Abstract:
Invariant manifolds of a periodic orbit at infinity in the planar circular RTBP are studied. To this end we consider the intersection of the manifolds with the passage through the barycentric pericenter. The intersections of the stable and unstable manifolds have a common even part, which can be seen as a displaced version of the two-body problem, and an odd part which gives rise to a splitting. The theoretical formulas obtained for a Jacobi constant C large enough are compared to direct numerical computations showing improved agreement when C increases. A return map to the pericenter passage is derived, and using an approximation by standard-like maps, one can make a prediction of the location of the boundaries of bounded motion. This result is compared to numerical estimates, again improving for increasing C. Several anomalous phenomena are described.
Citation:
Regina Martínez, Carles Simó, “Invariant Manifolds at Infinity of the RTBP and the Boundaries of Bounded Motion”, Regul. Chaotic Dyn., 19:6 (2014), 745–765
\Bibitem{MarSim14}
\by Regina Mart{\'\i}nez, Carles Sim\'o
\paper Invariant Manifolds at Infinity of the RTBP and the Boundaries of Bounded Motion
\jour Regul. Chaotic Dyn.
\yr 2014
\vol 19
\issue 6
\pages 745--765
\mathnet{http://mi.mathnet.ru/rcd196}
\crossref{https://doi.org/10.1134/S1560354714060112}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3284613}
\zmath{https://zbmath.org/?q=an:06507831}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000345996200011}
Linking options:
https://www.mathnet.ru/eng/rcd196
https://www.mathnet.ru/eng/rcd/v19/i6/p745
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Wang X., Cao X., “On the Persistence of Lower-Dimensional Tori in Reversible Systems With Hyperbolic-Type Degenerate Equilibrium Point Under Small Perturbations”, Acta Appl. Math., 173:1 (2021), 10
M. Alvarez-Ramirez, A. Garcia, J. F. Palacian, P. Yanguas, “Oscillatory motions in restricted n-body problems (vol 265, pg 779, 2018)”, J. Differ. Equ., 267:7 (2019), 4525–4536
A. Delshams, V. Kaloshin, A. Rosa, T. M. Seara, “Global instability in the restricted planar elliptic three body problem”, Commun. Math. Phys., 366:3 (2019), 1173–1228
W. Si, J. Si, “Response solutions and quasi-periodic degenerate bifurcations for quasi-periodically forced systems”, Nonlinearity, 31:6 (2018), 2361–2418
C. Simo, “Some questions looking for answers in dynamical systems”, Discret. Contin. Dyn. Syst., 38:12, SI (2018), 6215–6239
M. Alvarez-Ramirez, A. Garcia, J. F. Palacian, P. Yanguas, “Oscillatory motions in restricted n-body problems”, J. Differ. Equ., 265:3 (2018), 779–803
J. D. M. James, M. Murray, “Chebyshev-Taylor parameterization of stable/unstable manifolds for periodic orbits: implementation and applications”, Int. J. Bifurcation Chaos, 27:14 (2017), 1730050
I. Baldoma, E. Fontich, P. Martin, “Gevrey estimates for one dimensional parabolic invariant manifolds of non-hyperbolic fixed points”, Discret. Contin. Dyn. Syst., 37:8 (2017), 4159–4190
C. Simo, “Experiments looking for theoretical predictions”, Indag. Math.-New Ser., 27:5, SI (2016), 1068–1080
T. Zhang, A. Jorba, J. Si, “Weakly hyperbolic invariant tort for two dimensional quasiperiodically forced maps in a degenerate case”, Discret. Contin. Dyn. Syst., 36:11 (2016), 6599–6622
A. Haro, M. Canadell, J. Figueras, A. Luque, J. Mondelo, Parameterization Method For Invariant Manifolds: From Rigorous Results to Effective Computations, Applied Mathematical Sciences Series, 195, Springer, 2016, 267 pp.
Àlex Haro, Applied Mathematical Sciences, 195, The Parameterization Method for Invariant Manifolds, 2016, 1
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