Return Maps, Dynamical Consequences and Applications

Abstract

After reviewing some general settings for return maps in problems reducible to 2D symplectic maps, details on the construction of return maps are presented. Different forms of such maps close to splitted separatrices (separatrix maps) are introduced, taking into account the size and shape of the splitting function and also the return time to the domains of interest. Then it is shown how to derive approximations by suitable standard-like maps. Dynamical consequences concerning the existence of invariant rotational curves (IRC) are derived. An application is made to theoretically estimate the location of the outermost IRC in the Sitnikov problem, which is in good agreement with numerical data. To compare with the cases which are approximated by the classical standard map, some details on the properties of the standard-like map with two harmonic terms are included. Finally a method to estimate the amount of chaos depending on the form of the separatrix map is introduced. Except otherwise stated all the systems we consider are assumed to be analytic, despite several of the properties we study are no longer analytic.

Some questions looking for answers in dynamical systems

“…In many problems, for instance in Celestial Mechanics or in particle accelerators, we can apply the usual methods to obtain, from the splitting of suitable manifolds and the return time to a fundamental domain, a separatrix map (that is, the return map to this domain) which can be approximated by a standard-like map in a subdomain not too close to the broken separatrix. See, e.g., [83,95,117] and references therein.…”

Global dynamics and diffusion in the rational standard map

“…Models of multiharmonic SMs have been studied in several papers, like the seminal works of Ketoja and MacKay [14,15]. See also [16,17] and references therein.…”

“…5 presents in red the region of the (K , µ) space, Σ b , where the numerical evidence suggests that the iterates of initial data taken on W u 0 remain bounded by an invariant curve. We also include in the figure the stability interval for the two-periodic orbits given by (17). Note that the border of stability of the RSM is not smooth (fractal structures are clearly observed, see for instance [14][15][16][17] for related discussions), a large stability domain appears for K values at both sides of the line K = 4µ when µ < 0.6.…”

“…We also include in the figure the stability interval for the two-periodic orbits given by (17). Note that the border of stability of the RSM is not smooth (fractal structures are clearly observed, see for instance [14][15][16][17] for related discussions), a large stability domain appears for K values at both sides of the line K = 4µ when µ < 0.6. At µ = 0 we obtain the expected value of K = K c ≈ 0.9716354 (Greene's value) and only for µ ≳ 0.75 the stability border in K lies below K c .…”

“…We can compare these results for the RSM with the ones that one obtains for the SM with two harmonics, say SM2h. For instance with the ones presented as an example in [17]. There the function f in SM2h depends on a parameter α and is given by f (x, α) = cos(2π α) sin(x) + sin(2π α) sin(2x).…”