2002/07/15 by S Addas Zanata, Salvador Addas-Zanata · 10 citations
Mathematics · Physics and Astronomy · #Chaos control and synchronization #Invariant (physics) #Mathematical Dynamics and Fractals #Orbit (dynamics) #Periodic orbits #Quantum chaos and dynamical systems #Rotation number #Torus #Twist #Type (biology) #math.DS #msc:37E40 #msc:37E45
paper · pdf · doi:10.1088/0951-7715/15/5/303
published in Nonlinearity 15(5), 1399-1416 (IOP Publishing) · 20 pages. to appear in Nonlinearity 15(5) 1399-1416
openalex publication_date 2002/07/15 · arxiv created 2002/07/16 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We prove that for a large and important class of C 1 twist maps of the torus periodic and quasi-periodic orbits of a new type exist, provided that there are no rotational invariant circles (RICs). These orbits have a non-zero `vertical rotation number' (VRN), in contrast to what happens to Birkhoff periodic orbits and Aubry–Mather sets. The VRN is rational for a periodic orbit and irrational for a quasi-periodic. We also prove that the existence of an orbit with a VRN = a >0, implies the existence of orbits with VRN = b , for all 0< b < a . In this way, related to a generalized definition of rotation number, we characterize all kinds of periodic and quasi-periodic orbits a twist map of the torus can have. As a consequence of the previous results we obtain that a twist map of the torus with no RICs has positive topological entropy, which is a very classical result. At the end of the paper we present some examples, like the standard map, such that our results apply.