The general solution of the HENON-HEILES problem

Details The general solution of the HENON-HEILES problem The general solution of the HENON-HEILES problem

Feb 2005

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2005ap%26ss.295..375p&link_type=abstract

Astrophysics and Space Science, Volume 295, Issue 3, pp.375-396

Astronomy and Astrophysics

Astrophysics

4

Dynamical System, Axisymmetric Potential, Equilibrium Points, Periodic Solutions, Numerical Integration, Periodic Family Curves, General Solution, Stability, Stable/Unstable Periodic Family Arcs, Poincaré Surface Of Sections, Order, Chaos, “Third” Integral, Zooming Into Space Of Solution, Poincar&Eacute, Surface Of Sections, &Ldquo, Third&Rdquo, Integral

Scientific paper

The general solution of the Henon-Heiles system is approximated inside a domain of the ( x, C) of initial conditions ( C is the energy constant). The method applied is that described by Poincaré as ‘the only “crack” permitting penetration into the non-integrable problems’ and involves calculation of a dense set of families of periodic solutions that covers the solution space of the problem. In the case of the Henon-Heiles potential we calculated the families of periodic solutions that re-enter after 1-108 oscillations. The density of the set of such families is defined by a pre-assigned parameter ɛ (Poincaré parameter), which ascertains that at least one periodic solution is computed and available within a distance ɛ from any point of the domain ( x, C) for which the approximate general solution computed. The approximate general solution presented here corresponds to ɛ = 0.07. The same solution is further improved by “zooming” into four square sub-domain of ( x, C), i.e. by computing sufficient number of families that reduce the density parameter to ɛ = 0.003. Further zooming to reduce the density parameter, say to ɛ = 10-6, or even smaller, although easily performable in both areas occupied by stable as well as unstable solutions, was found unnecessary. The stability of all members of each and all families computed was calculated and presented in this paper for both the large solution domain and for the sub-domains. The correspondence between areas of the approximate general solution occupied by stable periodic solutions and Poincaré sections with well-aligned section points and also correspondence between areas occupied by unstable solutions and Poincaré sections with randomly scattered section points is shown by calculating such sections. All calculations were performed using the Runge-Kutta (R-K) 8th order direct integration method and the large output received, consisting of many thousands of families is saved as “Atlas of the General Solution of the Henon-Heiles Problem,” including their stability and is available at request. It is concluded that approximation of the general solution of this system is straightforward and that the chaotic character of its Poincaré sections imposes no limitations or difficulties.

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