If there exists a continuum of equilibria (see black curves in the following figure) of a dynamical system, is it possible that in this set of equilibria, there exists isolated asymptotic stable equilibrium (see figure A) or a continuous subset of asymptotic stable equilibria (see figure B)?
I think it is not possible to have isolated asymptotic stable equilibrium on this continuum of equilibria, since any neighborhood of that equilibrium (lets say Eq1) contains another equilibrium (lets say Eq2), and trajectory arrives at Eq2 will be fixed at Eq2, so it will not converge to Eq1.
For the same reason, it is also not possible to have the case illustrated in figure B.
Thus, the only possible cases are figure C (the entire curve is unstable) and figure D (the entire curve is attractor).
Am I right? I am very not sure about this.
Thanks for any suggestion!
Note that there is a difference between stable and asymptotically stable. Your argument shows that any asymptotically stable equilibrium point $p$ must be isolated, i.e. there is a neighbourhood of $p$ that contains no other equilibrium point. But it is quite possible to have a continuum of equilibrium points that are stable but not asymptotically stable. For example, consider the two-dimensional system
$$ \eqalign{\dot{x} &= 0\cr \dot{y} &= (x^2-1) y \cr} $$
This has stable equilibria $(x,0)$ for $-1 < x < 1$, and unstable equilibria $(x,0)$ for $|x|\ge 1$ and $(\pm 1,y)$ for all $y$.