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WOLFRAM|DEMONSTRATIONS PROJECT

Stability of a Linear Two-Dimensional Autonomous System

​
.
x
=
a
1
x +
b
1
y
a
1
1
b
1
1
.
y
=
a
2
x +
b
2
y
a
2
2
b
2
2
stability character
Consider the two-dimensional linear autonomous system
x
=
a
1
x+
b
1
y
y
=
a
2
x+
b
2
y
and define the matrix
M=
a
1
b
1
a
2
b
2
.
This Demonstration plots the phase portrait and the vector field of directions around the critical point
(0,0)
. This steady-state is indicated by the green dot in the phase plane diagram. In addition, the eigenvalues of
M
, the trace
tr(M)
, the determinant
M
, and
Δ=
2
tr
(M)-4M
are computed.
A plot of the curve obeying the equation
Δ=0
is obtained if you select the stability character tab. This parabola is indicated in solid and dashed black curves for positive and negative values of
tr(M)
, respectively.
The stability character of the steady-state is obtained by looking at the position of the black dot relative to the different colored regions (numbered 1–6).
region
eigenvalues
origin
tr(M)
M
Δ
1
complexconjugateswithnegativerealpart
asymptoticallystable
-
+
-
2
complexconjugateswithpositiverealpart
unstablespiral
+
+
-
3
realandpositive
unstableimpropernode
+
+
+
4,5
realwithoppositesigns
saddlepoint
-
+
6
realandnegative
asymptoticallystableimpropernode
-
+
+
green,-xaxis
……
marginallystable
-
+
yellow,+xaxis
……
unstable
+
0
+
blue,+yaxis
pureimaginary
centerwithalimitcycle
0
+
-
solidblackcurve
real,positive,andequal
unstablenodes
+
+
0
dashedblackcurve
real,negative,andequal
stablenodes
-
+
0
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