Pm. Gade et Re. Amritkar, WAVELENGTH-DOUBLING BIFURCATIONS IN ONE-DIMENSIONAL COUPLED LOGISTIC MAPS, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 49(4), 1994, pp. 2617-2622
We discuss in detail the interesting phenomenon of wavelength-doubling
bifurcations in the model of coupled-map lattices reported earlier [P
hys. Rev. Lett. 70, 3408 (1993)]. We take nearest-neighbor coupling of
logistic maps on a one-dimensional lattice. With the value of the par
ameter of the logistic map, mu, corresponding to the period-doubling a
ttractor, we see that the wavelength and the temporal period of the ob
served pattern undergo successive wavelength- and period-doubling bifu
rcations with decreasing coupling strength epsilon. The universality c
onstants alpha and delta appear to be the same as in the case of the p
eriod-doubling route to chaos in the uncoupled logistic map. The phase
diagram in the epsilon-mu plane is investigated. For large values of
mu and large periods, regions of instability are observed near the bif
urcation fines. We also investigate the mechanism for the wavelength-d
oubling bifurcations to occur. We find that such bifurcations occur wh
en the eigenvalue of the stability matrix corresponding to the eigenve
ctor with periodicity of twice the wavelength exceeds unity in magnitu
de.