Bifurcation Diagram If we are drawing a pitchfork bifurcation of $x'=\lambda x-x^3$ then my concern is that what is the difference if $\lambda$ is piece-wise constant? That is what is difference between the original bifurcation diagram for general $\lambda$ and the bifurcation diagram where λ varies with time? Piece-wise constant means $\lambda$ takes a value for some period of time and it will have another value for another period of time.
A similar problem can be found in Ordinary Differential Equations with Applications (Texts in Applied Mathematics) by Carmen Chicone, Second Edition on page number 13 Exercise 1.16 and my question is exactly part (c) of this exercise.
Thanks and hope to get your help soon.
 A: Given the system:
$$x'=\lambda x-x^3$$
What happens when we have a:


*

*fixed $\lambda$?

*time-Varying $\lambda$


Analysis


*

*There are three fixed (equilibrium points) $x = 0, x = \pm \sqrt{\lambda}$.

*As the parameter $\lambda$ passes through the bifurcation value $\lambda = 0$, the equilibrium at the origin loses its stability by giving it up to two new stable equilibria which bifurcate at the origin.

*The equilibrium point $x=0$ has become unstable for $\lambda \gt 0$, while the other two fixed points are stable. 


Fixed Bifurcation Diagram
Lets take a bunch of values for $\lambda$ and plot them on a $\lambda - x$ ($\lambda = r$ in the plots) chart. We can think of these as discrete points in time and each point represents a fixed $\lambda$. The blue represent stable and red unstable, that is unstable region is when $\lambda = [0, +\infty)$.
The resulting plot is:

Time-Varying Bifurcation
We can join all of these points since the qualitative behavior is the same for them from the given analysis and would have:

Aside: we can look at the long term behaviors if we were to plot this as a Logistic Map. We have:

Now, what if we could write a time-varying $\lambda$ that smoothly traversed those points and then played a movie of the bifurcation diagram? We would see the system go from a stable system to an unstable or stable system depending on which path of the bifurcation is taken. The long term behavior of the system shows that we can get some wild and chaotic swings.
The danger here is that we still have the original DEQ to solve, so creating a time-varying $\lambda$ can create an ODE that is very difficult or impossible to solve analytically, but we can still solve it numerically and observe the behaviors. 
In the book example you cite, $\lambda$ is representing the harvest rate of fish. If we knew that we could use a particular $\lambda$ over time to give the system time to recover (for example, only in this month are you are allowed to harvest), this would give the fish population time to recover so the system never gets into an unstable condition (population is gone). It would be helpful to figure out a time-varying $\lambda$ that does that as we could then run the simulations and see if the parameter is tuned so we always avoid extinction with some margin (as nature is very unpredictable) and keep the population well within a stable region.
For our original system, the solution is:
$$x(t) = \pm \dfrac{\sqrt{a}e^{a (c_1+t)}}{\sqrt{e^{2 a c_1+t}}-1}$$
If we change $\lambda = t$, then the solution becomes:
$$x(t) = \pm  \dfrac{-e^{t^2/2}}{\sqrt{c-1 + \sqrt{\pi}erfi(t)}}$$
where $erfi(t)$ is the imaginary error function (ugly).
Imagine if we had a more complex $\lambda(t)$ function, numerical methods to the rescue (if solvable at all). It is also worth noting that we were dealing with an equilibrium that is has the $\sqrt{\lambda}$ and care is also needed there with the choice of time-varying function.
So, you can see that choosing an appropriate $\lambda(t)$ would be tricky indeed. However, I think the point of the exercise is that we could use it to run a simulation over time to see if for the parameters choices when can avoid the unstable regions for our model and then adapt them accordingly to give us margin for indeterminate things in the real world. For the problem you are asking about, it is sufficient to think about $\lambda$ varying smoothly over the range of the bifurcation diagram in a time-varying function. The movie shows the qualitative behaviors over time.
Summary from above


*

*One has to be careful that $\lambda(t)$ is not negative in this particular example given the equilibrium points.

*There is no clear way to choose a $\lambda(t)$ to avoid issues or analyze the behaviors.

*$\lambda(t)$ could be changing so quickly as to make the bifurcation change too quickly and be sending you to stable-unstable-semi stable too quickly. 

*Different choices of $\lambda(t)$ could change the dynamics of your system and in fact change your DEQ entirely for different choices as demonstrated above.


This is not claiming that it is not possible. If you could somehow choose a $\lambda(t)$ that moved slowly, maybe even emulated the points on the non-time-varying case or did not change the nature of the dynamical system, it might be possible.
There has been research in this area as pointed out in the other answer, and, these items may also be of interest as examples:


*

*CUN-CAI HUA and QI-SHAO LU, Int. J. Bifurcation Chaos 11, 3153 (2001). DOI: 10.1142/S0218127401004091, "Time-Dependent Bifurcation: A New Method and Applications"

*I.Flegar, D. Pelin, D. Zacek, "Bifurcation diagrams of the buck converter"

*Yong-Gang Wang, Hui-Fang Song, Dan Li, Jing Wang, "Bifurcations and chaos in a periodic time-varying temperature-excited bimetallic shallow shell of revolution"

*Bifurcation Theory of Chaotic and Quasiperiodic Systems- see KAM Theory

A: Why exactly piecewise-constant function? 
Anyway, I think that to ask this question of p. 13 of a general book on ODE is not exactly appropriate. I read through @Amzoti's answer, and while it has some points, I do not quite find any answer either to your question or to the question in Exercise 1.16. 
Let me assume that $\lambda$ is not piecewise continuous but simply time dependent slowly varying function. What can be expected in this case? There will be a bifurcation, albeit dynamical in this particular case. What is the difference with the original bifurcation diagram? It depends on the type of the system you consider. If the system is analytical, then it is possible to have what is called delayed loss of stability, when it look like we already passed the bifurcation value long ago, but still stay in the vicinity of the origin. Here are a few references on this:


*

*A. I. Nejshtadt: Asymptotic investigation of the loss of stability of an
equilibrium upon slow passage of a pair of eigenvalues through the
imaginary axis (in Russian). Usp. Mat. Nauk. 40:5 (1985), 300301.

*A. I. Nejshtadt: On delay of loss of stability under dynamic bifurcations.
I. Differ. Uravn. 23 (1987), 2060-2067 (English translation
under the title: Persistence of stability loss for dynamical bifurcations.
I. Differ. Equations 23 (1987), 1385-1391).

*A. I. Nejshtadt: On delay of loss of stability under dynamic bifurcations.
II. Differ. Uravn. 24 (1988), 226-233 (English translation under
the title: Persistence of stability loss for dynamical bifurcations.
II. Differ. Equations 24 (1988), 171-176).


Interestingly, if you have $C^{\infty}$ system, then the delay basically ceases to exist. Anyway, the underlying theory is well beyond first fifteen pages of a textbook on ODE.
The most important lesson here is that

one has to be very careful making predictions on the system with
  time-varying parameter by looking at the bifurcation diagram, in which
  the parameter is supposed to be constant.

