Computational method for phase space transport with applications to lobe dynamics and rate of escape
Lobe dynamics and escape from a potential well are general frameworks introduced to study phase space transport in chaotic dynamical systems.While the former approach studies how regions of phase space get transported by reducing the flow to a two-dimensional map, the latter approach studies the phase space structures that lead to critical events by crossing certain barriers. Lobe dynamics describes global transport in terms of lobes, parcels of phase space bounded by stable and unstable invariant manifolds associated to hyperbolic fixed points of the system. Escape from a potential well describes how the critical events occur and quantifies the rate of escape using the flux across the barriers. Both of these frameworks require computation of curves, intersection points, and the area bounded by the curves. We present a theory for classification of intersection points to compute the area bounded between the segments of the curves. This involves the partition of the intersection points into equivalence classes to apply the discrete form of Green’s theorem. We present numerical implementation of the theory, and an alternate method for curves with nontransverse intersections is also presented along with a method to insert points in the curve for densification.
Keywordschaotic dynamical systems numerical integration phase space transport lobe dynamics
MSC2010 numbers37J35 37M99 65D20 65D30 65P99
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- 6.Beigie, D., Codimension-One Partitioning and Phase Space Transport in Multi-Degree-of-Freedom Hamiltonian Systems with Non-Toroidal Invariant Manifold Intersections. Decidability and Predictability in the Theory of Dynamical Systems, Chaos Solitons Fractals, 1995, vol. 5, no. 2, pp. 177–211.MathSciNetCrossRefzbMATHGoogle Scholar
- 9.Dellnitz, M., Junge, O., Koon, W.S., Lekien, F., Lo, M.W., Marsden, J.E., Padberg, K., Preis, R., Ross, Sh. D., and Thiere, B, Transport in Dynamical Astronomy and Multibody Problems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2005, vol. 15, no. 3, pp. 699–727.MathSciNetCrossRefzbMATHGoogle Scholar
- 18.Samelson, R.M. and Wiggins, S., Lagrangian Transport in Geophysical Jets and Waves: The Dynamical Systems Approach, Interdiscip. Appl. Math., vol. 31, New York: Springer, 2006.Google Scholar
- 22.Rom-Kedar, V., Transport in a Class of n-D.o.F. Systems, in Hamiltonian Systems with Three or More Degrees of Freedom: Proc. of the NATO Advanced Study Institute (S’Agaró, 1995), C. Simó (Ed.), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 533, Dordrecht: Kluwer,1999, pp. 538–543.CrossRefGoogle Scholar
- 33.Koon, W. S., Lo, M.W., Marsden, J. E., and Ross, Sh. D., Dynamical Systems, the Three-Body Problem and Space Mission Design, Marsden Books, 2011.Google Scholar