I have a question concerning this document, section 12.4 ("The Bistable Equation") on pp. 197.
Consider the bistable equation given by $$ u_t=u_{xx}+f(u) $$ with the bistable function $f$ having zeros at $0,1$ and $a$ for some $a\in (0,1)$, cf. Figure 12.2.
1.) First of all, I think there is a typo in assumption (12.16) which to my understanding should be replaced by
\begin{equation} \int_0^1 f(u)\, du \textbf{>}0 \end{equation}
since the author wants to ensure the existence of a homoclinic orbit for the saddle $(0,0)$ and this is only possible if $F(0)<F(1)$.
2.) The author considers the line $L=\{(v,w): v=a\}$, the unstable manifold $M^U$ of the saddle $(0,0)$ and the stable manifold $M^S$ of the saddle $(1,0)$. By $P=(a,p(s))$, he denotes the intersection of $L$ and $M^U$, by $Q=(a,q(s))$ the intersection of $L$ and $M^S$.
For me it is not clear that, for each $s\leq 0$, the intersection points $P$ and $Q$ exist. For $s=0$, the "energy landscape" explains the existence of $P$ and $Q$. But for $s<0$ I do not see the existence of $P$ and $Q$.
For $s<0$, in order to clarify the existence of $P$ and $Q$, the author suggests to consider the line $$ w=-f(v)/s $$ since on this line, we have $w'=0$, i.e. we only have changes in the $v$-direction. In other words, this curve is a nullcline with respect to $w$.
But I do not see how this implies the existence of $P$ and $Q$.
Could you give an explanation why consiering the curve $w=-f(v)/s$ helps to see the existence of $P$ and $Q$?
Let's look at this mechanically. Then the equation is $v''+sv'+F'(v)=0$ where $F$ is an anti-derivative of $f$, $F'(v)=f(v)$, which is a function that has a graph that can be described as double-valley with minima at $0,1$ and a hill top at $a$. The motion is of a particle that moves in the potential field $F$ and has friction $s$.
If the friction coefficient $s$ is negative, so that moving around "gathers up" energy (with the energy function $E=\frac12(v')^2+F(v)$), then any small oscillation on one of the valley floors gets magnified until it reaches the hill top.
For a positive, "normal" friction coefficient $s$, you need sufficient initial speed to reach the hill top. If it is barely enough, then the energy will shortly after fall below the potential energy of the hill top, so the motion will never reach it again. The motion will oscillate down to the minimum of the other valley.