Stability analysis of a 2-d dynamical system
January 23, 2014
The system is given by a pair of couple non-linear dierential equations:
= f (x1 , x2 )
ẋ2
= g (x1 , x2 )
(1)
f (x⋆1 , x⋆2 ) = g (x⋆1 , x⋆2 ) = 0. We would like to analyze the
⋆
⋆
stability of the system in the neigborhood of (x1 , x2 ). To do so, we can expand the nonlinear functions
⋆
⋆
about their values at the xed point. Dene ξ1 = x1 − x1 and ξ2 = x2 − x2 . Using this, the Taylor
A point
(x⋆1 , x⋆2 )
ẋ1
is said to be a xed point if
series to rst order are:
∂f ∂f ≈ f
+
ξ1 +
ξ2
∂x1 x⋆
∂x2 x⋆
∂f ∂f =
ξ1 +
ξ2
∂x1 x⋆
∂x2 x⋆
∂g ∂g ≈
ξ1 +
ξ2
∂x1 ⋆
∂x2 ⋆
(x⋆1 , x⋆2 )
f (ξ1 , ξ2 )
g (ξ1 , ξ2 )
x
where
means that the derivative is evaluated at the xed point. Since
x⋆
can write the dynamics (close to the xed point) in a compact matrix form:
(
The vector
ξ1
ξ2
ξ˙1
ξ˙2
(
)
=
(3)
x
∂f ∂x1 (
(2)
∂f
∂x2
∂g
∂x2
∂f
∂x1
∂g
∂x1
)
(
x⋆
ξ1
ξ2
ξ˙1 = ẋ1
ξ˙2 = ẋ2
we
)
.
(4)
(
)
can be decomposed to the eigenvectors
and
v1,2 of the matrix M =
∂f
∂x1
∂g
∂x1
∂f
∂x2
∂g
∂x2
)
dened
x⋆
by the equation:
Mv1,2 = λ1,2 v1,2
where
where
λ1,2
are the eigenvalues of the matrix
v1 (t)
is the component of the
M.
(5)
The solution to the linear dynamics is:
v1 (t) = eλ1 t v1 (0)
v2 (t) = eλ2 t v2 (0)
(
)
ξ1
vector
along the rst
ξ2
(6)
(7)
eigenvector of
M.
We usually don't
care about the full explict solution because it only applies very close to the xed point. We do care
however about the stability of the xed point which is given by the eigenvalues
There is a convenient formula for the eigenvalues of a
λ1,2 =
2×2
)
√
1(
T ± T 2 − 4D
2
1
λ1,2 .
matrix:
(8)
Figure 1: Stability regions in a 2-d dynamical system
where
T = trace (M) and D = det(M).
T as a function of D and
We can plot
separate the space into regions with dierent behaviors
around the xed point. Let's go over all the cases:
If T < 0, both λ1 and λ2 have negative real parts so the xed point is stable. If in addition
T 2 ≥ 4D, both eigenvalues are real so there is a regular stable xed point (also called stable node).
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If T
< 4D the eigenvalues are a complex conjugate pair, and the xed point is a stable spiral (also
called stable focus).
If T > 0, both λ1 and λ2 have
T 2 ≥ 4D, both eigenvalues are real
2
node). If T
< 4D the eigenvalues
positive real parts so the xed point is unstable.
If in addition
so there is a regular unstable xed point (also called unstable
are a complex conjugate pair, and the xed point is an unstable
spiral (also called unstable focus).
If
T = 0 we separate into three cases.
and one negative, so the xed point is a
T 2 > 4D, both eigenvalues are real, one of them is positive
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saddle. If T
> 4D the eigenvalues are purely imaginary so
If
there are oscillations around the xed point. The stability of the oscillations cannot be determined
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from linear analysis. If T = 4D , linear analysis is insucient to determine the dynamics around the
xed point.
All the cases are summarized in the plot below.
For a more detailed derivation, including explanation of the dierent types of bifurcations that
exist when transitioning from one region to the other, refer to Eugene Izhikevich's book: Dynamical
Systems in Neuroscience (chapters 3,4). The eBook is available for free through roger.ucsd.edu.
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