Existence and Stability of Time-Periodic Dissipative Structures in Parabolic Systems of Reaction-Diffusion Type

Abstract

On the interval 0 ≤ x ≤ π the parabolic system

$$\frac{{\partial u}}{{\partial t}} = \nu D\frac{{\partial ^2 u}}{{\partial x^2 }} + F\left( u \right),\left. {\frac{{\partial u}}{{\partial x}}} \right|_{x = 0} = \left. {\frac{{\partial u}}{{\partial x}}} \right|_{ = x} = 0,$$

is considered, where u ∈ $\mathbb{R}$ ⁿ, n ≥ 2; ν > 0 is a parameter; D is a Hurwitz matrix; the vector-function F(u) ∈ C ^∞($\mathbb{R}$ ⁿ;$\mathbb{R}$ ⁿ) is such that the system u = 3DF(u) has an orbitally stable cycle u ₀(t), u₀(t) ≢ 0. It is shown that under some conditions and for ν → 0 in the vicinity of a homogeneous cycle u = u ₀(t) we can observe an infinite sequence of bifurcations of birth and death of time-periodic dissipative structures. These structures depend nontrivially on x. It is also shown that the quantity of coexisting stable periodic dissipative structures can grow indefinitely.