March 7, 2017

A primer on determinism by John Earman

By John Earman

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Roughly speaking, (t− , t+ ) are the points furthest apart on the real axis that can be connected by a path of constant Re Ψ , satisfying certain regularity assumptions. 13. Assume that a(Yt ) = t − t and ω(Yt ) = ω0 > 0. 73) is constant on hyperbolas centred in (t , −iω0 ). We have Π(t) = 2t − t, and the buffer times are given by t± = t ± ω0 . 67) is usually not analytic, even when the original system is, because of lack of regularity of centre manifolds. This is why the analysis in [Ne˘ı87, Ne˘ı88] does not use centre manifolds, but keeps track of all original variables.

It is in fact possible to give a rather complete classification of the scaling laws that may occur for one-dimensional bifurcations. Such a classification involves two steps: Firstly, analyse the power-law behaviour of slow manifolds near a bifurcation point; secondly, study the equation governing the distance between solutions of the slow–fast system and the slow manifold. The first step is best carried out with the help of the bifurcation point’s Newton polygon. 46) where f is of class C r for some r 2, with all derivatives uniformly bounded in a neighbourhood N of the origin.

A saddle–node bifurcation occurs if ∂xx f (0, 0) = 0 and ∂y f (0, 0) = 0 . 10) hold. This implies in particular that the slow manifold exists for y < 0. 11) guarantees that trajectories starting near the stable slow manifold are driven towards the bifurcation point. 12) where the slow manifold consists of a stable branch M− = {(x, y) : x = √ √ −y, y < 0} and an unstable branch M+ = {(x, y) : x = − −y, y < 0}. √ The linearisation of f at these branches is given by ∓2 −y. 11), a qualitatively similar behaviour holds in a neighbourhood of the bifurcation point (0, 0).

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