Hopfield networks are gradient dynamical systems, used (among other things) to solve combinatorial optimization problems, because stable equilibria are at vertices of the hypercube $[-1,1]^n$. They have been compared to interior point algorithms, since trajectories remain bounded $x_i \in (-1,1)$.
The main difficulty (numerically, geometrically...) is precisely that asymptotically stable equilibria lie on the boundary of the phase space, which is bounded. Now, in order to generalize the results on Hopfield systems, I would like to know about more systems with this property.
For instance, consider the system of ODEs: $$x'=(1-x^2)\, (w y -a)\\y'=(1-y^2)\,(w x-b)$$ which is stable because $V= -w x y + a x + b y$ is a Lyapunov function. However, this is only so if we consider the system RESTRICTED to the square $[-1,1]^2$. To give a hint of my concerns, think about applying a standard numerical method to the system. If a point with $x>1$ results from the approximation, everything is spoilt: the system is not stable, trajectories go to infinity, etc.
So, I would like to hear about physical/mathematical relevant systems, only defined on a bounded set that possess stable points on the border of such set: maybe continuous Ising models?...
Thanks in advance.