I was looking at some exercises in Strogatz' book Non-Linear Dynamics and Chaos and came across this particular question in one of the exercises.
Here is my attempt:
By plotting the two functions $y=r$ and $y=\cosh(x)$, we see that as we vary $r$, we go from two to one to zero fixed points. We are interested in the critical point where the number of fixed points goes from zero to one. In order to do this, not only must we set the functions equal to each other, but we must do the same for the derivatives as well. By doing so, we obtain the equations $$r=\cosh(x),\quad (r)'=(\cosh(x))',$$ the second equation gives $0=\sinh(x)$ from which we get $x=0$. Then, substituting this into the first equation gives $r=1$.
Now, we distinguish three cases
- When $r<1$. Then, we get the situation where there are no fixed points.
- When $r=1$. Then, we get the situation where the fixed point $0$ is half-stable.
- When $r>1$. Then, we get the situation that there are two fixed points, where one is stable and one is unstable.
Is this correct? There are no solutions in Strogatz' book so I would appreciate a verification of sorts. I am new with this bifurcation stuff. Thanks in advance.