Diffusion chaos in the reaction–diffusion boundary problem in the dumbbell domain

Abstract

We consider a boundary problem of the reaction–diffusion type in the domain that consists of two rectangular areas connected by a bridge. The bridge width is a bifurcation parameter of the problem and is changed in such way that the measure of the domain is preserved. The conditions on chaotic oscillations emergence have been studied and the dependence of invariant characteristics of the attractor on the bridge width has been constructed. The diffusion parameter has been chosen such that, in the case of widest possible bridge (corresponding to a rectangular domain), the spatially homogeneous cycle of the problem is orbitally asymptotically stable. By decreasing the bridge width, the homogeneous cycle loses stability; then, a spatially inhomogeneous chaotic attractor emerges. For the obtained attractor, we have calculated the Lyapunov exponents and Lyapunov dimension and observed that the dimension grows as the parameter decreases, but is bounded. We have shown that the dimension growth is connected with the growing complexity of the distribution of stable solutions with respect to the space variable.