Chapter 12.3, Problem 22E

Consider the system

$\frac{dx}{dt}=\left(\lambda +\alpha \right)x+\beta y$,

$\frac{dy}{dt}=-\beta x+\left(\lambda +\gamma \right)y$,

where $\alpha$, $\beta$, $\gamma$, and $\lambda$ are real numbers. The characteristic equation for the unperturbed system, where $\alpha =\beta =\gamma =0$, has a repeated root $\lambda$. Show that if we perturb the system, keeping $\alpha$, $\beta$, and $\lambda$ small, then the characteristic equation for the perturbed system can have the following:

a. Equal roots for $\beta =0$ and $\alpha =\gamma$.

b. Distinct roots for $\beta =0$ and $\alpha \ne \gamma$.

c. Complex roots for $\beta \ne 0$ and $\alpha =\gamma =0$.

d. What does this suggest about the type of critical point for almost linear system if the characteristic equation for the corresponding linear system has equal real roots?