Poincare Les methodes nouvelles, Section 36. Poincare's use of probabilistic concepts to argue that certain classes of solution were infinitely unlikely (have zero measure as we would now say) was itself a radical innovation (Aubin and Dalmedico (2002)). Poincare described this idea, used by Tisserand in 1886, and other implications of periodic solutions for the satellite systems of the outer planets in Sections 49-50 of Les methodes nouvelles.

Birkhoff's proof is outlined in Barrow-Green (1997), Section 7.4.2.

for periodic solutions to the restricted three-body problem and discussed their stability.42

In the third volume of New Methods, Poincare described what has become a basic tool in the qualitative study of differential equations: the Poincare return map. Suppose we have a periodic orbit within phase space which, for simplicity, we will assume to be three-dimensional (as it is in the planar-restricted three-body problem). Take a surface S (called a 'Poincare section'), such that the periodic orbit crosses the surface at M, for example. Viewed on S, the periodic orbit takes the form of a fixed point. Now, consider a curve in phase space that passes through a point M0 on S. Provided M0 is sufficiently close to M, this trajectory will cross S again, e.g. at M1. Poincare calls M1 the 'consequent of M0', and we refer to the function mapping M0to M1 as 'Poincare's return map'. The point M1 will have its own consequent, M2, and so on, and the collection M0, Mi, M2,..., is called the 'orbit of the point M0'. With this device, the study of the region of phase space near periodic orbits becomes the study of the region of the Poincare section near a fixed point.

The new problem is set in a space of one dimension less than the original one, but the nature of the possible behaviour near fixed points is now much more complicated. For one particular class of fixed points (which Poincare referred to as unstable), there is a family of points that get closer and closer to the fixed point under the action of the return map (referred to by Poincare as the 'second family of asymptotic solutions' and now called the 'stable manifold', S, for example) and there is a different family of points (Poincare's first family of asymptotic solutions; the unstable manifold, U, for example) that approach the fixed point under the action of the inverse of the return map, i.e. as time runs backwards. The curves S and U are invariant in the sense that, for any point on S, its consequent is also on S, and similarly for U.

If S and U intersect at some point T, then the orbit of T must lie on both S and U and so must approach the unstable fixed point both as time increases and as it decreases. Poincare called these 'doubly asymptotic solutions' and T a 'homoclinic point'. If T is on both S and U, then so is its consequent, and so on. Thus, if S and U intersect once, they must intersect infinitely many times. The simplest possibility (and the only one for completely integrable systems) is

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