M-furcations in coupled maps

Details M-furcations in coupled maps M-furcations in coupled maps

1994-11-06

arxiv.org/abs/chao-dyn/9411003v1

Physics

Condensed Matter

18 pages + 2 figures (available upon request), Revtex 3.0

Scientific paper

10.1142/S0217979295000689

We study the scaling behavior of $M$-furcation $(M\!=\!2, 3, 4,\dots)$ sequences of $M^n$-period $(n=1,2,\dots)$ orbits in two coupled one-dimensional (1D) maps. Using a renormalization method, how the scaling behavior depends on $M$ is particularly investigated in the zero-coupling case in which the two 1D maps become uncoupled. The zero-coupling fixed map of the $M$-furcation renormalization transformation is found to have three relevant eigenvalues $\delta$, $\alpha$, and $M$ ($\delta$ and $\alpha$ are the parameter and orbital scaling factors of 1D maps, respectively). Here the second and third ones, $\alpha$ and $M$, called the ``coupling eigenvalues'', govern the scaling behavior associated with coupling, while the first one $\delta$ governs the scaling behavior of the nonlinearity parameter like the case of 1D maps. The renormalization results are also confirmed by a direct numerical method.

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