Chemistry Reference
In-Depth Information
In Chap. 2 (Theorem 2.1) we will see that the eigenvalues of this matrix lie inside
the unit circle if and only if a
k
<4for all k,and
X
N
a
k
4
a
k
<1:
kD1
In the case of linear systems local and global asymptotic stability are the same, so
the equilibrium is globally asymptotically stable if and only if the above conditions
are satisfied.
In the case of continuous time scales the dynamical system for partial adjustment
(1.24) can be written as
0
1
2
X
l
1
A
c
k
2B
x
k
.t/
@
A
;
x
k
.t/
D
a
k
x
l
.t/
C
(1.26)
¤
k
which is again a linear system with coefficient matrix
0
1
a
2
:::
a
2
a
1
@
A
a
2
a
2
a
2
:::
:
:
:
:
:
:
:
:
:
:
a
2
a
2
:::
a
N
In Chap. 2 (Theorem 2.2) we will see that all eigenvalues of this matrix always have
negative real parts so the equilibrium is locally asymptotically stable. The linear-
ity of the system implies that the Nash equilibrium is also globally asymptotically
stable.
Introducing the non-negativity conditions and the capacity limits into the model
makes the best reply functions nonlinear. Nonlinearity can also occur by assum-
ing nonlinear cost or price functions. Then the corresponding dynamical systems
become nonlinear, and local asymptotic stability does not imply global asymptotic
stability. This observation points to the need to perform detailed global analysis of
the dynamical behavior. The next section will present the foundation of the relevant
methodology.
In models (1.20)-(1.21), for the best response dynamics with adaptive expec-
tations, and (1.23) and (1.24) for the dynamics of partial adjustment towards the
best response with naive expectations, we have used simple linear adjustment rules.
However these can be easily extended to the nonlinear case by introducing sign-
preserving adjustment functions. A real-variable, real-valued function ˛
W R ! R
is called sign-preserving, if ˛.x/ has the same sign as x,thatis,
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