| dc.description.abstract | We study two identical oscillators each with an asymptotically stable
limit cycle coupled together in a line to its nearest neigbour by a linear diffusion
like path with a time lag. The system of equations is inbuilt
with symmetries which we exploit to get an analytic understanding of the
dynamics of the system. The symmetries help us get two two-dimensional
invariant manifolds for the system. One manifold contains an in-phase
periodic orbit while the other has an out-of-phase periodic orbit. The
thesis contains eight main parts: the introduction, literature review, and
six other chapters. The introduction in Chapter one, deals with the motivation
of the problem and basic properties of delay differential equations
(DDEs). In chapter two, we review what has been done in systems of
coupled oscillators with a delay in the coupling. We also review what
has been done on the existence of periodic solutions of a system of delay
differential equations. In Chapter three, we exploit the symmetries in the
coupling terms to show the existence of two invariant manifolds. For each
manifold, we determine the renewal equation of the equations describing
motion on it. In Chapter four, we show that .for small delays, there exists,
on each manifold, a stable periodic orbit. This is done by considering the
delay system as a perturbed ordinary differential equation. In Chapters
five and six we use a method of -cone maps and ejective fixed points to
show that the equations describing motion on the manifolds in Chapter 3 ,
under certain conditions, describe periodic motions. In Chapter seven it is
shown how a system of delay equations can be approximated by a system
of first order ordinary differential equations. Using this approximation,
we compute and study the stability of the periodic solutions of Chapter
five and six. The periodic solutions are found to be limit cycles for certain
values of the coupling strength. Chapter eight is the conclusion. | en |
