Stable Quadratic Polynomials - Nonlinear Oligopolies: Stability and Bifurcations - page 310

Clearly 2<p<2, otherwise this relation cannot hold for both signs of the
square root. Notice that (F.3) is equivalent to
p p 2

4q < min f 2 C p;2 p g ;

or equivalently

p 2 4q < min f 4 C 4p C p 2 ;4 4p C p 2 g ;

which can be rewritten as

1 C p C q>0 and 1 p C q>0:

The cases of real and complex roots are shown in Figure F.1. The assertion can
be obtained by combining the two cases.

Consider next a real matrix

ab
cd

;

the characteristic polynomial of which is

'./ D .a /.d / bc D 2

.a C d/ C .ad bc/:

Let Tr D a C d denote the trace and Det D ad bc the determinant of this

matrix, then

'./ D 2

Tr C Det :

The eigenvalues of this matrix are inside the unit circle if and only if

q

Complex roots

p

Real roots

Fig. F.1 The stability region (shaded) of the quadratic polynomial (F.1) in the .p;q/ plane. It
shows the bounding lines 1 q>0and q<1. Also shown are the regions

where the roots of (F.1) are real and where they are complex, with the boundary between the two
regions being the parabola q

C p C q>0, 1 p C

p 2 =4. D