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The equilibrium point is .x
1
D 0;x
2
D 0/. It holds that V.x/>08 .x
1
;x
2
/¤.0;0/
and V.x/D 0 for .x
1
;x
2
/ D .0;0/. Moreover, it holds
V.x/D 2x
1
x
1
C 2x
2
x
2
D 2x
1
x
2
C 2x
2
.x
1
x
2
/)
V.x/D2x
2
<08 .x
1
;x
2
/¤.0;0/
(2.43)
Therefore, the system is asymptotically stable and lim
t!1
.x
1
;x
2
/ D .0;0/.
Example 2.
Consider the system
x
1
Dx
1
.1 C 2x
1
x
2
/
x
2
D x
1
x
2
(2.44)
The following Lyapunov function is considered
1
2
x
1
C x
2
(2.45)
V.x/D
The equilibrium point is x
1
D 0;x
2
D 0. It holds that
V.x/D x
1
x
1
C 2x
2
x
2
Dx
1
.1 C 2x
1
x
2
C 2x
2
.x
1
x
2
//)
V.x/Dx
1
<08 .x
1
;x
2
/¤.0;0/
(2.46)
Therefore, the system is asymptotically stable and lim
t!1
.x
1
;x
2
/ D .0;0/.
2.3.3
Stability Analysis of the Morris-Lecar Nonlinear Model
Local stability of the Morris-Lecar model can be studied round the associated
equilibria. Local linearization can be performed round equilibria. Using the set
of differential equations that describe the Morris-Lecar neuron that was given
in Eq. (
1.57
) and performing Taylor series expansion, that is x D h.x/)x D
h.x
0
/j
x
0
Cr
x
h.x x
0
/ C one obtains the system's local linear dynamics.
The Morris-Lecar neuron model has the generic form
x
1
x
2
f.x
1
;x
2
/
g.x
1
;x
2
/
D
(2.47)
where f.x
1
;x
2
/ D I
app
g
L
.V E
L
/ g
k
.V E
k
/ g
Ca
m
1
.V/.V E
Ca
/ D
I
app
I
ion
.V;n/ and g.x
1
;x
2
/ D .
1
.V/ /=
.V/. The fixed points of this
model are computed from the condition
x
1
D 0 and
x
2
D 0. The Jacobian matrix
r
x
h D M is given by
