Section 3. Two Dimensional Systems
A two dimensional system defines a two dimensional vector field and a vector flow. The dsolve/numeric procedure outputs approximate solution values and curves. Linear systems can be solved using standard methods by reduction to one second order equation or by matrix methods (featured in the next section).
Load the plots and DEtools packages via with(plots): with(DEtools):
1. Use DEplot to obtain the phase portrait for the linear system
x' = -x + 2 y
y' = 4 x + y
Note. The phase portrait is a sketch of some solution curves and the direction field when the system is autonomous. Compare the picture you get to Figure 6.3.1 in Ledder (reproduced below).
2. Continuing 1. Use dsolve to obtain the general solution to the system in Exercise 1. Call the solution soln and represent it as a column vector using the entry
subs(soln, <x(t),y(t)> );
3. Add the two nullclines to the phase portrait in Exercise 1. (What is the stationary point?)
Hint. Go back and add the entry PP := %: at the same input prompt and directly after the DEplot entry creating the phase portrait. Execute the entries. Then make the nullclines using implicitplot (both can be drawn at the same time). Call the nullcline plot NP and then use
display( PP, NP );
4.* The model
x' = x (1 - y - x/a)
y' = y (1 - x - y/b)
is used to study competing species. See Ledder, Section 5.5, Exercise 11. Use DEplot to do the following.
a. Draw the direction field when 1/a = 1.9 and 1/b = 1.5. Use the window x = 0..1 , y = 0..1 .
b. Find the stationary point and add the nullclines to the direction field. Discuss the solutions based upon the picture you see.
c. Add solution curves corresponding to the following set of initial conditions (a circle of points around the stationary point)
{ [x(0)=0.3+0.2*cos(Pi/6*k),y(0)=0.5+0.2*sin(Pi*k/6)] }
5. Use DEplot to obtain the phase portrait for the system
x' = x - 2 y - 1
y' = x - y - 2
Add the stationary point and nullclines. Find the solution formulas using dsolve.
This is a linear system with periodic solutions (closed curves). Use the solution formulas to determine the period of the trajectories.