Analytical and numerical manifolds in a symplectic 4-D map

Abstract

We study analytically the orbits along the asymptotic manifolds from a complex unstable periodic orbit in a symplectic 4-D Froeschlé map. The orbits are given as convergent series. We compare the analytic results by truncating the series at various orders with the corresponding numerical results and we find agreement along a more extended length, as the order of truncation increases. The agreement is improved when the parameters approach those of the stability domain. Along the manifolds no terms with small divisors appear in the series. The same result is found if we use a parametrization method along the asymptotic curves. In the case of orbits starting close to the manifolds small divisors appear, but the orbits remain close to the manifolds for an extended period of time. If the parameters of the map are close to the stable domain the orbits recede and approach the origin several times and remain confined in a certain volume around the origin for a long time before escaping to large distances. For special sets of parameters we see resonance phenomena and the orbits take particular forms near every resonance.

Homoclinic points of 2D and 4D maps via the parametrization method

“…The validity of our findings was checked again by finding parameter values at which homoclinic orbits disappear through a tangency of the corresponding manifolds. What is remarkable, both in the 2D and in the 4D case it was possible to accurately locate homoclinic points just using the parametrization method without any additional application of the mapping or its inverse (see [34,37] for another approach).…”

Geometry of complex instability and escape in four-dimensional symplectic maps

“…The occurrence of the plateaus can be explained by the alternating spiraling in and out of the dynamics already observed in Refs. [6,11,36,38]: An orbit initially started near the complex unstable fixed point moves away from it on a spiral along the unstable manifold until it reaches a maximal distance to the fixed point. This behavior corresponds to the first expansion phase up to approximately 100 iterations.…”

“…A distinctive feature is the spiraling motion in the surrounding of a complex unstable periodic point [6,30]. Moreover, it was found that commonly an extended region around a complex unstable fixed point emerges to which the dynamics is confined for rather long times [11,[36][37][38]. Important approaches to understand the complex unstable dynamics are based on computations of the invariant manifolds [36,38,39] and normal form descriptions [15,40,41].…”

“…Moreover, it was found that commonly an extended region around a complex unstable fixed point emerges to which the dynamics is confined for rather long times [11,[36][37][38]. Important approaches to understand the complex unstable dynamics are based on computations of the invariant manifolds [36,38,39] and normal form descriptions [15,40,41]. Hamiltonian-Hopf bifurcations have also been studied in much detail for reversible maps, see e.g.…”

Chaoticity in the vicinity of complex unstable periodic orbits in galactic type potentials

“…In galactic dynamics it characterizes periodic orbits of many three dimensional (hereafter 3D) models in a large volume of their parameter space (Magnenat, 1982a,b;Pfenniger, 1984Pfenniger, , 1985bContopoulos and Magnenat, 1985;Contopoulos, 1986;Martinet and Pfenniger, 1987;Pfenniger, 1987;Martinet and de Zeeuw, 1988;Zachilas, 1988;Patsis and Zachilas, 1990;Zachilas, 1993;Patsis and Zachilas, 1994;Olle and Pfenniger, 1998;Katsanikas, Patsis and Contopoulos, 2011;Patsis and Katsanikas, 2014). However, considerable insight in the role of complex instability for the dynamics of Hamiltonian systems has been gained by works on several other kinds of potentials (Heggie, 1985;Papadaki, Contopoulos and Polymilis, 1995;Ollé and Pacha, 1999) or 4-dimensional symplectic mappings (Pfenniger, 1985a;Contopoulos and Giorgilli, 1988;Olle and Pfenniger, 1995;Jorba and Ollé, 2004;Delis and Contopoulos, 2016;Stöber and Bäcker, 2021). These results have also been used for understanding the behaviour of stellar orbits in galactic type potentials.…”