
We consider a distributed delay neural network representation of a conceptual model of spatial memory in the form of the system

We consider nonlinear equations with a distributed delay which involve integrodifferential equations. In the case of a single variable concentrated delay we have the equation x¢(t) = r(t) [ f ( x ( th(t) ) )  x(t) ] which is compared to the difference equation x_{n+1} = f(x_{n}). In particular, under certain conditions global attractivity of the latter equation implies stability of the former one, independently of a bounded delay. We also present some delaydependent stability, oscillation and nonoscillation conditions. The results are applied to the Nicholson's blowflies equation and the MackeyGlass equation with a distributed delay.
This is a joint work with L. Berezansky, D. Kinzebulatov and S. Zhukovskiy.
We consider a model for regenerative chatter in a drilling process. The model is a nonlinear delay differential equation where the delay arises from the fact that the cutting tool passes over the metal surface repeatedly. For any fixed value of the delay, a large enough increase in the width of the chip being cut results a Hopf bifurcation from the steady state, which is the origin of the chatter vibration. We show that for zero delay the Hopf bifurcation is degenerate and that for small delays this leads to a canard explosion. That is, as the chip width is increased beyond the Hopf bifurcation value, there is a rapid transition from a small amplitude limit cycle to a large relaxation cycle. Our analysis relies on perturbation techniques and a small delay approximation of the DDE model. We use numerical simulations and numerical continuation to support our analysis and to determine when the small delay approximation fails. We discuss how our results may apply to other systems with time delays.
This is joint work with Emily Stone and Thomas Erneux.
This talk concerns with the equivariant normal form of delay differential equations with parameters in the presence of symmetry. We present and justify a process that involves center manifold reduction and normalization preserving the symmetry, and that yields normal forms explicitly in terms of the coefficients of the original system. We observe that the form of the reduced vector field relies only on the information of the linearized system at the critical point and on the inherent symmetry, and the normal forms give critical information about not only the existence but also the stability and direction of bifurcated spatiotemporal patterns. We illustrate our general results with application to equivariant Hopf bifurcation.
We apply the recently developed global Hopf bifurcation theory to examine global continuation with respect to the parameter for periodic solutions of functional differential equations with a statedependent delay. As major components toward this goal, we establish sufficient geometric conditions to ensure the uniform boundedness of periodic solutions and then prove the existence of rapidly oscillating periodic solutions by obtaining an upper bound of the period of the nontrivial periodic solutions in a connected bifurcation branch.
Although not widely considered in the literature, several problems, including travelling waves on lattices, naturally give rise to advanceddelay boundary value problems. Since there is very little theory to underpin these problems, numerical results are needed to provide intuition, but are themselves difficult to obtain. Problems are compounded by statedependency of the delay terms. We highlight ongoing work in this area, illustrating the issues using a Bando discrete carfollowing model with reaction time delay, and some model problems with linear statedependency.
The study of travelling waves in a 1D lattice with Z \ltimes Z_{2} symmetry reduces to the study of solutions of a "reversible" advanceddelay equation (ADE). We present an extension of Lin's method for ODEs by which we prove that any Z_{2}invariant localized travelling wave (biasymptotic to the trivial state) is accumulated by a oneparameter family of periodic travelling waves.
This is joint work with Marc Georgi (Free University Berlin), Alice Jukes and Kevin Webster (Imperial College London).