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Determine whether the given equilibrium point for the given system is Liapunov
stable, is attractive, is asymptotically stable, or is not stable. Give a brief
explanation.
\((x,y,z)=(0,0,0)\) for \[ \begin {cases} \dot {x}=y-x,\\ \dot {y}=-y-xz,\\ \dot {z}=xy-z. \end {cases} \]
\((x,y)=(0,0)\) for \[ \begin {cases} \dot {x}=-\dfrac {\partial V}{\partial x}(x,y),\\ \dot {y}=-\dfrac {\partial V}{\partial y}(x,y), \end {cases} \] where \(V(x,y)=x^4-x^2+y^2\) .
The zero solution \(x(t)=0\) for \(\ddot {x}+\dot {x}+g(x)=0\) where \(g\in C^1(\mathbb {R})\) and \(xg(x)>0\) for all \(x\ne 0\) .
Determine whether the given equilibrium point for the given system is Liapunov stable, is
attractive, is asymptotically stable, or is not stable. Give a brief explanation.
\begin{enumerate}
\item
\((x,y,z)=(0,0,0)\) for
\[
\begin{cases}
\dot{x}=y-x,\\
\dot{y}=-y-xz,\\
\dot{z}=xy-z.
\end{cases}
\]
\item
\((x,y)=(0,0)\) for
\[
\begin{cases}
\dot{x}=-\dfrac{\partial V}{\partial x}(x,y),\\
\dot{y}=-\dfrac{\partial V}{\partial y}(x,y),
\end{cases}
\]
where \(V(x,y)=x^4-x^2+y^2\).
\item
The zero solution \(x(t)=0\) for \(\ddot{x}+\dot{x}+g(x)=0\) where
\(g\in C^1(\mathbb{R})\) and \(xg(x)>0\) for all \(x\ne0\).
\end{enumerate}