Closed Orbits — Structural Stability

Abstract

After singular points (that is, equilibrium positions) the most significant elements of the phase portrait of a vector field are the periodic or closed orbits (known also under the poetic name of nonlinear oscillations). They are studied by a method that goes back to Poincaré. This consists of choosing a point a of the closed orbit Ω, taking a small piece of hypersurface W through o and transverse to Ω, and for each x € W considering the first point p(x) at which the orbit of x cuts W again (we say that p is the Poincaré map, or the first-return map). The fact that W was chosen transverse to Ώ implies that p is well defined in a neighbourhood of the point a in W (clearly p(a) = a) and that p is a local diffeomorphism. The destiny of x is reflected in the succes- sive intersections with W : … ,p−¹{p−¹{x}), p−^x(x), x, p(x), p(p[x)}),… and knowledge of p enables us to reconstruct the dynamics in a neighbourhood of i?, at least if we decide to disregard the time between two successive intersections, which amounts to working to within orbital equivalence (6.11.5, 6.11.6)