Environmental Engineering Reference
In-Depth Information
Box 4.3: Stability of multivariate deterministic systems
In this box we review the conditions determining the linear stability of a deterministic
n
-dimensional system. We indicate with
φ
1
,φ
2
,...,φ
n
) the state variables and
express their dynamics with a set of first-order differential equations:
d
φ
=
(
φ
i
d
t
=
f
i
(
φ
1
,φ
2
,...,φ
n
)
,
(
i
=
1
,
2
,...,
n
)
.
(B4.3-1)
To evaluate the stability of the equilibrium state
φ
st
=
(
φ
1
,
st
,φ
2
,
st
,...,φ
n
,
st
) with respect
to a small perturbation of amplitude
=
(
1
,
2
,...,
n
), we investigate the properties of
the dynamics when its state
φ
has a small displacement
from the equilibrium point:
φ
=
φ
st
+
.
(B4.3-2)
For (
B4.3-2
) to be a state of the system it has be a solution of (
B4.3-1
). Inserting
(
B4.3-2
)into(
B4.3-1
), applying a Taylor expansion (truncated to the first order) of
φ
φ
st
, and using equilibrium condition
f
i
(
φ
st
)
=
=
,
,...,
about
0(
i
1
2
n
), we obtain
n
d
i
d
t
=
a
i
,
j
,
j
,
(B4.3-3)
j
=
1
where
a
i
,
j
are the coefficients of the Jacobian matrix
A
[or
community matrix
in the
ecology literature (May, 1973)]:
∂
f
i
∂φ
j
a
i
,
j
=
φ
=
φ
st
.
(B4.3-4)
The solution of the set of differential equations (
B4.3-3
) is in the form
n
C
i
,
j
e
λ
j
t
i
(
t
)
=
,
(B4.3-5)
j
=
1
where coefficients
C
i
,
j
depend on the initial conditions and
λ
j
expresses how the
perturbation varies with time. The equilibrium point
φ
=
φ
st
is stable if the perturbation
decays with time, i.e., if all the values of
λ
j
(
j
=
1
,
2
,...,
n
)arenegative.
We find the values of
λ
j
by inserting (
B4.3-5
)into(
B4.3-3
):
n
λ
i
(
t
)
=
a
i
,
j
j
(
t
)
.
(B4.3-6)
j
=
1
This system of equations can be rewritten in the form
(
A
−
λ
I
)
(
t
)
=
0
,
(B4.3-7)
where
I
is the identity matrix. This homogeneous, linear set of equations has nontrivial
solutions only if the determinant of its matrix is zero:
|
A
−
λ
I
|=
0
.
(B4.3-8)
Thus
λ
j
(
j
=
1
,
2
,...,
n
) are eigenvalues of the matrix
A
. The stability of the
equilibrium point
φ
=
φ
st
requires that the real parts of these eigenvalues be negative.
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