The existence of a Smale horseshoe in a planar circular restricted four-body problem. (English) Zbl 1293.70049

Summary: In this paper we study the existence of a Smale horseshoe in a planar circular restricted four-body problem. For this planar four-body system there exists a transversal homoclinic orbit, but the fixed point is a degenerate saddle, so that the standard Smale-Birkhoff homoclinic theorem cannot be directly applied. We therefore apply the Conley-Moser conditions to prove the existence of a Smale horseshoe. Specifically, we first use the transversal structure of stable and unstable manifolds to make a linear transformation and then introduce a nonlinear Poincaré map \(P\) by considering the truncated flow near the degenerate saddle; based on this Poincaré map \(P\), we define an invertible map \(f\), which is a composite function; by carefully checking the satisfiability of the Conley-Moser conditions for \(f\) we finally prove that \(f\) is a Smale horseshoe map, which implies that our restricted four-body problem has the chaotic dynamics of the Smale horseshoe type.


70F10 \(n\)-body problems
70K44 Homoclinic and heteroclinic trajectories for nonlinear problems in mechanics
70K55 Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics
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