Holmes, P. Poincaré, celestial mechanics, dynamical-systems theory and “chaos”. Physics Reports 193, 137-163, doi:http://dx.doi.org/10.1016/0370-1573(90)90012-Q (1990)
As demonstrated by the success of James Gleick’s recent book , there is considerable interest in the scientific community and among the general public in “chaos” and the “new science” which is supposed to accompany it. However, as usual, it is not easy to separate hyperbole from fact. In an attempt to do this, I will offer a precise definition of chaos in the context of differential equations: mathematical models which, since Newton, have played a vital role in scientific discovery. I will show how the classical problems of celestial mechanics led Poincaré to ask fundamental questions on the qualitative behavior of differential equations, and to realize that chaotic orbits would provide obstructions to the conventional methods of solving them. In a major paper which appeared almost exactly one hundred years ago, Poincaré studied mechanical systems with two degrees of freedom and identified an important class of solutions, now called transverse homoclinic orbits, the existence of which implies the system has no analytic integrals of motion other than the total (Hamiltonian) energy. I will explain these terms and outline the history of subsequent developments of these ideas by Birkhoff, Cartwright, Littlewood, Levinson and Smale, and describe how the ideas of Melnikov have made possible an “analytical algorithm” for the detection of chaos and proof of nonintegrability in wide classes of perturbed Hamiltonian systems. I will discuss the physical implications of the mathematical statements that these methods afford. In the process, I will point out that, while there is a precise vocabulary and grammar of chaos, developed largely by mathematicians and steaming from Poincaré’s work, it is not always easy to use it in speaking of the real world.