New Phenomena in the Spatial Isosceles Three-Body Problem with Unequal Masses

Abstract

This work studies periodic orbits as action minimizers in the spatial isosceles three-body problem with mass [Formula: see text]. In each period, the body with mass [Formula: see text] moves up and down on a vertical line, while the other two bodies have the same mass [Formula: see text], and rotate about this vertical line symmetrically. For given [Formula: see text], such periodic orbits form a one-parameter set with a rotation angle [Formula: see text] as the parameter. [Formula: see text]Two new phenomena are found for this set. First, for each [Formula: see text], this set of periodic orbits bifurcate from a circular Euler (central configuration) orbit to a Broucke (collision) orbit as [Formula: see text] increases from [Formula: see text] to [Formula: see text]. There exists a critical rotation angle [Formula: see text], where the orbit is a circular Euler orbit if [Formula: see text]; a spatial orbit if [Formula: see text]; and a Broucke (collision) orbit if [Formula: see text]. The exact formula of [Formula: see text] is numerically proved to be [Formula: see text]. Second, oscillating behaviors occur at rotation angle [Formula: see text] for all [Formula: see text]. Actually, the orbit with [Formula: see text] runs on its initial periodic shape for only a few periods. It breaks the first periodic shape and becomes irregular in a moment. However, it runs close to a different periodic shape after a while. In a short time, it falls apart from the second periodic shape and runs irregularly again. Such oscillation continues as time [Formula: see text] increases. Up to [Formula: see text], the orbit is bounded and keeps oscillating between periodic shapes and irregular motions. Further study implies that, for each [Formula: see text], the angle between any two consecutive periodic shapes is a constant. When [Formula: see text], similar oscillating behaviors are expected.