# 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.