The transition to chaos via mode locking, quasiperiodicity have been found, both experimentally and numerically, exist in a large number of nonlinear systems. Circle map provides a basic mathematical model in describing this phenomenon, and it reveals rich universal scaling behavior in such a transition.;Hopf bifurcations in two and higher 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 as one-dimensional circle map. We numerically investigated a two dimensional map (the coupled logistic map), and found an excellent agreement of the map with the critical scaling predictions for a circle map with smooth cubic inflection point. This occurs in spite of the fact that within mode locking intervals on the critical line, which occupy a set of full measure, the induced map has no cubic inflection point.;A new approach to the study of critical scaling, based on the thermodynamic formalism is presented. The critical scaling of several thermodynamic quantities in the circle map is studied numerically. New scaling relations are discovered which seem to be universal, as indicated by our numerical results on both one-dimensional circle maps and the two-dimensional coupled logistic map. In particular, one of the thermodynamic quantities has a saddle point near the critical line inside each tongue. Its existence and the related scaling properties may provide a basis for systematic study of the universal behavior of the transition to chaos via quasiperiodicity in real systems and numerical models in higher dimensional space.
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