Circle-map scaling in a two-dimensional setting

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Abstract
Hopf bifurcations in two-dimensional maps give rise to closed invariant curves and circle maps induced on these curves. It is not obvious whether the induced maps will exhibit the full array of scaling phenomena familiar from the study of one-dimensional maps. In the present work we numerically study a variable Jacobian map (the coupled logistic map). A detailed investigation along its critical line shows an excellent agreement of the map with the critical scaling predictions for a circle map with a smooth cubic inflection point. This occurs in spite of the fact that within mode-locking intervals on the critical line, which together occupy a set of full measure, the induced map has no cubic inflection point.