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This work studies periodic orbits as action minimizers in the spatial isosceles three-body problem with mass . In each period, the body with mass moves up and down on a vertical line, while the other two bodies have the same mass , and rotate about this vertical line symmetrically. For given , such periodic orbits form a one-parameter set with a rotation angle as the parameter.Two new phenomena are found for this set. First, for each , this set of periodic orbits bifurcate from a circular Euler (central configuration) orbit to a Broucke (collision) orbit as increases from to . There exists a critical rotation angle , where the orbit is a circular Euler orbit if ; a spatial orbit if ; and a Broucke (collision) orbit if . The exact formula of is numerically proved to be . Second, oscillating behaviors occur at rotation angle for all . Actually, the orbit with 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 increases. Up to , the orbit is bounded and keeps oscillating between periodic shapes and irregular motions. Further study implies that, for each , the angle between any two consecutive periodic shapes is a constant. When , similar oscillating behaviors are expected. [ABSTRACT FROM AUTHOR]

In the three-body problem, it is known that there exists a special set of periodic orbits: spatial isosceles periodic orbits. In each period, one body moves up and down along a straight line, and the other two bodies rotate around this line. In this work, we revisit this set of orbits by applying variational method. Two unexpected phenomena are discovered. First, this set is not always spatial. It actually bifurcates from the circular Euler (central configuration) orbit to the Broucke (collision) orbit. Second, one of the orbits in this set encounters an oscillating behavior. By running its initial condition, the orbit stays periodic for only a few periods before it becomes irregular. However, it moves close to another periodic shape in a while. Shortly it falls apart again and starts running close to a third periodic shape after a moment. This oscillation continues as t increases. Actually, up to t = 1.2 × 105, the orbit is bounded and keeps oscillating between periodic shapes and irregular motions. [ABSTRACT FROM AUTHOR]