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 4-periodic 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 4-periodic solutions is established. Sufficient conditions for the uniqueness of special symmetric 4-periodic solutions are also given. A related singularly perturbed system is considered, where it is shown that the limiting profile of a special symmetric 4-periodic solution is pulse-like 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 2-periodic solutions are considered. It is observed that special symmetric 4-periodic solutions of the positive feedback system have line segment-like waves as limiting profiles in the plane, while special symmetric 2-periodic solutions of the negative feedback system approach a diamond-like wave.