Should the fixed points cross each other for a transcritical bifurcation?

Question

Consider the dynamical system
$$
dx/dt = -cos(r)sin(x)
$$
Clearly $x=0$ and $x=pi$ are two fixed points of this system.
The stability of these two fixed points change as r is varied. Starting from $r=0$, the zero fixed point is initially stable but becomes unstable at $r=pi/2$ and the opposite happens for the other fixed point.
Is this a transcritical bifurcation? I have so far only seen systems where one of the fixed point approaches the other fixed point and their stability switch as they cross each other.

However in my system, the fixed points remain at the same values of $x$. Only stability changes.

stafusa · Answer

Since the system is a ODE and not a map, I believe your stability analysis is correct. For instance, for $r=0$, $dot{x}$ is simply a negative sine function, so positive $xapprox 0$ values have negative derivatives (and decrease, i.e., move towards zero) and negative values have positive derivatives (so increase and also move towards zero), i.e., $x=0$ is stable for $r<pi/2$. See figure below:

But it does look like it goes through some sort of transcritical bifurcation or something similar as $r$ is varied - though a weird one due to the singular nature of the system at $r=pi/2$: For this value of $r$ all points $xin[0,2pi]$ are (neutrally stable) fixed points, since the equation is simply $dx/dt=0$. That's how I'd sketch its bifurcation diagram:

It looks like we could say there's a collision on $r=pi/2$.

Should the fixed points cross each other for a transcritical bifurcation?

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