The effect of delay and diffusion on spontaneous symmetry breaking in functional differential equations (Q1898333)

The paper is devoted to the study of the following one parameter family of retarded differential equations (1) \(x'(t)= f(x_t, \alpha)\), where \(f: C\times \mathbb{R}\mapsto \mathbb{R}^n\) is continuously differentiable and \(f(0, \alpha)= 0\) for \(\alpha\in \mathbb{R}\). In the first part of the paper, the existence of smooth branches of periodic solutions of the equations (1) is studied. It is established a local Hopf bifurcation theorem proving that, under certain nonresonance, genericity and transversality assumptions maximal isotropy groups with maximal dimensional fixed-point subspace lead to periodic solutions with certain spatial-temporal symmetry. In the second part of the paper the author illustrates the previous result by discrete waves in Turing rings with retarded diffusion along the sides of a polygon. The effect of delay and diffusion on the occurrence of spontaneous symmetry breaking is demonstrated.