When Linearization Fails. Most of the time, the linearization accurately predicts the beha...

When Linearization Fails. Most of the time, the linearization accurately predicts the behavior of a nonlinear system at an equilibrium point. There are exceptions, most notably when the the matrix A has purely imaginary eigenvalues, or when one of the eigenvalues is zero. For example, consider the system

Enter the system in pplane6, set the display window so that −1 ≤ x ≤ 1 and −1 ≤ y ≤ 1, select Arrows for the direction field, then click Proceed. In the PPLANE6 Display window, select Solutions→Show nullclines to overlay the nullclines on the vector field and note the presence of an equilibrium point at (0, 0).

a) Use Keyboard input to start a solution trajectory at (0.5,0) and note that the origin behaves as a spiral source. If the solution takes too long to stop on its own use the Stop button.

b) Select Solutions→Find an equilibrium point and find the equilibrium point at (0, 0). Note that the eigenvalues of the Jacobian are purely imaginary, indicating that the linearization has a center, not a spiral source, at the origin. Display the linearization and draw some solution trajectories.

In Exercises 21–30, perform steps i) and ii) for the linear system

where A is the given matrix.

i) Find the type of the equilibrium point at the origin. Do this without the aid of any technology or pplane6. You may, of course, check your answer using pplane.

ii) Use pplane6 to plot several solution curves — enough to fully illustrate the behavior of the system. You will find it convenient to use the “linear system” choice from the Gallery menu. If the eigenvalues are real, use the “Keyboard input” option to include straight line solutions starting at ±10 times the eigenvectors. If the equilibrium point is a saddle point, compute and plot the separatrices.