Diffusion Chaos in Reaction – Diffusion Boundary Problem in the Dumbbell Domain

We consider a boundary problem of reaction-diffusion type in the domain consisting 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 were studied and the dependence of invariant characteristics of the attractor on the bridge width was constructed. The diffusion parameter was 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 looses stability and then the spatially inhomogeneous chaotic attractor emerges. For the obtained attractor we compute Lyapunov exponents and Lyapunov dimension and notice that the dimension grows as the parameter decreases but is bounded. We show that the dimension growth is connected with the growing complexity of stable solutions distribution with respect to the space variable.