Oscillating solutions of Delay Differential Equations

The behaviour of Delay Differential Equations has been analysed in detail. There are many relevant web links and some classic reference books are Delay Equations. Functional, complex and nonlinear analysis, by O. Diekmann, S.A. van Gils, S. M. Verduyn-Lunel and H.-O. Walther, Springer-Verlag, 1995, Introduction to functional differential equations, by J. K. Hale and S. M. Verduyn-Lunel, Springer-Verlag, 1993, Differential-Difference Equations by R. Bellman and K. L. Cooke, Academic Press, London, 1963.
The system of n+1 nonlinear delay differential equations, that is model for the pupil reflex response to light with n neuronal units modelled as leaky integrators with shunting, represents a dynamical system. A point in the phase space of this dynamical system is a function defined on the interval [tM , 0] and with values in Rn+1, where tM is the maximum of all time delay parameters. This phase space, with the supremum norm, is an infinite dimensional Banach space.

The existence and stability of periodic solutions for autonomic Delay Differential Equations can be studied with methods like those used for Ordinary Differential Equations, modified to suit the structure of the infinite dimensional phase space of the DDEs. One method is to use a geometric approach similar to the Poincaré map for ODEs, and define a map A given by the intersection of the solution to a convex closed subset K of the phase space. A closed orbit representing a periodic solution will contain a fixed point of this map. The problem is that in the infinite dimensional phase space, the closed convex subset K contains the equilibrium point, that is a fixed point of the map. A nontrivial, nonconstant orbit does not contain the equilibrium point. It is necessary to prove that the origin is an ejective point and that the map has at least another fixed point than the equilibrium point that is not ejective. The existence of this point implies the existence of a nontrivial, nonconstant periodic solution. This solution is stable, as in the case of the Poincaré map, if the map A has an eigenvalue v with v < 1.

Another method, that analyses the behaviour of solutions in a neighbourhood of the equilibrium point under certain conditions of continuity and differentiability, is using centre manifold theory and normal form reduction. The characteristic equation of the linearized system has eigenvalues the roots of a transcendental polynomial. There is an infinite number of complex roots with negative real part. The number of eigenvalues with positive real part is finite or zero and changes only when parameters vary such that an eigenvalue crosses the imaginary axis. Thus, the system can move from a region in the parameter space with stable equilibrium point (where all eigenvalues have negative real part) to a region where the equilibrium point is unstable and a nonconstant periodic solution can appear (Hopf bifurcation). Centre manifold theory takes into account the nonlinearities involved and can determine if this periodic solution represents a stable or unstable limit cycle. The centre manifold contains all the bounded and periodic solutions in a neighbourhood of a non-hyperbolic equilibrium point (that has pure imaginary eigenvalues) and is invariant under the semiflow of the system of equations. The centre manifold is tangent at the equilibrium point to the centre subspace generated by the pure imaginary eigenvalues. The centre subspace contains all the periodic solutions of the linearized system of equations and the centre manifold represents the distortion caused by nonlinearities of this centre subspace.

It is quite difficult to study global behaviour of solutions, especially if the system has multiple equilibrium points. The system of equations considered is quite complicated, with many parameters, time delays and unknown exact expression for nonlinearities. Some of these nonlinearities can be non-monotonous and with hysterezis, making the global behaviour of solutions very difficult to analyze, like the real biological process that is being modelled.