Generalisation of Index of a curve to higher dimensions

Question

Im studying Non Linear Dynamics and Chaos from Strogatz's textbook. In the sixth chapter, while talking about non linear flows in 2 dimensions he introduces the index of a curve in a vector field and shows some beautiful properties that the index has.

I've understood how to use the index to classify and or deduce properties about fixed points in two dimensions but what about three dimensions? In 3D, is there any generalisation of the index that is useful in analysing fixed points? I'm assuming it would have to be a property of surfaces rather than curves but I couldn't conclude anything further on my own.

I'm asking this because the (Lorenz-like) system of equations I'm dealing with for my project is a 3 dimensional system and the fixed points are tedious functions of the parameters and it would be really difficult to use linearisation or such other methods for them.

Thanks in advance.

Note: There is an answer to a similar question on math overflow, but I understood practically nothing of the answers there and it didn't seem too useful anyway.

Answer 1

Let me briefly explain the Conley index theory. The full theory can be found in the references of this wikipedia page. As I said in my comments, I learned it from Conley's original treatment which I quite liked.

Let $M$ be the manifold on which the vector field is defined, and let $\phi : M \times \mathbb{R} \to M$ be the flow generated by that vector field. I'll use notations like $x \cdot t = \phi(x,t)$.

The first idea is to focus not on fixed points per se, but instead on invariant subsets $C$, meaning that for all $x \in C$ and $t \in \mathbb{R}$ we have $x \cdot t \in C$.

The second idea is to focus only on compact invariant subsets $C$ which are isolated meaning that there exists an neighborhood $N$ of $C$ called an isolating neighborhood of $C$: by definition, this means that $C$ is the largest invariant subset of $N$. (In more detail, for each $x \in N-C$ there exists $t \in \mathbb{R}$ such that $x \cdot t \not \in N$. To say that "$N$ is a neighborhood of $C$" I mean that there exists an open subset $U \subset M$ such that $C \subset U \subset N$; the set $N$ itself is not required to be open.)

Conley proved that for each compact invariant subset $C$ there exists a very special kind of isolating neighborhood $B$ of $C$ called an "isolating block for $C$". To say what this means, the first requirement is that $B$ is a compact submanifold-with-boundary in $M$ of dimension $n$. Furthermore, each point $x \in \partial B$ is required to fall into one of three types:

1. $x$ is an exit point meaning that the vector field at $x$ is transverse to $\partial B$ and points outwards, which implies that there exists $\epsilon > 0$ such that $x \cdot (-\epsilon,\epsilon) \cap B = x \cdot (-\epsilon,0]$. It follows that the set of exit points forms an open subset of $\partial B$.
2. $x$ is an entry point meaning that the vector field at $x$ is transverse to $\partial B$ and points inwards, which implies that there exists $\epsilon > 0$ such that $x \cdot (-\epsilon,\epsilon) \cap B = x \cdot [0,\epsilon)$. It follows that the set of entry points forms an open subset of $\partial B$.
3. $x$ is an external tangency, which means two things. To say that $x$ is a "tangency" means that the vector field at $x$ is tangent to $\partial B$, but only up to first order (the vector field should be $C^2$ for this to make sense). And to say that $x$ is an external tangency means that there exists $\epsilon > 0$ such that $x \cdot (-\epsilon,\epsilon) = \{x\}$. It follows that the set of external tangencies is compact submanifold of dimension $n-2$ in $\partial B$.

To summarize, the boundary $\partial B$ of any isolating block $B$ contains a dimension $n-2$ submanifold consisting of the external tangencies which I'll denote $\tau B$. The submanifold $\tau B$ separates the boundary into two open pieces, the entry points and the exit points, and I'll denote their closures as $\partial_{in} B$ and $\partial_{out} B$, respectively. Notice that $$\partial(\partial_{in} B) = \partial\bigr(\partial_{out} B) = \tau B $$ and so if $\tau B$ is nonempty then both of $\partial_{in} B$ and $\partial_{out} B$ are nonempty.

Okay, so far all that's happened is that Conley has proved the existence of an isolating block $B$ for each isolated compact invariant subset. Now comes the interesting stuff.

Conley defines the index of the isolating block $B$ as follows. Pick an abstract base point, disjoint from $B$, which I'll denote $*$. Form the quotient space of $B \cup \{*\}$ by identifying $\partial_{out} B \cup \{*\}$ to a single point, and take that to be the base point of the quotient. The quotient is therefore an object in the category of base-pointed topological spaces. The index of $B$ is defined to be the homotopy type of the quotient in the category of base-pointed topological spaces.

In Conley's theory, the following things are proved:

• Index is well defined: Any two isolating blocks for $C$ have the same index, up to homotopy equivalence of pointed topological spaces. Thus the index of $C$ is well defined by taking it to be equal to the index of any of its isolating blocks.
• Index is stable under perturbation: For any isolating block $B$, if the vector field is perturbed a small amount (in the $C^2$ topology), then $B$ is still an isolating block, and although its set of exit points, entry points, and external tangencies may have themselves been perturbed, nonetheless the index of $B$ is unchanged by the perturbation.

It's interesting to work out some examples of index, in order to see different "kinds" of compact isolated invariant sets.

• For an attracting fixed point in dimension $n$, the index is equal to the homotopy type of a two point space, also known as the "$0$-sphere". This is true because there is an isolating block $B$ consisting of an $n$-ball with $\partial_{in} B = \partial B$.
• For a repelling fixed point in dimension $n$, the index is equal to the homotopy type of the $n$-sphere, because there is an isolating block $B$ consisting of an $n$-ball with $\partial_{out} B = \partial B$.
• Hyperbolic fixed points. In $\mathbb{R}^n$ the vector field $$\vec v(x_1,....,x_i,x_{i+1},...,x_n) = (x_1,...,x_i,-x_{i+1},...,-x_n) $$ defines a flow with an isolated fixed point at the origin, and this fixed point has an isolating block consistint of the $n$-ball with $\partial_{out} B$ being equal to a regular neighborhood of an embedded round sphere of dimension $i-1$. So the index is equal to the homotopy type of the $n$-ball with an $i-1$ dimensional subsphere of the boundary collapsed to a point; it's not hard to see that this is equal to the homotopy of the sphere of dimension $i$. This is a well known example where the integer $i$ was referred to as the "index" of the fixed point long before Conley, and here we see this integer recurring in Conley's theory as the dimension of his index.

• In the special case of a hyperbolic fixed point with $n=2$ and $i=1$, the isolating block is $B^2$ with $\partial_{out}(B)$ being a pair of disjoint intervals in the boundary circle. Collapsing these two intervals to the base point $*$, we obtain a pointed topological space which is homotopy equivalent to the circle. So, the index of a standard hyperbolic fixed point of a flow in dimension 2 is the homotopy type of the circle.

• The index of an empty compact invariant set is the homotopy type of a one point space, i.e. the contractible homotopy type. This is true because you can imagine the empty set to have an isolating block obtained by fattening up any Poincare section of the flow, and when you form the quotient by crushing the exit set to a point then the flow gives you a deformation retraction to that point. Let's call this the "trivial" index.

As a consequence of that last example, combined with Conley's theorems, you get the following great application:

Theorem: If $C$ is an isolated compact invariant set with nontrivial index, and if you perturb the vector field, then there will be a nearby nonempty isolated compact invariant set of the same index.