Biomedical Engineering Reference
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that the solution
x
(
t
;
x
0
) converges to
x
∗
as time approaches infinity. The
stability
of a steady state
x
∗
can be determined by computing the
Jacobian matrix and its
eigenvalues,
λ
±
at that point:
∂f
1
∂x
1
.
∂f
1
∂x
2
J
(
x
)=
∂f
2
∂x
1
∂f
2
∂x
2
The
steady state
x
∗
is locally stable
if all eigenvalues of
J
(
x
∗
) have a strictly
negative real part: Re(
λ
±
)
<
0. For two-dimensional systems, the stability can also
be established by looking at the trace and the determinant of the Jacobian matrix:
∂f
1
∂x
1
∂f
2
∂x
2
∂f
1
∂x
1
∂f
2
∂x
2
−
∂f
2
∂x
1
∂f
1
∂x
2
tr(
J
(
x
)) =
+
,
det(
J
(
x
)) =
.
The steady state
x
∗
is locally stable if: tr(
J
(
x
∗
))
<
0 and det(
J
(
x
∗
))
>
0.
Geometrically speaking, the equilibria in dimension two can be classified into
saddle (one positive and one negative real eigenvalue), stable sink (two real
negative eigenvalues), unstable sink (two real positive eigenvalues), stable focus
(two complex conjugate eigenvalues with negative real part), unstable focus (two
complex conjugate eigenvalues with positive real part), plus the non-generic cases.
n
2.2.2.2
Analysis of
-Dimensional Systems
This analysis can be extended to general systems of ordinary differential equations.
Consider now a system with
n
variables
x
=(
x
1
,...,x
n
)
t
∈
R
n
+
,
f
=(
f
1
,...,f
n
)
t
with
f
:
R
n
+
→
R
n
and
x
=
f
(
x
)
,x
(0) =
x
0
.
(2.2)
For large
n
, it becomes difficult to perform the stability analysis for a general set
of parameters, and so the steady states, the Jacobian matrix and its egenvalues
will typically be computed numerically, for given sets of parameters. As for the
two-dimensional systems, existence and uniqueness of solutions of Eq. (
2.2
)are
guaranteed by sufficient conditions on
f
: each
f
i
is continuously differentiable.
The invariance of the positive orthant may be checked by condition in Eq. (
2.1
)
for
i
=1
,...,n
.
The
nullclines
corresponding to each variable can be similarly computed:
Γ
i
=
.The
steady states
are given by all points
x
∗
such that
f
i
(
x
∗
)=0,for
i
=1
,...,n
.The
Jacobian matrix
is again obtained by computing
the partial derivatives of
f
i
:
n
{
x
∈
R
+
:
f
i
(
x
)=0
}
⎛
⎞
∂f
1
∂f
1
∂x
n
∂x
1
···
⎝
.
.
⎠
J
(
x
)=
.
∂f
n
∂f
n
∂x
n
∂x
1
···
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