Existence of echo wave in the coupled-Oregonator system

In this paper we suggest a new line of solving a problem (say, existence of echo waves) in a high-dimensional system, in the sense that we transform the problem on existence of echo waves in n identical oscillators coupled into that on existence of a periodic solution in a functional differential equation of a planar system. We first give the location of periodic solutions and then through successive lemmas we rigorously prove existence of echo waves in the n coupled Oregonators. This work, we think, is the key step toward solving Tysonʹs conjecture [Ann. N. Y. Acad. Sci. 316 (1979) 279].