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.

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    $\begingroup$ Don't have an answer, but two comments:1). Take a fluid dynamical system in any smooth 2D or 3D bounded domain $\Omega$. Since in most physical situation , we have the "no-slip" condition, that means there is no velocity at the boundary. Now consider the dynamics of a "passive" particle: $\dot{x}=v(x,t)$ in $\Omega$, with $v|_{\partial \Omega}=0$. The whole boundary is a fixed point, albeit not asymptotically stable. I believe adding some kind of dissipation would probably give you what you need. $\endgroup$ – Piyush Grover Aug 10 '15 at 15:32
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    $\begingroup$ 2). There has been some work in invariance-preserving discretization schemes for ODEs and PDEs, i.e. schemes that will ensure that the discretized numerical integration doesn't get out of the $[-1,1]^2$ domain. Here's one such work:arxiv.org/abs/1406.6755 $\endgroup$ – Piyush Grover Aug 10 '15 at 15:36
  • $\begingroup$ Thank you, Piyush, I'll have a look at the idea of fluids... after struggling with PDEs :( And also thank you for the reference. $\endgroup$ – Miguel Aug 10 '15 at 15:37

Based on your description, I assume that you seek dynamical systems $\dot{{\bf{x}}}={\bf{f}}({\bf{x}})$ on $\mathbb{R}^n$ for which the set $X=[-1,1]^n$ is invariant, and for which there exist stable equlibria, solely at ${\bf{x}}\in A$ where $A=\{ {\bf{x}} \in\mathbb{R}^n : x_i=\pm 1\}$. If these conditions are met, then any reasonable numerical approximation of the DS with initial condition in $X$ can be forced to stay within $X$ by chosing appropriately small step sizes. Additionally, if the $\Omega$-limit set of the dynamical system is precisely $A$, then any reasonable numerical approximation to the DS with initial conditions in $X$ should converge to some point in $A$. To avoid overshooting (i.e. obtaining numerical approxmiations outside the set) simply include an 'if' statement to eliminate the possibility (i.e. if approximation lies outside $X$, then recalculate the previous step of the approximation, but with a sufficiently small step length to remain in $X$).

  • $\begingroup$ Thank you, JCM, I tried that and saw some weird numerical behaviours. Anyway, I could dedicate more time to the issue if I were convinced it has wide applicability. $\endgroup$ – Miguel Aug 10 '15 at 15:47
  • $\begingroup$ Could you be more specific with regard to the weird behaviour? $\endgroup$ – JCM Aug 11 '15 at 14:36
  • $\begingroup$ It did not preserve energy dissipation, and trajectories were chaotic or blew up to infinity. But maybe there simply was some error in the code. $\endgroup$ – Miguel Aug 11 '15 at 15:31

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