Stability on Autonomous ODE By definition, given a general system $x'=f(x)$ and $f(x_0)=0$, we say that $x_0$ is stable if $\forall\;V$ neighborhood of $x_0$, $\exists\;W$ neighborhood of $x_0$ such that $\forall\;x\in W,\;\varphi(t,x)$ is defined $\forall\;t\geq0$ and $\varphi(t,x)\in V\quad\forall\;t\geq0$, where $\varphi(t,x)$ denotes the (maximal) solution passing by the point $(t=0,x)$. Also $x_0$ is unstable if it is not stable. Is the following negation true? $x_0$ is instable if $\;\exists\;V$ a neighborhood of $x_0$ such that $\forall\;W$ neighborhood of $x_0$, then there exists $x\in W$ such that $\varphi(t,x)\notin V \;\forall\;t\geq0$

Rather this:

> ...then there exists $x$ in $W$ and $t\geqslant0$ such that $\varphi(t,x)$ is either undefined or not in $V$.

Recall that for every proposition $P$, the negation of $(\forall x,\ P(x))$ is $(\exists x,\ \lnot P(x))$ and the negation of $(\exists x,\ P(x))$ is $(\forall x,\ \lnot P(x))$, and that, for every propositions $P$ and $Q$, the negation of $P\land Q$ is $(\lnot P)\lor(\lnot Q)$. Thus, the negation of $$\forall V,\ \exists W,\ \forall x\in W,\ \forall t\geqslant0,\ P(x,t)\land Q(x,t),$$ is the proposition $$\exists V,\ \forall W,\ \exists x\in W,\ \exists t\geqslant0,\ (\lnot P(x,t))\lor(\lnot Q(x,t)).$$