Chapter 3.3, Problem 34P

Dependence on a Parameter. Consider the system

$\text{x}\text{'}=\left(\begin{array}{cc}-1& -1\\ -\alpha & -1\end{array}\right)\text{x}$.

(a) Solve the system for $\alpha =0.5$. What are the eigenvalues of the coefficient matrix? Classify the equilibrium point at the origin as to type.

(b) Solve the system for $\alpha =2$. What are the eigenvalues of the coefficient matrix? Classify the equilibrium point at the origin as to type.

(c) In parts (a) and (b), solutions of the system exhibit two quite different types of behaviour. Find the eigenvalues of the coefficient matrix in terms of $\alpha$ and determine the value of $\alpha$ between $0.5$ and $2$ where the transition from one type of behaviour to the other occurs.