In contrast to Exercises 31–34, consider the systemfor the three values 0,10 and −10 of th...

In contrast to Exercises 31–34, consider the system

for the three values 0,10 and −10 of the parameter a.

a) Show that all three systems have the same Jacobian matrix at the origin. What type of equilibrium point at (0, 0) is predicted by the eigenvalues of the Jacobian?

b) Use pplane6 to find evidence that will enable you to make a conjecture as to the type of the equilibrium point at (0,0) in each of the three cases.

c) Consider the function h(x, y) = x2 + y2. In each of the three cases, restrict h to a solution curve and differentiate the result with respect to t (Recall: dh/dt = (∂h/∂x)(dx/dt) + (∂h/∂y)(dy/dt)). Can you use the result to verify the conjecture you made in part b)? Hint: Note that h(x, y) measures the distance between the origin and (x, y).

d) Does the Jacobian predict the behavior of the non-linear systems in this case?