These are just some of the considerations associated with the problem. I will continue to use Mathematica for now, as this is more convenient (and judging by your code above, you should be able to manage this in MATLAB, if not, I will try to convert it). However, if you have Mathematica and you can test them, that would be great.
Fixed point : the fixed point of the function f(x) is the point at which the solution f(x)=x ; in other words, the point at which the function maps it to itself.
The solution for your function, you get x=0 and x=3/4 as fixed points.
In:= Solve[Min[3/2 x, 3 - 3 x] - x == 0, x] Out= {{x -> 0}, {x -> 3/4}}
Indeed, the trajectory that begins at these points will remain at these points forever. You can also interactively observe effects when changing the initial and the number of iterations using
Manipulate[ CobwebDiagram[xstart, steps], {xstart, 0, 1, 1/1000}, {steps, 1, 200, 1}]
The nature of fixed points
Let's look at the nature of fixed points. If it is an attractor, points in an arbitrarily small Epsilon neighborhood of a fixed point remain in a neighboring similar size (do not have to be the same size), and if it is repellent, it repels and diverges to a completely arbitrary point outside the neighborhood (my definitions are pretty free here but guesswork will do).
So try the following
eps = 10^-16; CobwebDiagram[0.75 + eps, 200]
we get
fig. (1)
which, of course, does not seem to converge to a fixed point. Indeed, if you look at the evolution of x[t] , you will see that it diverges
Clear[f] f[1] = 0.75 + eps; f[t_] := f[t] = Piecewise[{{3/2 f[t - 1], 0 <= f[t - 1] <= 2/3}}, 3 (1 - f[t - 1])]; ListLinePlot[Table[f[n], {n, 1, 200}]]
fig. (2)
The result is similar if you break it in another direction, i.e. f[1]=0.75-eps .
For another fixed point (this time it can be perturbed in only one direction, since the function is defined for x>=0 ), you will see that the behavior is the same, and therefore two fixed points appear to be diverging.
fig. (3)
fig. (4)
Now consider the starting point x[1]=18/25 .
CobwebDiagram[18/25, 200]
fig. (5)
Wow !! It looks like a terminal cycle !
Limit cycle: The limit cycle is a closed trajectory of the system, from which it is not possible to reach a point not on the trajectory, even as t->Infinity . So, when you look at x[t] , you see something like
fig. (6)
which is repeated by only 3 points (image compression creates an image of moire , but in fact, if you built it for a small number of steps, you will see 3 points. I'm just too sleepy to go back and change it). Three points: 12/25 , 18/25 and 21/25 . Starting at any of these three points, you will end up in the same limit cycle.
Now, if the trajectories close enough to the limit cycle converge to it, this is an attractive / stable limit cycle, otherwise it is a repulsive / unstable limit cycle. Thus, still indignant at eps in any direction, we see that the trajectory diverges (I show only + ve the direction below).
fig. (7)
fig. (8)
Interestingly, starting with x[1]=19/25 18/25 it maps it to 18/25 in the next step, which then continues indefinitely in the trajectory of the limit cycle. It is easy to understand why this is happening, since the line from 19/25 on to y=x is simply an extension of the line from 12/25 to y=x (i.e., from the first part of the function). By the same logic, there should be points corresponding to 18/25 and 21/25 , but I'm not going to find them now. In light of this, I’m not entirely sure whether the limit cycle really attracts or repels here (in accordance with the strict definition of the limit cycle, there should be only one other path that spirals into it, you find three! Perhaps someone who knows more about this, can weigh it).
A few more thoughts
Starting point 1/2 also interesting, as it will take you to 3/4 in the next step, which is a fixed point and therefore stays there forever. In the same way, point 2/3 takes you to another fixed point at 0 .
CobwebDiagram[1/2, 200]
fig. (9)
CobwebDiagram[2/3, 200]
fig. (10)
Oscillation behavior also tells you about the system. If you look at the trajectory in fig. (2.4), the system takes longer to spiral into chaos for the case with a fixed point 0 than the other. In addition, in both graphs, when the trajectories approach 0 , it takes longer to recover than with 3/4 , where it simply oscillates quickly. They are similar to relaxation vibrations (think that the capacitor is slowly charging and instantly discharging by short circuiting).
That is all I can think of now. Finally, I believe that the exact nature of the fixed points should be analyzed in the general Lyapunov stability setting, but I am not going to get up to it. I hope this answer gave you some options to explore.