Henon map

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

Generally, the sequence of points, xo,yo), (xi,yi),. .., xi,yi),. .., either diverges to infinity (for Xq large) or settles onto an attractor (for (xo,yo) near the origin). An analysis similar to the one performed earlier for the logistic equation may be carried out here to determine the behavior of the map as a function of a 

In this section we discuss another two-dimensional map with a strange attractor. It was devised by the theoretical astronomer Michel Henon (1976) to illuminate the microstructure of strange attractors. 

According to Gleick (1987, p. 149), Henon became interested in the problem after hearing a lecture by the physicist Yves Pomeau, in which Pomeau described the numerical difficulties he had encountered in trying to resolve the tightly packed sheets of the Lorenz attractor. The difficulties stem from the rapid volume contraction in the Lorenz system after one circuit around the attractor, a volume in phase space is typically squashed by a factor of about 14,000 (Lorenz 1963). 

Henon had a clever idea. Instead of tackling the Lorenz system directly, he sought a mapping that captured its essential features but which also had an adjustable amount of dissipation. Henon chose to study mappings rather than differential equations because maps are faster to simulate and their solutions can be followed more accurately and for a longer time. 

As desired, the Henon map captures several essential properties of the Lorenz system. (These properties will be verified in the examples below and in the exercises.) 

The Henon map is invertible. This property is the counterpart of the fact that in the Lorenz system, there is a unique trajectory through each point in phase space. In particular, each point has a unique past. In this respect the Henon map is superior to the logistic map, its one-dimensional analog. The logistic map stretches and folds the unit interval, but it is not invertible since all points (except the maximum) come from two pre-images. 

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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. 

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

The Henon map is dissipative. It contracts areas, and does so at the same rate everywhere in phase space. This property is the analog of constant negative divergence in the Lorenz system. 

The next property highlights an important difference between the Henon map and the Lorenz system. 

Some trajectories ofthe Henon map escape to infinity. In contrast, all trajectories of the Lorenz system are bounded they all eventually enter and stay inside a certain large ellipsoid (Exercise 9.2.2). But it is not surprising that the Henon map has some unbounded trajectories far from the origin, the quadratic term in (1) dominates and repels orbits to infinity. Similar behavior occurs in the logistic map—recall that orbits starting outside the unit interval eventually become unbounded. 

The next three exercises deal with the fixed points of the Henon map. 

Calculate the Jacobian matrix of the Henon map and find its eigenvalues. 

A fixed point of a map is linearly stable if and only if all eigenvalues of the Jacobian satisfy A <1. Determine the stability of the fixed points of the Henon map, as a function of a and b. Show that one fixed point is always unstable, while the other is stable for a slightly larger than Show that this fixed point loses stability in a flip bifurcation (A = -1) at a, = (1 - b. ... 

Numerical experiments) Explore numerically what happen.s in the Henon map for other values of a, still keeping Z = 0.3. 

Some orbits of the Henon map escape to infinity. Find one that you can prove diverges. 

X11 Show that for a certain choice of parameters, the Henon map reduces to an effectively one-dimensional map. 

Computer project) Explore the area-preserving Henon map (6 = 1). 

This map is a canonical example illustrating the chaotic behavior. For certain parameter values the Henon map models the mechanism of the creation of the Smale horseshoe as illustrated in Fig. C.6.3, for the map and for its inverse ... 

The Jacobian of the Henon map is constant and equal to h. Therefore, when 6 > 0, the Henon map preserves orientation in the plane, whereas orientation is reversed when 6 < 0. Note also that if 6 < 1, the map contracts areas, so the product of the multipliers of any of its fixed or periodic points is less than 1 in absolute value. Hence, in this case the map cannot have completely unstable periodic orbit (only stable and saddle ones). On the contrary, when b > 1, no stable orbits can exist. When 6 = 1, the map becomes conservative. At b = 0, the Henon map degenerates into the above logistic map, and therefore one should expect some similar bifurcations of the fixed points when b is suflSciently small. 

When 6=1, the Henon map becomes conservative, as its Jacobian equals -1-1. At 6 = 1 and a = —1, it has an unstable parabolic fixed point with two multipliers +1 at 6 = 1 and a = 3, it is a stable parabolic fixed point with two multipliers —1. In between these points, for —1 < a < 3 (i.e. (a, 6) G T), the map has a fixed point with multipliers where cos > = 1 y/a -h 1. This is a generic elliptic point for tp 7r/2,27r/3,arccos(—1/4) [167]. Since the Henon map is conservative when 6=1, the Lyapunov values are all zero. When we cross the curve AH, the Jacobian becomes different from 1, hence the map either attracts or expands areas which, obviously, prohibits the existence of invariant closed curves. Thus, no invariant curve is born upon crossing the curve AH. ... 

Note that the curves SN and PD are close to the curves SN and PD of the original Henon map. 

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