Chemistry Reference
In-Depth Information
Example B.5.
In the case of the system x
D
0, the Jacobian is zero with zero
eigenvalue. All solutions are constant and all real numbers are equilibria. Clearly all
equilibria are marginally stable.
Example B.6.
Consider finally the system driven by the single dimensional differ-
ential equation
x
D
x
3
:
Here
x
D
0 is the only equilibrium, and the Jacobian is the derivative
d
dx
.
x
3
/
D
3x
2
;
giving zero value at the equilibrium. Since the equation is separable, one can easily
find the state trajectories
x.0/
p
1
C
2tx.0/
2
:
x.t/
D
Clearly x.t/
!
0 as t
!1
showing the global asymptotic stability of the equilib-
rium.
In the case of linear systems local and global asymptotic stability are equivalent.
A discrete time invariant linear system is asymptotically stable if and only if all
eigenvalues of the coefficient matrix are inside the unit circle, and a time invariant
continuous linear system is asymptotically stable if and only if all eigenvalues have
negative real parts.
Continuous systems based on certain adjustment principles can often be written
as
x D K
.
g
.
x
//
(B.6)
n
is an adjustment function with sign preserving components. If
x
is an equilibrium of this system, then the Jacobian of the right hand side has the
special form
n
where
K W R
! R
J
.
x
/
D J
K
.
g
.
x
//
J
g
.
x
/
D J
K
.
0
/
J
g
.
x
/;
where
J
K
and
J
g
are the Jacobians of
K
and
g
, respectively. In analyzing the
asymptotic behavior of this system the following result can be applied.
Theorem B.7.
Assume
J
K
.
0
/
is positive definite and
J
g
.
x
/
C J
g
.
x
/
T
is negative
definite. Then all eigenvalues of
J
.
x
/
have negative real parts implying the (local)
asymptotic stability of the equilibrium of system (B.6).
Proof.
Consider the time invariant linear continuous system
z
D J
K
.
0
/
J
g
.
x
/
z
(B.7)
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