Poincare rotation number

The condition that the functional is locally non-constant means that in the region of its definition there are no open sets in a neighborhood of X where it might take a constant value. FVom this point-of-view, the Poincare rotation number for typical diffeomorphisms of a cycle is not a modulus. 

If the mapping (11.6.2) is the Poincare map of an autonomous system of differential equations, then the invariant curve corresponds to a two-dimensional smooth invariant torus (see Fig. 11.6.3). It is stable if L < 0, or it is saddle with a three-dimensional unstable manifold and an (m -h 2)-dimensional stable manifold if L > 0. Recall from Sec. 3.4, that the motion on the torus is determined by the Poincare rotation number if the rotation number v is irrational, then trajectories on the torus are quasiperiodic with a frequency rate u] otherwise, if the rotation number is rational, then there are resonant periodic orbits on a torus. 

The invariant manifold depends continuously on p. At p = 0, it coincides with W, When /x < 0, it is the imion of the mist able manifold of the saddle periodic orbit L p) with the stable periodic orbit L p) (where L p) are the periodic orbits into which the saddle-node bifurcates ). In the case of torus, for p> 0, the Poincare rotation number on Tfj, tends to zero as /x -> +0. Thus, on the /x-axis there are infinitely many (practically indistinguishable as p -hO) resonant zones which correspond to periodic orbits on 7 with rational rotation numbers, as well as an infinite set (typically, a Cantor set) of irrational values of p for which the motion on is quasiperiodic. 

Therefore, in the orientable case, the Poincare rotation number on the torus depends monotonically on (see Sec. 4.4). Typically, each rational rotation number corresponds to an interval of values of fi (a resonant zone). In the simplest case, there exist only two periodic orbits on the torus in the... 

Figure 2. Schematic of typical data and consistent Poincare sections from the quasiperiodic regime of Rayleigh-B nard convection. The rotation number W (in arbitrary units) is plotted versus Rayleigh number R for two different values...

If the resonant tori, which are the invariant tori whose rotational numbers are rational, are broken under perturbations, the pairs of elliptic and hyperbolic cycles are created in the resonance zone. This fact is known as a result of the Poincare-Birkhoff theorem [4], which holds only if the twist condition, Eq. (2), is satisfied. Around elliptic cycles thus created, new types of tori, which are... 

According to the Poincare-Birkhoff fixed point theorem all resonant tori break up for arbitrarily small perturbations. If the rotation number is p/q the perturbation leaves q pairs of hyperbolic and elliptic periodic orbits. The unstable hyperbolic orbits are embedded in a layer filled by aperiodic, chaotic orbits that do not stay on an invariant surface, but cover a finite non-zero volume of the phase space in a chaotic layer around the original resonant torus. The elliptic points, however, are wrapped around by new concentric tori that form islands of regular orbits within the chaotic band (Fig. 2.5). 

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