Stable Quadratic Polynomials - Nonlinear Oligopolies: Stability and Bifurcations - page 311

1 C Tr C Det >0;
1 Tr C Det >0;
Det <1:

The continuous time counterpart of Lemma F.1 can be formulated in the follow-
ing way.

Lemma F.2. The roots of (F.1) have negative real parts if and only if p;q > 0:

Proof. Assume first that the roots are complex, 1;2 D a ˙ ib with a<0.Then

C b 2 >0:If the roots are real
and negative, then p D . 1 C 2 />0and q D 1 2 >0:
Assume next that p;q > 0: If the roots are complex, then Re 1;2 D p=2< 0:

D a 2 p D . 1 C 2 / D 2a > 0 and q D 1 2

If the roots are real, then from (F.2), both roots are negative, since p p 2

4q < p: