Abstract
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.
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Acknowledgments
This work was partly supported by NSFC-61003021, NSFC-11290141, NSFC-11371047, and SKLSDE-2013ZX-10. In particular, the authors would like to express their profound thanks to the editor and the two anonymous reviewers for their helpful and detailed comments that greatly improved the presentation of this paper.
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She, Z., Cheng, X. The existence of a Smale horseshoe in a planar circular restricted four-body problem. Celest Mech Dyn Astr 118, 115–127 (2014). https://doi.org/10.1007/s10569-013-9528-1
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DOI: https://doi.org/10.1007/s10569-013-9528-1