Resonant zones, inner and outer splittings in generic and low order resonances of area preserving maps

Abstract

We consider a one-parameter family of APM in a neighbourhood of an elliptic fixed point. As the parameter evolves hyperbolic and elliptic periodic orbits of different periods are created. The exceptional resonances of order less than 5 have to be considered separately. The invariant manifolds of the hyperbolic periodic points bound islands containing the elliptic periodic points. Generically, these manifolds split. It turns out that the inner and outer splittings are different under suitable conditions. We provide accurate formulae describing the splittings of these manifolds as a function of the parameter and the relative values of these magnitudes as a function of geometric properties. The numerical agreement is illustrated using mainly the Hénon map as an example.

Transient chaos in the Sitnikov problem

“…We note that transient phenomena and the stickiness effect in problems of Celestial Mechanics were mainly studied by Benet et al (1996Benet et al ( , 1998; Efthymiopoulos et al (1999); Contopoulos et al (1999); Contopoulos and Efstathiou (2004); Sun et al (2005); Contopoulos and Patsis (2006); Benet and Merlo (2009); Simó and Vieiro (2009). Several concepts and methods on this subject are discussed in the textbook of Contopoulos (2002).…”

Return Maps, Dynamical Consequences and Applications

“…In case of symmetry just the same expression can be used for A + → A − and for A − → A + . Note that there are simple cases, like what happens in a resonant zone around an elliptic fixed point, for which one can expect some symmetry in the inner and outer parts but in [33] it is proved that, in fact, this symmetry does not exist.…”

“…Using (33) the chaotic factor has been computed from data like the ones shown in Fig. 12 but using a small step in κ.…”

“…While for the classical standard map (α = 0) and for the case of two harmonics with α small the measure is close to the ratio a(ε)/γ (ε), for other values of α can be three times the ratio. The same idea can be applied to produce estimates for arbitrary shape functions, using suitable changes in (33). Similar ideas can be used to estimate the measure in the upper loop of a case like the one in Fig.…”

Escape of particles in a time-dependent potential well

“…This power law arises from combined effects of an infinite hierarchy of islands with a corresponding hierarchy of time scales. Recent investigations of resonance splittings and homoclinic tangles in the vicinity of islands may provide insight into these mechanisms [25][26][27][28].…”