Pattern Formation on Trees

Details Pattern Formation on Trees Pattern Formation on Trees

2001-05-04

arxiv.org/abs/nlin/0105008v1

Nonlinear Sciences

Pattern Formation and Solitons

Submitted to Phys. Rev. E, 15 pages, 9 figs

Scientific paper

10.1103/PhysRevE.64.026208

Networks having the geometry and the connectivity of trees are considered as the spatial support of spatiotemporal dynamical processes. A tree is characterized by two parameters: its ramification and its depth. The local dynamics at the nodes of a tree is described by a nonlinear map, given rise to a coupled map lattice system. The coupling is expressed by a matrix whose eigenvectors constitute a basis on which spatial patterns on trees can be expressed by linear combination. The spectrum of eigenvalues of the coupling matrix exhibit a nonuniform distribution which manifest itself in the bifurcation structure of the spatially synchronized modes. These models may describe reaction-diffusion processes and several other phenomena occurring on heterogeneous media with hierarchical structure.

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