Lyapunov stability check

Question

It is stated that for a system $\dot{x}=f(x)$ asymptotic stability is true if there is a Lyapunov function $V(x)$ such that $V(x)$ is positive definite and its time derivative is negative definite. My question is, isn't it possible that $V(x)$ is always decreasing but does not converge to zero but rather to certain asymptote. For example, is it possible that we have 2D parabola, and $V$ is always decreasing along some spiral path but its rate of decrease is also decreasing by time and goes to zero when time goes to infinity. And eventually $V$ goes to some circular path (asymptote).