Stable Quadratic Polynomials - Nonlinear Oligopolies: Stability and Bifurcations - page 309

Appendix F
Stable Quadratic Polynomials

Consider the quadratic polynomial

2 C p C q D 0 (F.1)

with real coefficients.
In examining the asymptotic stability of two-dimensional dynamical systems the
following result can be used.

Lemma F.1. All roots of (F.1) are inside the unit circle if and only if

1 C p C q>0
1 p C q>0

and

q<1:

Proof. The roots are

p ˙ p p 2 4q

1;2 D : (F.2)

2

Assume first complex roots. This is the case when q>p 2 =4,then

2 ˙ i p 4q p 2

p

1;2 D ;

2

So j 1;2 j <1if and only if

p 2
4 C

4q p 2
4

D q<1:

Assume next that the roots are real. Then q p 2 =4; and j 1;2 j <1if and only

if

2 C p< ˙ p p 2 4q < 2 C p: (F.3)

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