Pattern Formation on Trees

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.Comment: Submitted to Phys. Rev. E, 15 pages, 9 fig

Spectral Properties and Synchronization in Coupled Map Lattices

Spectral properties of Coupled Map Lattices are described. Conditions for the stability of spatially homogeneous chaotic solutions are derived using linear stability analysis. Global stability analysis results are also presented. The analytical results are supplemented with numerical examples. The quadratic map is used for the site dynamics with different coupling schemes such as global coupling, nearest neighbor coupling, intermediate range coupling, random coupling, small world coupling and scale free coupling.Comment: 10 pages with 15 figures (Postscript), REVTEX format. To appear in PR

Some generic aspects of bosonic excitations in disordered systems

We consider non-interacting bosonic excitations in disordered systems, emphasising generic features of quadratic Hamiltonians in the absence of Goldstone modes. We discuss relationships between such Hamiltonians and the symmetry classes established for fermionic systems. We examine the density \rho(\omega) of excitation frequencies \omega, showing how the universal behavior \rho(\omega) ~ \omega^4 for small \omega can be obtained both from general arguments and by detailed calculations for one-dimensional models

Wavelength-doubling bifurcations in one-dimensional coupled logistic maps

We discuss in detail the interesting phenomenon of wavelength-doubling bifurcations in the model of coupled-map lattices reported earlier [Phys. Rev. Lett. 70, 3408 (1993)]. We take nearest-neighbor coupling of logistic maps on a one-dimensional lattice. With the value of the parameter of the logistic map, μ , corresponding to the period-doubling attractor, we see that the wavelength and the temporal period of the observed pattern undergo successive wavelength- and period-doubling bifurcations with decreasing coupling strength ε. The universality constants α and δ appear to be the same as in the case of the period-doubling route to chaos in the uncoupled logistic map. The phase diagram in the ε -μ plane is investigated. For large values of μ and large periods, regions of instability are observed near the bifurcation lines. We also investigate the mechanism for the wavelength-doubling bifurcations to occur. We find that such bifurcations occur when the eigenvalue of the stability matrix corresponding to the eigenvector with periodicity of twice the wavelength exceeds unity in magnitude

Wavelength doubling bifurcations in coupled map lattices

We report an interesting phenomenon of wavelength doubling bifurcations in the model of coupled (logistic) map lattices. The temporal and spatial periods of the observed patterns undergo successive period doubling bifurcations with decreasing coupling strength. The universality constants α and δ appear to be the same as in the case of period doubling route to chaos in the uncoupled logistic map. The analysis of the stability matrix shows that period doubling bifurcation occurs when an eigenvalue whose eigenvector has a structure with doubled spatial period exceeds unity