For the initial value problem in Problem 18, plot u' versus u for , , and ; that is, draw the phase plot of the solution for these values of . Use a t interval that is long enough so the phase plot appears as a closed curve. Mark your curve with arrows to show the direction in which it is traversed as t increases.
The initial value problem in Problem 18 is:
The MATLAB module Ch03Sec09Prob20.m will plot u' versus u for a specified value of over a specified t-interval. We were asked to create phase plots for ,, and . Below is a set of phase plots for these three cases and the additional cases , , and . The module asks for the largest t-value: t = 70 works well for the assigned problems. The "colored graph" option is discussed below, so answer "n" for now. The module prints 'w' for omega for simplicity.
We were also asked to determine the direction that these curves are being traversed. With the limitations of the software, putting arrows on the graph isn't possible. However, we can look at the direction using color. If we answer "y" to the question "Would you like the graph colored?", then the graph will be colored in the order red, yellow, green, aqua, blue, magenta, black. We can then determine the direction of the plot from the progression of colors. This will not be effective if an unnecessarily large t-interval was chosen.
For example, consider the graph below, where . If we follow the colors from red to yellow to green, etc., we see that the graph is traversed in a generally clockwise direction.
This module creates a phase plot of u' -vs- u for
u'' + u = 3 cos(w t).
Enter the value of omega => .7
Enter the maximum value for t => 70
Would you like the graph colored? (y or n) => y
Note: Some of the phase plots for different values of are very interesting. Experiment with module Ch03Sec09Prob20 and see what you get. For example, try = 2.7.