Qualitative Characteristics of Dynamical Systems

Abstract

We compare some basic families of orbits in a number of galactic models and in the restricted three body problem. In plane rotating barred galaxies the orbits around the Lagrangian points L₄, L₅, L₁, L₂ are qualitatively similar to the orbits around L₄, L₅, L₃ of the restricted three body problem. The long period orbits are connected to the short period orbits (SPO) by an infinity of bridges. The limit of the sequence of bridges is a family of heteroclinic (in the galactic problem), or homoclinic (in the restricted problem) orbits. In various galactic models the SPO families are connected with the families around L₁ L₂ by one or more bridges. All these families end at retrograde orbits around the centre. This behaviour is compared with the restricted problem, in which the SPO families join the family around L₃, which ends at retrograde orbits around the larger primary.

In 3-dimensional systems we describe briefly the consequences of some collisions of bifurcations, like the nonuniqueness of certain families of periodic orbits. Then we examine the behaviour of nonperiodic orbits starting close to unstable periodic orbits, especially in the case of complex instability. We define a LCN (Lyapunov characteristic number) for orbits at different distances from the periodic orbit. There is a “local LCN” equal to the real part of the characteristic exponent, then LCN’s appropriate for a small or large neighbourhood of the periodic orbit. We follow the change of the LCN’s as certain parameters of the system vary.