Michel Hénon (1931-2013) has put numerical experiments at the center of his scien- tific practice to which he accorded the same status as physics experiments. He was interested in astrophysics – an area where experiments with controlled conditions are obviously impossible and numerical computations the only way to experiment.
In the 1950s, Hénon worked on the construction of analogic computers be- fore making his own one. This was just before the advent and democratization of digital computers. At Meudon’s ob- servatory, near Paris, he worked on an IBM 750 and, later, on an IBM 7040 at Nice Observatory. He also used the very first programmable pocket calculator, the HP-65.
In the 1960s, Hénon was interested in various problems arising in astrophysics and, during his 1962 stay at Princeton, he studied the movement of a star gravitating in a galaxy with cylindrical symmetry. Several numerical experiments with this model revealed some irregular behaviors. He asked Carl Heiles, a graduate student at that time, to redo the program and the experiments by himself on another computer, just as for a physics experiments which has to be reproducible [1].These experiments resulted in a paper of Hénon and Heiles in 1964 which revealed a striking mixture of quasi-periodic and “chaotic” behaviors in a model that seemed very simple [2], see next figure.
Hénon’s approach to astrophysics was to focus on simple mathematical models to shed light on basic phenomena and not to study realistic systems which are both intractable and not enlightening. He systematically applied some basic ideas of Henri Poincaré and George Birkhoff which lead to replace differential equations by iterations of maps using appropriately chosen “Poincaré sections” to the orbits in phase space. Although seemingly simple, the models so obtained are still very hard to analyze mathematically, but numerical experiments are both easy to make and instructive. A shining illustration of his approach is the so-called “restricted three body problem” where he explored systematically the behavior of possible orbits [16]. Let us mention in passing the discovery by numerical experiments of a remarkable “figure-eight” shaped solution by the physicist Cris Moore [3].
The work for which Hénon is undoubtedly famous outside astronomy is the attractor which bears his name. Tuning the parameters of Lorenz’s equations we alluded to above, and using a Poincaré section, Yves Pomeau and Jose-Luis Ibanez observed Smale’s horseshoe mechanism. Pomeau gave a talk on this observation at Nice Observatory to which Hénon assisted. This prompted him to propose a very simple model based on a quadratic map in the plane.
In this model, the tuning of a parameter displays Smale’s horseshoe mechanism ; this is the so-called Hénon map :
$$ \begin{cases} x_{n+1}=y_n+1-ax_n^2\\ y_{n+1}=bx_n \end{cases} $$
where $a,b$ are positive parameters, and given an initial condition $(x_0,y_0)$. For $a=1.4, b=0.3$, there is a strange attractor displayed in the adjacent figure.
Hénon made his computations with a HP-65 and turned to a IBM 7040 computer for more extensive computations. The fact that it is not a numerical artifact was an open problem till 1991 when the mathematicians Michael Benedicks and Lennart Carleson published a (lengthy and complicated) proof. [4]
Digital interactive experiment :
[1] They also redid their calculations with different numerical integration methods.
[2] M. HÉNON & C. HEILES. The applicability of the third integral of motion : Some numerical experiments. The Astrophysical Journal 69 (1964), 73-79.
[3] C. MOORE. Braids in classical dynamics. Physical Review Letters 70 (24) 3675–3679, 1993.
[4] Notice that the values of the parameters used by Hénon are not covered by their result.