# Positive and Negative Stability of Fixed Points

I am trying to prove that a positively asymptotically stable fixed point cannot also be classified as negatively stable.

I am employing the definition of positive (negative) stability as: A point p is said to be positively stablie if

(1) There exists $r>0$ such that when $|\zeta -p|< r$ the solution $x(t, \zeta )$ is defined for all $t \ge 0$ (or $t \le 0$ for negatively stability)

(2) Given $\epsilon \ge 0$ there exists $\delta \ge 0$ such that $|x(t, \zeta ) -p|< \epsilon$ for all $t \ge 0$ (or $t \le 0$ for negatively stability) when $|\zeta -p|< \delta$

Furthermore a point is said to be positively asymptotically stable when the additional condition is satisfied

(3) There exists $\gamma \ge 0$ such that $\lim_{t\to\infty}x(t,\zeta)=p$ whenever $| \zeta -p| < \gamma$

Do I prove this by contradiction by allowing p to be positively asymptotically stable and assume that it is also negatively stable and show the two are not compatible? I'm not entirely sure how to start the proof.

Assume both, and take the $\epsilon$ in the definition of negatively stable smaller than the $\gamma$ in the definition of positively asymptotically stable. Then think of a solution which starts within distance $\gamma$ from the fixed point, but not within distance $\epsilon$. After a while, it will be within distance $\delta$. Now what happens if you start from that position and run time backwards?
• If we take $\epsilon$ <γ, then choose β such that ϵ<β<γ.Then when|ζ−p|<β, $x(t,\zeta )$is defined for all $t \ge 0$. As time increases given $\beta > \epsilon \ge 0$ there exists $\delta(\beta) >0$ such that $|x(t,\zeta )-p|< \beta$ for all $t \ge 0$ when $|\zeta -p|< \delta(\beta)$. Now reversing the time will give the contradiction, but here I am unsure of the specifics. When time reverses from this point, then given $0 \le \epsilon < \gamma$ then we have found that $|x(t,\zeta )-p|> \epsilon$ for all $t \le 0$ when $|\zeta -p|< \delta(\beta)$. Is this the desired contradiction? Feb 10, 2014 at 5:44