics. Delay-dependent and delay-independent stability criteria are derived for this system. Conditions are given in terms of linear matrix inequalities, and for the first time refer to neutral systems with discrete and distributed delays. The proposed criteria are less conservative than other existing criteria since they are based on an equivalent model transformation and they require bounds for fewer terms. Examples are given to illustrate advantages of our approach.

A model in a coupled system of simple neural oscillators with delays

We consider a coupled system of simple neural oscillators. Using the symmetric functional differential equation theories of Wu [J. Wu, Symmetric functional differential equations and neural networks with memory, Transactions of the American Mathematical Society 350 (12) (1998) 4799–4838], we demonstrate the multiple Hopf bifurcations of the equilibrium at the origin. The existence of multiple branches of bifurcating periodic solution is obtained. Then some numerical simulations support our analysis results.

A Neutral-Type Delayed Projection Neural Network for Solving Nonlinear Variational Inequalities

A neutral-type delayed projection neural network is proposed to deal with nonlinear variational inequalities. Compared with the existing delayed neural networks for linear variational inequalities, the proposed approach apparently has the larger application domain. By the theory of functional differential equation, a delay-dependent sufficient stability condition is derived. This stability condition is easily checked, and can guarantee that the proposed neural network is convergent to the solution of nonlinear variational inequality problem exponentially, which improves the existing stability criteria for the neutral-type delayed neural network. Moreover, many related problems, such as the projection equation and optimization problems, can also be dealt with by the proposed method. Finally, simulation examples are given to illustrate the satisfactory performance of the proposed method.

Center manifolds for periodic functional differential equations of mixed type

We study the behaviour of solutions to nonlinear functional differential equations of mixed type (MFDEs), that remain sufficiently close to a prescribed periodic solution. Under a discreteness condition on the Floquet spectrum, we show that all such solutions can be captured on a finite dimensional invariant center manifold, that inherits the smoothness of the nonlinearity. This generalizes the results that were obtained previously in [H.J. Hupkes, S.M. Verduyn Lunel, Center manifold theory for functional differential equations of mixed type, J. Dynam. Differential Equations 19 (2007) 497–560] for bifurcations around equilibrium solutions to MFDEs.

Global Asymptotic Stability of Recurrent Neural Networks With Multiple Time-Varying Delays

In this paper, several sufficient conditions are established for the global asymptotic stability of recurrent neural networks with multiple time-varying delays. The Lyapunov-Krasovskii stability theory for functional differential equations and the linear matrix inequality (LMI) approach are employed in our investigation. The results are shown to be generalizations of some previously published results and are less conservative than existing results. The present results are also applied to recurrent neural networks with constant time delays.