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Stability theory, including global stability (in dynamical systems, where it can notably be used in combination with ds.dynamical-systems)

3 votes

Nonlinear ODE system: stability

Another approach would be to expand in series form around some equilibrium point, and observe whether the lower order terms resemble some "known" behaviour, which is known as a normal form on the cent …
Miguel's user avatar
  • 274
2 votes
Accepted

Quadratic stability of linear time varying system

Sure: take the scalar ODE $\dot x=-x$, with $A=-1$, which is exponentially stable and the state transition matrix is $\Phi(t)=e^{-t}$ (i.e. $x(t)=\Phi(t) x_0$). Assume $Q=e^{2t}$, which is not bounded …
Miguel's user avatar
  • 274
2 votes
Accepted

Show that 0 is Lyapunov stable by using the given Hamiltonian $H(z)$ as a Lyapunov-function

With your setting, it seems you are right and this cannot be a Lyapunov function, but I cannot check the original paper, which could have some relevant context. Note that different authors provide sli …
Miguel's user avatar
  • 274
3 votes
1 answer
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Examples of systems with stable equilibria at the boundary of the phase space

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 ha …