Sue Ann Campbell

Stability of a class of linear switching systems with time delay

We consider a switching system composed of a finite number of linear delay differential equations (DDEs). It has been shown that the stability of a switching system composed of a finite number of linear ordinary differential equations (ODEs) may be achieved by using a common Lyapunov function method switching rule. We modify this switching rule for ODE systems to a common Lyapunov functional method switching rule for DDE systems and show that it stabilizes our model. Our result uses a Riccati-type Lyapunov functional under a condition on the time delay.

Stability, Bifurcation, and Multistability in a System of Two Coupled Neurons with Multiple Time Delays

A system of delay differential equations representing a model for a pair of neurons with time-delayed connections between the neurons and time delayed feedback from each neuron to itself is studied. Conditions for the linear stability of the trivial solution of this system are represented in a parameter space consisting of the sum of the time delays between the elements and the product of the strengths of the connections between the elements. It is shown that the trivial fixed point may lose stability via a pitchfork bifurcation, a Hopf bifurcation, or one of three types of codimension-two bifurcations. Multistability near these latter bifurcations is predicted using center manifold analysis and confirmed using numerical simulations.

Frustration, stability, and delay-induced oscillations in a neural network model

The effect of time delays on the linear stability of equilibria in an artificial neural network of Hopfield type is analyzed. The possibility of delay-induced oscillations occurring is characterized in terms of properties of the (not necessarily symmetric) connection matrix of the network.Such oscillations are possible exactly when the network is frustrated, equivalently when the signed digraph of the matrix does not require the Perron property. Nonlinear analysis (centre manifold computation) of a three-unit frustrated network is presented, giving the nature of the bifurcations taking place. A supercritical Hopf bifurcation is shown to occur, and a codimension-two bifurcation is unfolded.

Stability and bifurcations of equilibria in a multiple-delayed differential equation

The influence of multiple negative delayed feedback loops on the stability of a single-action mechanism are considered. A characteristic equation for the linearized stability of the equilibrium is completely analyzed, as a function of two parameters describing a delay in one loop and a ratio of the gains in the two feedback loops. The bifurcations occurring as the linear stability is lost are analyzed by the construction of a centre manifold. In particular, the nature of Hopf and more degenerate, higher codimension bifurcations are explicitly determined.

BIFURCATION ANALYSIS OF A PREDATOR-PREY SYSTEM WITH NONMONOTONIC FUNCTIONAL RESPONSE ∗

We consider a predator-prey system with nonmonotonic functional response:

Equivariant Hopf bifurcation in a ring of identical cells with delayed coupling

p(x)=\frac{mx}{ax^2+bx+1}

Limit Cycles, Tori, and Complex Dynamics in a Second-Order Differential Equation with Delayed Negative Feedback

. By allowing b to be negative (

Delayed Coupling Between Two Neural Network Loops

b > -2\sqrt a

QUALITATIVE ANALYSIS OF A NEURAL NETWORK MODEL WITH MULTIPLE TIME DELAYS

), p(x) is concave up for small values of x > 0 a...

Dynamics of an Inverted Pendulum with Delayed Feedback Control

We consider a ring of identical elements with time delayed, nearest-neighbour coupling. The individual elements are modelled by a scalar delay differential equation which includes linear decay and nonlinear delayed feedback. The bifurcation and stability of nontrivial asynchronous oscillations from the trivial solution are analysed using equivariant bifurcation theory and centre manifold construction.