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 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 2 ,...,φ n )
,
( i
=
1
,
2
,...,
n )
.
(B4.3-1)
To evaluate the stability of the equilibrium state
φ st =
(
φ 1 , st 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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