Local Linearization - Nonlinear Oligopolies: Stability and Bifurcations - page 284

equation we have

D x e .x / 2 ;

x

showing that x D 0, which is the unique equilibrium of the system. Hence the
zero equilibrium is globally asymptotically stable. In this case the Jacobian is the
derivative of the right hand side:

d
dx .xe x 2 / D e x 2

C xe x 2 . 2x/

which equals 1 at x D 0.

We next turn our attention to continuous time systems (B.2), Theorem B.4 can
be modified as follows (see for example, Bellman (1969)).

Theorem B.5. Assume that all eigenvalues of J . x / have negative real parts, then
x is locally asymptotically stable.

The global stability version of this theorem is not valid in general. Cima et al.
(1997) provide a counterexample for all n 3 where all eigenvalues of the Jacobian
of g have negative real parts for all x

n , but the equilibrium of the continu-
ous time system is unstable. There has been intensive research on this problem for
the case of n D 2. Many authors proved the global asymptotic stability with different
additional conditions, and finally Gutierrez (1995) proved the global asymptotic sta-
bility of the equilibrium without additional assumptions in the case of continuously
differentiable functions. This result was extended without requiring the continuity
of the Jacobian by Fernandes et al. (2004). Their main result is the following:

2 R

2 , g . x / D 0 and g is differentiable everywhere.
Assume furthermore that all eigenvalues of the Jacobian of g have negative real
parts on

Theorem B.6. Assume D D R

D .Then x is the of system (B.2) and it is globally asymptotically stable.

The eigenvalues of the Jacobian at the equilibrium might also indicate the insta-
bility of the equilibrium, since similarly to the discrete time case we can show that
if at least one eigenvalue of J . x / has positive real part, then x is unstable. If all
eigenvalues of J . x / have non-positive real parts and at least one eigenvalue is zero
or pure complex, then x may be unstable, marginally stable, or even asymptotically
stable. Such examples are given next.

Example B.4. Consider first the two-dimensional linear system

01
00

x D x

which has both eigenvalues equal to zero. It is easy to show that the fundamental
matrix (which is the matrix exponential) is given as
1t
01

;

thus showing the instability of the zero equilibrium.