3. 3E: Runge-Kutta Usuli (Mashqlar) Matematika

Exercise 13
The direction field for the equation [frac
= 5 y (1-y) - left( 1 + frac <2> ight)] may appear chaotic at first glance. We’ll try to make sense of it in this exercise.
Plot this direction field for (0 < t < 6) , (0 < y < 1) . Can you tell what is the long-term behavior of solutions from the picture?
Use ode45 to generate solutions to the equation with initial conditions (y(0) in < 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1 >) , again on the range (0 < t < 6) . Plot them all on the same graph, and speculate as to what’s going on.
Overlay the figures from part (a) and part (b) to see how the solutions followed the direction field.
Qaror
Here’s the direction field. If you can see what’s going on, you have a keener eye than I do. It just kind of looks wavy to me.

The figure is produced below, as well as the commands that generated it. It appears that there is an equilibrium solution that is sinusoidal, and all solutions that start nearby are attracted to it.

There’s actually more to the story: there’s a second sinusoid which has the same shape as the one that we’ve detected, which starts around (y(0) = 0.3) , and is a repeller: solutions that start near it are pushed away. Unfortunately the arrows below that second sinusoid indicate immediate divergence to negative infinity, so if you’re a solution that starts under it you get sucked into a black hole very quickly. Thus using ode45 with (y(0) = 0.1) or (0.2) will produce an error message due to numbers getting too large for it to handle. This makes it difficult to see the repelling behavior going on. The commands
give some indication of this, though. (The first ode45 call runs a backwards simulation the computer draws the graph in reverse.)
Here’s the piece of art we just created. Does the direction field seem a little less chaotic with some trajectories overlaid on it?

The bonus trajectory discussed in the solution to part (b) is included in this graph, so your figure probably has one fewer squiggle than this image does.