CHAOTIC ORBITS IN A PLANAR THREE-CENTER PROBLEM OF SLIGHTLY NEGATIVE ENERGY

Abstract

The orbital fates in a planar three-center problem of slightly negative energy are studied numerically, by classifying the dynamics into colliding with one of the centers, escaping to infinity and orbiting around the centers. We consider three energy levels, for each of them, the set of initial values leading to collision with the centers is determined. It presents an intriguing fractal structure. The complementary set, which corresponds to those initial values whose orbits are either regular or chaotic winding around the fixed centers, also exhibits a fractal structure. These fractal structures found here might lead to some observable physical feature in the future. The fractal structures imply the sensitivity to initial conditions, thus enable us to find initial values with which the orbits are chaotic, having a positive Lyapunov exponent.