
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 onedimensional real interval [0..1] onto itself, the Hénon map is defined on the twodimensional 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:
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). The Hénon map is dissipative, i. e. a small volume in the state space (= a piece of the x,yplane) 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 twodimensional dissipative maps are for example the Ikeda map, the KaplanYorke map, the Tinkerbell map or the Zaslavskij map.
2. Hénon´s quadratic twist map There is another twodimensional 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 x_{n}^{2} << y_{n} this map just means rotation by the angle . To this rotation a quadratic (therefore nonlinear) perturbation is added whose effect becomes stronger for larger x_{n}. 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.
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 Hénonattractor is discussed in [Grassberger89]. For more details about the twist map see [Henon69] or [Lichtenberg83], e.g.




