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The second condition implies the absence of the negative instability (i.e.,
α
b
>
0).
Following the strategy in Section 2.7.2.1, only the case 0
<α
b
≤
γ<
1 is taken
into consideration. There is a possible dynamic instability if
1
+
2
1
γ
p
>
(2.171)
2
2
x
(
t
)
Hence, the instability does not depend on
ϑ
x
w
, only on the weight modulus; so
unlike the other neurons, the FENG stability does not improve when approaching
the MC direction (
can be taken
very low, and this choice improves the dynamic stability. By considering that
p
ϑ
x
w
→±
π/
2). However, near convergence,
γ
/λ
n
[see eq. (2.31)], it follows that an autocorrelation matrix with too low
a value of
→
1
λ
n
(near-singular matrix) can cause dynamic instability.
2.7.4 OJA
+
For OJA
+
the analysis is the same as above. It holds that
2
p
(
1
−
α
q
)
ρ(α)
=
(2.172)
2
H
p
−
2
α (
1
−
p
)(
u
−
p
)
+
α
where
2
2
q
=
x
(
t
)
−
u
−
1
+
p
(2.173)
and
+
(
1
+
u
)
(
1
−
p
)(
p
−
u
)
−
p
2
H
=
u
(
q
+
p
)
+
p
3
(2.174)
Then
ρ(α)>
1 (dynamic instability) iff
w
2
q
2
>
0
2
2
sin
2
2
α (α
−
α
)
p
x
(
t
)
ϑ
−
x
(
t
)
(2.175)
b
x
where
2
2
q
−
x
(
t
)
α
b
=
(2.176)
2
2
The dynamic instability condition is then
2
2
q
−
x
(
t
)
2
2
α >
∧
2
q
−
x
(
t
)
>
0
(2.177)
2
The second condition implies the absence of the negative instability (i.e.,
α
>
0).
b
It can be rewritten as
1
2
p
+
1
x
(
t
)
p
−
1
p
cos
2
ϑ
x
w
≤
(2.178)
2
2
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