Henon

In many ways, May s sentiment echoes the basic philosophy behind the study of CA, the elementary versions of which, as we have seen, are among the simplest conceivable dynamical systems. There are indeed many parallels and similarities between the behaviors of discrete-time dissipative dynamical systems and generic irreversible CA, not the least of which is the ability of both to give rise to enormously complicated behavior in an attractive fashion. In the subsections below, we introduce a variety of concepts and terminology in the context of two prototypical discrete-time mapping systems the one-dimensional Logistic map, and the two-dimensional Henon map. 

A simple two-dimensional analogue of the logistic equation, introduced by Henon in 1976 [henou76], is defined by... 

Fig. 4.8 Four snapshots of the Henon attractor for a = 1.4 and (3 = 0.3. Plots (b), (c) and (d) provide enlargements of the small window regions shown in plots (a), (b) and (c), respectively see text.

The table below gives a few selected values of for the logistic and Henon map attractors ... 

S Riviere, S Henon, J Meunier, DK Schwartz, MW Tsao, CM Knobler. J Chem Phys 101 10045-10051, 1994. 

Figure 18. Trajectories in the Henon-Heiles potential for the initial condition xo ya = —0.37. The marks ( ) indicate the location of caustics. Taken from Ref. [29].

The Driven Henon-Heiles System V. Outlook Open Problems... 

The diagnostic power of the time-dependent invariant manifolds Wu and Ws as separatrices between reactive and nonreactive trajectories was illustrated in Ref. 40 for the example of a driven Henon-Heiles system described by the Hamiltonian... 

Figure 6. The deterministic potential (80) of the driven Henon-Heiles system. A central well is separated from three asymptotic channels by three saddle points SA, SB, and Sr. Also indicated is the surface, Eq. (82), on which initial conditions are sampled. (From Ref. 40.)...

M. Henon and C. Heiles, The applicability of the third integral of motion some numerical experiments, Astron. J. 69, 73 (1964). 

J. Kaidel, P. Winkler, and M. Brack, Periodic orbit theory for the Henon-Heiles system in the continuum region, Phys. Rev. E 70, 066208 (2004). 

J. Aguirre, J. C. Vallejo, and M. A. F. Sanjuan, Wada basins and chaotic invariant sets in the Henon-Heiles system, Phys. Rev. E 64, 066208 (2001). 

Henon-Heiles system, transition state trajectory, deterministically moving manifolds, 228-231... 

Lynden-Bell, D. 1973, in G. Contopulos, M. Henon D. Lynden-Bell, Dynamical Structure and Evolution of Stellar Systems, Geneva Geneva Observatory, p. 132. Lynden-Bell, D. 1975, Vistas in Astr., 19, 299. 

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