Nonlinear Systems of Differential Equations
Consider the nonlinear system
1. Find the equilibrium points of the system.
(xe1,ye1) =
(xe2,ye2) =
2. To understand solutions that start near these equilibrium points, we must compute the Jacobian
matrix.
=
3. Evaluate the Jacobian matrix at each equilibrium point.
=
=
4. Write the linearized systems around (xe1,ye1) and (xe2,ye2) (i.e.,
).
and
5. By finding the eigenvalues and eigenvectors of the linear systems in #4, we can sketch the phase
portraits around each of the equilibrium points. We have learned how to calculate the
eigenvalues and eigenvectors by hand. Here we will use simple commands in Mathematica to
calculate these values. Use the following template below to find the eigenvalues and
eigenvectors for the Jacobian matrix at each equilibrium:
In[1]:= J={{1,0},{-3,2}}
Out[1]= {{1,0},-3,2}}
In[2]:= Eigenvalues[J]
Out[2]= {2,1}
In[3]:= Eigenvectors[J]
Out[3]= {{0,1},{1,3}}
Note that the first eigenvector, (0,1), corresponds to λ1 = -2 and the second eigenvector, (1,3),
corresponds to λ2 = 1. You can also use the command Eigensystem to compute the eigenvalues
and eigenvectors at the same time:
In[4]:= Eigensystem[J]
Out[4]= {{2,1},{0,1},{1,3}}
6. Sketch the local phase portrait near each equilibrium point.
7. In HPGSystemSolver, type in our original nonlinear system
Change the minimum and maximum values of x and y so that you can zoom into the area close to
(xe1,ye1). Compare the phase portrait to the one you sketched in #6 for the linearized system. Do
the same thing for (xe2,ye2).