In Exercise, we consider the effect of a non-linear perturbation which leaves the Jacobian...

In Exercise, we consider the effect of a non-linear perturbation which leaves the Jacobian at (0,0) unchanged. The theory predicts that, in (maybe very small) neighborhoods of equilibrium points where the eigenvalues of the Jacobian are non-zero and not equal, a non-linear system will act very much like the linear system associated with the Jacobian. We will use pplane6 to verify this by performing steps i), ii), and iii) below for the system in the indicated exercise.

i) Make up a perturbation for the system that vanishes to second order at the origin. As an example, instead of

the system in Exercise 21 consider the perturbed system

ii) Show that the Jacobian of the perturbed system at the origin is the same as that of the unperturbed system.

iii) Starting with the display rectangle −5 ≤ x ≤ 5, −5 ≤ y ≤ 5, zoom in square to smaller squares centered at (0, 0) until the solution curves look like those of the linear system.

Exercise 23.