Wavelength-doubling bifurcations in one-dimensional coupled logistic maps

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Abstract
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.