Question

In the problems below, the coecient matrix contains a parameteralpha. For both of these

matrices, carry out the following steps.

(i) Determine the eigenvalues in terms of alpha.

(ii) Find the critical value or values of alpha where thequalitative nature of the phase

portrait for the system changes.

(iii) Draw a phase portrait for a value of alpha slight below,and for another value slightly

above, each critical value.

(iv) Draw a phase portrait when alpha is exactly the criticalvalue.

a. x'=[alpha 1] b.x'=[-1 alpha]

[-1 alpha] x [-1 -1] x