Smooth dynamical systems as infinitesimal discrete dynamical systems

Here I am working with nonstandard analysis in the sense of Robinson, taking an ultrapower of the real numbers and building things out of that. But in general I am going to be a bit sloppy and not worry too much about the details.

Recall that a standard function is differentiable if, for every standard \(x\) and for every infinitesimal \(\epsilon\), \(f(x+\epsilon)-f(x)/\epsilon \approx a\), where \(a\) is also standard.

Fix an infinitesimal \(h\). Let \(M\) be a standard differentiable manifold, fix \(p \in M\), and consider the tangent space \(T_pM\). Recall that the elements of this vector space (may be taken to be) smooth paths \(\gamma: I \to M\), with \(0 \in I \subset \mathbb{R}\) some open interval, up to the equivalence relation of having the same first derivative at \(p\) (which may be checked in any choice of local coordinates, all coordinates giving the same answer). In particular we may compute this derivative, letting \(\phi: U \to \mathbb{R}\), \(p \in U \subset M\) being the local coordinates of choice, as the standard part of \((\phi(\gamma(h))-\phi(p))/h\). Having fixed \(h\), we may thus replace each curve with just the choice of \(\gamma(h)\), the only condition on this being that in any choice of local coordinates, the above quotient has a standard part (eg it is not unbounded). But since the change-of-coordinate maps are in particular differentiable, it suffices to verify this for one such choice. We may call this property “being at \(O(h)\)-order infinitesimal distance from from \(p\)”. For example, suppose that \(M = \mathbb{R}\), \(p=0\) and \(\phi = 1_\mathbb{R}\). Then if \(h = (1/n)\), it will not do to take \(\gamma(h) = (1/\sqrt{n})\), for in that case we will get the unbounded nonstandard real \((\sqrt{n})\) for the difference quotient.

Let \(\widetilde{D_pM}\) be the set of such points in \(M\). We don’t quite have a map \(T_pM \to \widetilde{D_pM}\), because e.g the path \(\gamma(t)=t^2\), which has local derivative zero, has \(\gamma(h) = h^2 \neq 0\). In order to make this map well-defined, we need to consider a quotient by the relation of “being at distance \(o(h)\)”. More precisely, let \(v \approx v'\) if, if any coordiante chart, \(d(v,v')/h\) is infinitesimal. (The local differentiability ensures this does not depend on the choice of coordinates.) Then let \(D_pM = \widetilde{D_pM}/\approx\) - now the map \(T_pM \to D_pM\) is well-defined.

We can also define a vector space structure on \(D_pM\) that makes this map linear. We simply lift the addition and scalar multiplication from any coordinate chart. Local differentiability imply that \(\psi(h(v+v')) = \psi'(0) h(v+v') + h\epsilon\), wher \(\psi\) is a change-of-coordinate map, so that addition is well-defined up to an infinitesimal times \(h\), which is quotiented out by in any case. Scalar multiplication works similarly. Note that we are defining a vector space over \(\mathbb{R}\), not over the full \(*\mathbb{R}\). Multiplication by a number of order \(o(h)\) is not invertible (in fact such products are always zero), and similarly scaling by an unbounded number may take you out of the coordinate patch, and is thus not well-defined.

In fact with this, we have an isomorphism \(T_pM \to D_pM\) - any \(v \in D_pM\) is the image of \(\gamma(t) = p + tv/h\), and two curves have the same derivative exactly if \(\gamma(h) \approx \gamma'(h)\).

Recall that smooth dynamical system on \(M\) is a smoth section \(M \to TM\). We can exploit our isomorphism above, by observing that each \(D_p\) is actually a quotient of a subset of \(M\). Hence we may ask for a function \(s: *M \to *M\) so that \(s(p) \in \widetilde{D_p} \subset *M\) - this induces a smooth dynamical system. Every (standard) smooth dynamical system has this form, and \(s,s'\) induce the same system if \(s(p) \approx s'(p)\) for each \(p\).

What’s cool about this is that these are essentially discrete dynamical systems, albeit nonstandard ones. A discrete dynamical system is a set with an “advance one timestep” function \(s: X \to X\). So a smooth dynamical system is a manifold with an “advance time \(h\)” function \(s: *M \to *M\), subject to the condition that \(s(p)\) is \(h\)-close to \(p\), and up to a certain equivalence relation. This suggests a way to use the same conceptual tools to study smooth and discrete dynamical systems.

The paper Differential geomtry via infinitesimal displacements, by Nowik and Katz, provides a more in-depth analysis of this idea.