Local Linearization - Nonlinear Oligopolies: Stability and Bifurcations - page 283

given, since x can be unstable, marginally stable, and even locally (or globally)
asymptotically stable. Such examples are given next.

Example B.1. Consider first the two-dimensional linear system

11
01

x .t C 1/ D x .t/;

with both eigenvalues of the coefficient matrix (which is also the Jacobian) being on
the unit circle. It is easy to show by finite induction that for all t 1,

11
01

t 1t

01

:

D

Consequently, with any x .0/ D .x 1 .0/;x 2 .0// T ,

1t
01

x .t/ D x .0/; (B.5)

and when x 2 .0/ > 0, x 1 .t/ !1 . Therefore the zero equilibrium is unstable.

Example B.2. Consider next the single dimensional system

x.t C 1/ D x.t/

with the unique equilibrium x D 0. Notice that the Jacobian of the right hand side
is 1 with unit absolute value. Clearly, for all t 0,

x.t/ D . 1/ t x.0/

showing that the zero equilibrium is only marginally stable.

Example B.3. Consider now the simple nonlinear system

x.t C 1/ D x.t/e x.t/ 2 :

Notice first that x.0/ D 0 implies that x.t/ D 0 for all t 1,ifx.0/>0,then
x.t/>0 and if x.0/<0,thenx.t/<0 for all t 0. Furthermore if x.0/ ¤ 0,then

j x.t C 1/ j < j x.t/ j ;

showing that the positive state sequence is strictly decreasing and the negative state
sequence is strictly increasing. Therefore in both cases the state sequence is bounded
by zero, so convergent. Let x be the limit. Letting t !1 in the defining difference