Mathematics – Dynamical Systems
Scientific paper
2011-05-23
Mathematics
Dynamical Systems
29 pages, 12 Postscript figures corrected typos
Scientific paper
A less studied numerical characteristic of periodic orbits of area preserving twist maps of the annulus is the twist or torsion number, called initially the amount of rotation [Mather]. It measures the average rotation of tangent vectors under the action of the derivative of the map along that orbit, and characterizes the degree of complexity of the dynamics. The aim of this paper is to give new insights into the definition and properties of the twist number, and to relate its range to the order properties of periodic orbits. We derive an algorithm to deduce the exact value or a demi--unit interval containing the exact value of the twist number. We prove that at a period--doubling bifurcation threshold of a mini-maximizing periodic orbit, the new born doubly periodic orbit has the absolute twist number larger than the absolute twist of the original orbit after bifurcation. We also show that the periodic orbits of absolute twist number greater than 1/2, that are born through a saddle--center bifurcation, are badly ordered orbits. Thus we are led to a converse KAM criterion, formulated in terms of the twist number.
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