Volume 9, Number 4, Winter 2001
SPECIAL SYMMETRIC PERIODIC SOLUTIONS
OF DELAYED MONOTONE FEEDBACK SYSTEMS
YUMING CHEN AND JIANHONG WU
Abstract. A delay differential system with monotone
feedback is considered. For the positive feedback nonlinearity,
it is shown that the existence of a special symmetric 4periodic
solution is equivalent to the existence of a fixed point
for a mapping defined on a cone in a Banach space. The coexistence
of multiple (including infinitely many) special symmetric
4periodic solutions is established. Sufficient conditions for
the uniqueness of special symmetric 4periodic solutions are
also given. A related singularly perturbed system is considered,
where it is shown that the limiting profile of a special
symmetric 4periodic solution is pulselike of unbounded amplitude
and that the product of each component of the solution
with the singular parameter approaches a sawtooth wave with
the slope determined by the limit of the nonlinearity at infinity.
Similar conclusions can be drawn for a negative feedback
nonlinearity by the consideration of involved symmetry and,
in particular, special symmetric 2periodic solutions are considered.
It is observed that special symmetric 4periodic solutions
of the positive feedback system have line segmentlike
waves as limiting profiles in the plane, while special symmetric
2periodic solutions of the negative feedback system approach
a diamondlike wave.
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