D. del-Castillo-Negrete, J. M. Greene, P. Morrison
Renormalization and transition to chaos in area preserving
non twist maps.
(90K, Latex 2.0)
ABSTRACT. The problem of transition to chaos, i.e.\ the destruction of
invariant circles or KAM (Kolmogorov-Arnold-Moser) curves, in area
preserving {\em nontwist} maps is studied within the renormalization group
framework. Nontwist maps are maps for which the twist condition is violated
along a curve known as the shearless curve. In renormalization language this
problem is that of finding and studying the fixed points of the
renormalization group operator ${\cal R}$ that acts on the space of maps. A
simple period-two fixed point of ${\cal R}$, whose basin of attraction
contains the nontwist maps for which the shearless curve exists, is found.
Also, a critical period-twelve fixed point of ${\cal R}$, with two unstable
eigenvalues, is found. The basin of attraction of this critical fixed point
contains the nontwist maps for which the shearless curve is at the threshold
of destruction. This basin defines a new universality class for the
transition to chaos in area preserving maps.