Undergrad Using a bifurcation diagrams and time dependent parameters

[MathCreature]

Hello,

I'm lost at where to go after drawing bifurcation diagram of
$$\dot{x} = r + x - x^3.$$ If we also assume our parameter is time dependent such that
$$\dot{r} = -\delta x.$$ How could we use our initial bifurcation diagram to sketch solutions for small δ?

 

[S.G. Janssens]

Is there a reason why you would not want to consider the planar system
$$
\left\{
\begin{aligned}
\dot{x} &= r + x - x^3\\
\dot{r} &= -\delta x
\end{aligned}
\right.
$$
in a neighborhood of $(x, r) = (0,0)$ and with (time-independent) parameter $\delta$ (which I suppose is small and positive)?

Another thing to note is that for time-independent $r$ the equation for $\dot{x}$ is the cusp normal form with parameters $\beta_1 = r$ and $\beta_2 = 1$, but although I have heard of it, I am not sufficiently familiar with "slowly time-varying parameters" to see immediately how that normal form could provide an easier approach than when we just start with the planar system.

 

[MathCreature]

Thanks for the reply.

I believe I found someone who has the same problem: https://math.stackexchange.com/questions/2935059/bifurcation-of-time-dependent-parameters

I've gone through my "Strogatz's Nonlinear Dynamics and Chaos" and haven't found any problems like this, so I'm at a loss at how we can qualitatively sketch solutions. If $$\delta \approx 0,$$ then our parameter is almost constant for small x, though as x→±∞ we should eventually hit the bifurcations.
I don't believe we are using the system, but I've been looking done that path just to get an intuition of what's going on.