TITLE: Algebraic interpretation of Lyapunov functions
QUESTION [14 upvotes]: I have recently learned the method of Lyapunov functions to rule out periodic solutions in two-dimensional nonlinear systems. My understanding is that there is some Lyapunov function for any nonlinear 2D system with no closed orbits, and basically that these functions always asymptotically converge to some fixed point.
The usual assumption of their form is something like $ax^2+by^2$, which is I guess convenient because both terms are always positive for positive coefficients, but I am wondering how this relates (if at all) to the idea of norm.
Is there an algebraic interpretation of such functions in 2D systems that might provide more insight into what is going on, or alternatively help someone have some better intuition about how to construct these guys? (Aside from being told to find one in a problem set, what about a 2D system might make you think there could be a Lyapunov function?)

REPLY [7 votes]: From what little I know, the notion of a Lyapunov function is nowadays widely used for almost all dynamical systems where the notion of certain stability makes sense. For example, the are even used in computer science and software verification - see e.g. this article, and also pay attention to the comment about ranking functions left there.
Notions of stability crucially depend on the dynamical system that is considered: if you have  a vector field on some manifold $\mathscr M$, e.g. $\mathscr M = \Bbb R^n$, you often talk about the stability of a point (equilibrium point), limit cycle or in general of some invariant submanifold - the set of points which is invariant under the vector flow. 
Let's think about a simplest example - that is the stability of an equilibrium point: the lowest dimension where it makes sense to deal with it is $1$, that is $\Bbb R$ (e.g. it does not make sense to talk about limit cycles in $\Bbb R$). The idea of a Lyapunov function
$$
  f:\mathscr M\to \Bbb R
$$
is to push the original dynamics to a simpler set, and study stability there. The condition on the decreasing behavior of $f$ further assures that the stability over $\Bbb R$ can be brought back to $\mathscr M$.  The nice feature of such approach is that it not only applies to a particular situation above. In hybrid dynamical systems classical notions of stability are enriched with Zeno stability; as Zeno behavior is not performed by smooth dynamical systems, the construction of Lyapunov functions goes as
$$
  f':\mathscr M'\to H
$$
where $\mathscr M'$ is a hybrid manifold, and $H$ is a 2d simple hybrid system. For example, real-valued Lyapunov functions are not anymore suitable to study this particular notion of stability. 
Finally, the notion of an incremental stability talks about stability of trajectories w.r.t. themselves rather then w.r.t. some fixed set. A similar ideas can also be applied to that case. As a result, one may think of Lyapunov functions as certain morphism-like maps between dynamical systems that preserve stability structures. 
With regards to the construction of Lyapunov functions, it yet again depends on the stability problem you are interested in. It often holds that their existence is equivalent to the stability. One direction is always easy to prove, as it often leads to the necessary conditions in the definition of a Lyapunov function. The reverse direction (*stability*$\,\Rightarrow\,$*existence*) is much harder, as it requires the construction of a Lyapunov function - and often invokes existence of diffeomorphisms. For example, one of the proofs I've seen discussed that it required the Poincare conjecture to hold true for the low-dimensional cases.
In general, there are many ad-hoc rules in construction of Lyapunov functions within certain classes. As an example, for non-linear system the Sum Of Squares (SOS) polynomial functions can be algorithmically constructed. The latter can be regarded as an algebraic method as it follows the symbolical computations.
What shall hint you upon the existence of such function - is a local stability/attractivity of a set of interest. I am not sure whether there are any receipts for this, since such kind of analysis always depends on a particular system you have in hands. If you are interested, I can look for some comprehensive references that discuss both ideas and the construction of Lyapunov functions in some particular classes of problems.