The Structure of Phase Space Close to Fixed Points in a 4d Symplectic Map

Abstract

We study the dynamics in the neighborhood of fixed points in a 4D symplectic map by means of the color and rotation method. We compare the results with the corresponding cases encountered in galactic type potentials and we find that they are in good agreement. The fact that the 4D phase space close to fixed points is similar to the 4D representations of the surfaces of section close to periodic orbits, indicates an archetypical 4D pattern for each kind of (in)stability, not only in 3D autonomous Hamiltonian systems with galactic type potentials but for a larger class of dynamical systems. This pattern is successfully visualized with the method we use in the paper.

Analytical and numerical manifolds in a symplectic 4-D map

“…4 shows that as we go away from the stability area of parameters the manifold becomes more and more chaotic. The above results are in good agreement with those in Zachilas et al (2013).…”

“…So, the study of Hamiltonian systems with three degrees of freedom can be reduced to the study of a four dimensional mapping. Therefore, various authors have chosen to study this map as a suitable model of four-dimensional maps (Froeschlé 1971;Chen 1989;Zachilas et al 2013).…”

Dynamics of a spinning particle in a linear in spin Hamiltonian approximation

“…The method of color and rotation [30] is used for the first time in a relativistic system. Until now this method was used in 3D galactic Hamiltonian systems ( [31-33, 35, 36], the 3D circular restricted three body problem [39] and a 4D symplectic map [38]. We encountered three types of orbits in our study, which, though studied in detail in a 3D galactic system [31,33], have never been investigated in other 3D systems in the framework of general relativity.…”

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

“…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].…”