Hénon system
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1. The Henon map

Like the logistic map the Hénon system is a system with a discrete time scale n=1, 2, ... (i. e. it is a map). Whereas the logistic map maps a one-dimensional real interval [0..1] onto itself, the Hénon map is defined on the two-dimensional real plane. And whereas there is only one control parameter r in the logistic map, there are two control parameters a and b in the Hénon map:

xn+1 = yn + 1 - axn2

yn+1 = bxn

xi and yi are the coordinates of a point in the real plane. For suitable values of the control parameters a and b and the initial condition (x0 , y0) the iterations will outline the boomerang-shaped fractal structur  of the Hénon-attractor:
How to use this applet: Zoom into the attractor by clicking a point with the mouse. The Draw-button adds further iterations to the structure so the attractors shape becomes better visible at large zoom levels. The number of added iterations per touch of the button can be chosen at the listbox (up to 1,000,000 if zoomfactor > 100). While zooming into the attractor the puff pastry-like structure of this fractal becomes clearly visible. On the left side you can adjust the control parameters a and b. Try carefully, since the attractor exists only in a small vicinity around a=1.4 and b=0.3 .

Two fundamental characteristics of chaotic systems can be illustrated very well at the Hénon system. The first one is called sensitive dependence on the initial conditions. This causes systems having the same values of control parameters but slightly differing initial conditions to diverge exponentially (on the average) during their evolution in time. The second characteristic is called ergodicity . Ergodicity means that a large set of identical systems which only differ in their initial conditions will be distributed after a sufficient long time on the attractor exactly the same way as the series of iterations of  one single system (for almost every initial conditions of this system).

How to use the applet: In the beginning  a set of initial conditions is drawn which is located on the circumference of a small circle. This circle can be relocated using the mouse. By pressing the Iterate-button the Hénon map is applied to each point in this set. From iteration to iteration the circle is distorted and stretched more and more (sensitive dependence on initial conditions!) and soon the whole shape of the attractor becomes visible. A totally different cloud-like set of initial conditions is generated by pressing the Randomize-button. The iterates of the most of these points will converge to the Hénon attractor. A part of them, however, diverges and leaves the frame of the applet. The reason of this divergence is the fact that the basin of attraction is only a small (roughly U-shaped) part of the x, y-plane. After only a few iterations the successors of both sets of initial conditions outline the shape of the attractor, the same shape which is drawn also by the series of iterations of one single point (ergodicity!).

The Hénon map is dissipative, i. e. a small volume in the state space (= a piece of the x,y-plane) is contracted by this map. Otherwise no attractor could arise. In order to proof this you can calculate the determinant of the Jacobian. This matrix reads:

So the determinant is -b. That means that a volume will be compressed by the factor |-b|, if  b is smaller than 1.

Other two-dimensional dissipative maps are for example the Ikeda map, the Kaplan-Yorke map, the Tinkerbell map or the Zaslavskij map.

 

2. Hénon´s quadratic twist map

There is another two-dimensional map that has been investigated by M. Hénon and whose properties are totally different from the Hénon map, it is the quadratic twist map [Henon69]:

In the limit xn2 << yn this map just means rotation by the angle . To this rotation a quadratic (therefore non-linear) perturbation is added whose effect becomes stronger for larger xn. Unlike the Hénon map this twist map is conservative since its Jacobian

 

has the determinant 1. Therefore no attractors exist for this map. Instead it exhibits the behaviour of Hamiltonian chaotic systems.

 

How to use the applet: At the beginning ten points with different colors are distributed randomly on the displayed part of the x,y -plane.  Each of these points is iterated 10 000 times and the iterates are drawn in the same color. You can change the value of the angle “psi”  (measured in radian). By pressing “New” you can generate 10 more randomly distributed points and again 10 000 iterations for each of them. The left mouse button zooms into the structure for a investigation of substructures. Use the “new” button after zooming-in for more details.

It can be easily checked using the applet that for suitable values of there are chains of “islands” which are seperated from each other by “disturbed” areas. In the center of each island there is an elliptic fixed point. Around this point there is a set of quasiperiodic orbits winding. When starting on a quasiperiodic orbit the system wanders around the fixed point but does never return exactly to the starting point. Between the islands there are hyperbolic fixed points, that means fixed points with a stable manifold and an unstable one which intersect at a positive angle.Close to these fixed points you will find chaotic areas with totally irregular iterations.

Literature

Of course the Hénon system as well as the logistic map is a main topic in many textbooks, e.g. [Guckenheimer83 ] or [Schuster88]. It has been introduced and studied for the first time in [Henon76 ].  In [Feit78] the dependence of the largest lyapunov exponent on the control parameters has been studied. Herein you also will find diagrams of the basin of attraction. The stable and unstable manifolds of the Hénon system (and related systems) are treated in [ Franceschini81] and [Tel82]. How to find unstable periodic orbits in the Hénon-attractor is discussed in [Grassberger89]. For more details about the twist map see [Henon69] or [Lichtenberg83], e.g.

 

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