Homoclinic dynamics in a spatial restricted four-body problem: blue skies into Smale horseshoes for vertical Lyapunov fa
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(2020) 132:38
ORIGINAL ARTICLE
Homoclinic dynamics in a spatial restricted four-body problem: blue skies into Smale horseshoes for vertical Lyapunov families Maxime Murray1
· J. D. Mireles James1
Received: 13 January 2020 / Revised: 27 June 2020 / Accepted: 6 July 2020 © Springer Nature B.V. 2020
Abstract The set of transverse homoclinic intersections for a saddle-focus equilibrium in the planar equilateral restricted four-body problem admits certain simple homoclinic orbits which form the skeleton of the complete homoclinic intersection—or homoclinic web. In the present work, the planar restricted four-body problem is viewed as an invariant subsystem of the spatial problem, and the influence of this planar homoclinic skeleton on the spatial dynamics is studied from a numerical point of view. Starting from the vertical Lyapunov families emanating from saddle-focus equilibria, we compute the stable/unstable manifolds of these spatial periodic orbits and look for intersections between these manifolds near the fundamental planar homoclinics. In this way, we are able to continue all of the basic planar homoclinic motions into the spatial problem as homoclinics for appropriate vertical Lyapunov orbits which, by the Smale tangle theorem, suggest the existence of chaotic motions in the spatial problem. While the saddle-focus equilibrium solutions in the planar problems occur only at a discrete set of energy levels, the cycle-to-cycle homoclinics in the spatial problem are robust with respect to small changes in energy. Keywords Gravitational four-body problem · Blue sky catastrophes · Smale tangle · Vertical Lyapunov families · Invariant manifolds · Boundary value problems Mathematics Subject Classification 70K44 · 34C45 · 70F15
The J. D. Mireles James was partially supported by NSF Grant DMS-1813501. Maxime Murray and J.D. Mireles James were partially supported by NSF Grant DMS-1700154 and by the Alfred P. Sloan Foundation Grant G-2016-7320.
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Maxime Murray [email protected] J. D. Mireles James [email protected]
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Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, USA 0123456789().: V,-vol
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M. Murray, J. D. Mireles James
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The restricted four-body problem . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Parameterization of stable/unstable manifolds . . . . . . . . . . . . . . . . . . . . 4 Cycle-to-cycle connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Results: homoclinic connections for the vertical Lyapunov families in the CRFBP 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: Obtaining a polynomial field by automatic differentiation of the CRFBP Appendix B: Orbit data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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