IEEE Circuits and Systems Magazine - Q2 2018 - 23

The local activity theorem precisely predicts the presence of complexity in a
nonlinear system where complex phenomena and information
processing will most likely emerge.

YC (s, V ) =

^s - z 1h
^ s - p 1 h^ s - p 2 h

(29)

where
z1 = p1 =

b0
,
b1

- a 1 + a 21 - 4a 2 a 0
,
2a 2

p2 =

-a 1 - a 21 - 4a 2 a 0
.
2a 2

(30a)
(30b)
(30c)

The loci of the real parts of poles vs. imaginary parts
of poles of YC (s, V ) as a function of input voltage V is
shown in Fig. 8(a) over the range - 8.7 V # V # 8.7 V
where the arrowheads show the movement of the poles.
The poles of YC (s, V ) lie on both the left-hand side and
the right-hand side of the imaginary axis and the pairs of
complex conjugate poles of the YC (s, V ) lie on the imaginary axis at V = -7 V and V = -5 V. They are known as
Hopf bifurcation points in bifurcation theory. Moreover,
observe that Re p 1 2 0 and Re p 2 2 0 for input voltage
- 7 V 1 V 1 -5 V.
Fig. 8(b) shows the loci of the Im p 1 and Im p 2 of
the poles vs. Re p 1 and Re p 2 of the poles of YC (s, V )
as a parameter of inductance L for an input voltage
V = -7 V where the arrowheads show the movement of
the poles. The pole diagram of the admittance function
as a parameter of L contains a pair of complex conjugate poles located at Im p 1 = 2.4495 and Im p 2 = -2.4495
for an inductance value of L = L) = 41.67 mH. Moreover,
observe from Fig. 8(b) that when L " 0, Re p 1 " - 1
and Re p 2 " - 3 whereas for L " 3, the poles tend to
Re p 1 " 6 and Re p 2 " 0, respectively.
B. Frequency Response of Composite 1-port N
The frequency response of YC (s, VQ) associated with the
composite 1-port N in Fig. 7 at an equilibrium point Q is
computed by substituting s = i~ in (27) [3]:
YC (i~, VQ) =

b 0 (a 0 - a 2 ~ 2 ) + a 1 b 1 ~ 2
(a 0 - a 2 ~ 2 ) 2 + a 21 ~ 2
~ [(a 0 - a 2 ~ 2 ) b 1 - a 1 b 0]
G.
+ i=
(a 0 - a 2 ~ 2 ) 2 + a 21 ~ 2

(31)

The real and imaginary part of the admittance function YC (i~, VQ) at V = -7 V is shown in Fig. 9 over the
sEcOnd quartEr 2018

range -10 rad/s # ~ # 10 rad/s. Observe from Fig. 9
that the Re YC (i~) = - 24 and Im YC (i~) = 0 at ~ = 0
w h e r e a s a t f r e q u e n c i e s ~ = ~ ) = ! 2.4495 rad/s,
Re YC (i~ ) ) " ! 3 and Im YC (i~ ) ) " ! 3. Since the frequency response of the admittance of the composite
1-port YC (i~ ) ) " ! 3 at ~ = ~ ) = ! 2.4495 rad/s, it follows that the impedance of composite 1-port N tends to
Z C (i~ )) " 0 at ~ = ~ ) = ! 2.4495 rad/s for an operating
voltage V = -7 V. Hence, the prime condition of an oscillator is satisfied, namely, the total small-signal impedance of the oscillating circuit must be equal to zero at
the operating point.
IV. Local Activity, Edge of Chaos
And Hopf Bifurcation
The local activity theorem precisely predicts the presence of complexity in a nonlinear system [11] where complex phenomena and information processing will most
likely emerge from the small subset of the locally-active
domain, namely, the edge of chaos in the parameter space of a dynamical system [5], [12]. To be locally
active, a nonlinear system either has a pole of Y (s) in

Re YC (iω, VQ), Im YC (iω, VQ)
∞

V = -7 V

Im YC (0) = 0 S

200

L∗ = 41.67 mH

Re YC (0) = -24 S
5

-10

10 ω

-5

ω∗ = -2.4495 rad/s

-200

ω∗ = 2.4495 rad/s

-∞
Figure 9. Frequency response of the 4-lobe chua corsage
memristor oscillator for an operating voltage V = - 7 V and
L = L * = 41.67 mH.

IEEE cIrcuIts and systEms magazInE

23



Table of Contents for the Digital Edition of IEEE Circuits and Systems Magazine - Q2 2018

Contents
IEEE Circuits and Systems Magazine - Q2 2018 - Cover1
IEEE Circuits and Systems Magazine - Q2 2018 - Cover2
IEEE Circuits and Systems Magazine - Q2 2018 - Contents
IEEE Circuits and Systems Magazine - Q2 2018 - 2
IEEE Circuits and Systems Magazine - Q2 2018 - 3
IEEE Circuits and Systems Magazine - Q2 2018 - 4
IEEE Circuits and Systems Magazine - Q2 2018 - 5
IEEE Circuits and Systems Magazine - Q2 2018 - 6
IEEE Circuits and Systems Magazine - Q2 2018 - 7
IEEE Circuits and Systems Magazine - Q2 2018 - 8
IEEE Circuits and Systems Magazine - Q2 2018 - 9
IEEE Circuits and Systems Magazine - Q2 2018 - 10
IEEE Circuits and Systems Magazine - Q2 2018 - 11
IEEE Circuits and Systems Magazine - Q2 2018 - 12
IEEE Circuits and Systems Magazine - Q2 2018 - 13
IEEE Circuits and Systems Magazine - Q2 2018 - 14
IEEE Circuits and Systems Magazine - Q2 2018 - 15
IEEE Circuits and Systems Magazine - Q2 2018 - 16
IEEE Circuits and Systems Magazine - Q2 2018 - 17
IEEE Circuits and Systems Magazine - Q2 2018 - 18
IEEE Circuits and Systems Magazine - Q2 2018 - 19
IEEE Circuits and Systems Magazine - Q2 2018 - 20
IEEE Circuits and Systems Magazine - Q2 2018 - 21
IEEE Circuits and Systems Magazine - Q2 2018 - 22
IEEE Circuits and Systems Magazine - Q2 2018 - 23
IEEE Circuits and Systems Magazine - Q2 2018 - 24
IEEE Circuits and Systems Magazine - Q2 2018 - 25
IEEE Circuits and Systems Magazine - Q2 2018 - 26
IEEE Circuits and Systems Magazine - Q2 2018 - 27
IEEE Circuits and Systems Magazine - Q2 2018 - 28
IEEE Circuits and Systems Magazine - Q2 2018 - 29
IEEE Circuits and Systems Magazine - Q2 2018 - 30
IEEE Circuits and Systems Magazine - Q2 2018 - 31
IEEE Circuits and Systems Magazine - Q2 2018 - 32
IEEE Circuits and Systems Magazine - Q2 2018 - 33
IEEE Circuits and Systems Magazine - Q2 2018 - 34
IEEE Circuits and Systems Magazine - Q2 2018 - 35
IEEE Circuits and Systems Magazine - Q2 2018 - 36
IEEE Circuits and Systems Magazine - Q2 2018 - 37
IEEE Circuits and Systems Magazine - Q2 2018 - 38
IEEE Circuits and Systems Magazine - Q2 2018 - 39
IEEE Circuits and Systems Magazine - Q2 2018 - 40
IEEE Circuits and Systems Magazine - Q2 2018 - 41
IEEE Circuits and Systems Magazine - Q2 2018 - 42
IEEE Circuits and Systems Magazine - Q2 2018 - 43
IEEE Circuits and Systems Magazine - Q2 2018 - 44
IEEE Circuits and Systems Magazine - Q2 2018 - 45
IEEE Circuits and Systems Magazine - Q2 2018 - 46
IEEE Circuits and Systems Magazine - Q2 2018 - 47
IEEE Circuits and Systems Magazine - Q2 2018 - 48
IEEE Circuits and Systems Magazine - Q2 2018 - 49
IEEE Circuits and Systems Magazine - Q2 2018 - 50
IEEE Circuits and Systems Magazine - Q2 2018 - 51
IEEE Circuits and Systems Magazine - Q2 2018 - 52
IEEE Circuits and Systems Magazine - Q2 2018 - 53
IEEE Circuits and Systems Magazine - Q2 2018 - 54
IEEE Circuits and Systems Magazine - Q2 2018 - 55
IEEE Circuits and Systems Magazine - Q2 2018 - 56
IEEE Circuits and Systems Magazine - Q2 2018 - 57
IEEE Circuits and Systems Magazine - Q2 2018 - 58
IEEE Circuits and Systems Magazine - Q2 2018 - 59
IEEE Circuits and Systems Magazine - Q2 2018 - 60
IEEE Circuits and Systems Magazine - Q2 2018 - 61
IEEE Circuits and Systems Magazine - Q2 2018 - 62
IEEE Circuits and Systems Magazine - Q2 2018 - 63
IEEE Circuits and Systems Magazine - Q2 2018 - 64
IEEE Circuits and Systems Magazine - Q2 2018 - 65
IEEE Circuits and Systems Magazine - Q2 2018 - 66
IEEE Circuits and Systems Magazine - Q2 2018 - 67
IEEE Circuits and Systems Magazine - Q2 2018 - 68
IEEE Circuits and Systems Magazine - Q2 2018 - 69
IEEE Circuits and Systems Magazine - Q2 2018 - 70
IEEE Circuits and Systems Magazine - Q2 2018 - 71
IEEE Circuits and Systems Magazine - Q2 2018 - 72
IEEE Circuits and Systems Magazine - Q2 2018 - 73
IEEE Circuits and Systems Magazine - Q2 2018 - 74
IEEE Circuits and Systems Magazine - Q2 2018 - 75
IEEE Circuits and Systems Magazine - Q2 2018 - 76
IEEE Circuits and Systems Magazine - Q2 2018 - 77
IEEE Circuits and Systems Magazine - Q2 2018 - 78
IEEE Circuits and Systems Magazine - Q2 2018 - 79
IEEE Circuits and Systems Magazine - Q2 2018 - 80
IEEE Circuits and Systems Magazine - Q2 2018 - 81
IEEE Circuits and Systems Magazine - Q2 2018 - 82
IEEE Circuits and Systems Magazine - Q2 2018 - 83
IEEE Circuits and Systems Magazine - Q2 2018 - 84
IEEE Circuits and Systems Magazine - Q2 2018 - 85
IEEE Circuits and Systems Magazine - Q2 2018 - 86
IEEE Circuits and Systems Magazine - Q2 2018 - 87
IEEE Circuits and Systems Magazine - Q2 2018 - 88
IEEE Circuits and Systems Magazine - Q2 2018 - 89
IEEE Circuits and Systems Magazine - Q2 2018 - 90
IEEE Circuits and Systems Magazine - Q2 2018 - 91
IEEE Circuits and Systems Magazine - Q2 2018 - 92
IEEE Circuits and Systems Magazine - Q2 2018 - 93
IEEE Circuits and Systems Magazine - Q2 2018 - 94
IEEE Circuits and Systems Magazine - Q2 2018 - 95
IEEE Circuits and Systems Magazine - Q2 2018 - 96
IEEE Circuits and Systems Magazine - Q2 2018 - 97
IEEE Circuits and Systems Magazine - Q2 2018 - 98
IEEE Circuits and Systems Magazine - Q2 2018 - 99
IEEE Circuits and Systems Magazine - Q2 2018 - 100
IEEE Circuits and Systems Magazine - Q2 2018 - 101
IEEE Circuits and Systems Magazine - Q2 2018 - 102
IEEE Circuits and Systems Magazine - Q2 2018 - 103
IEEE Circuits and Systems Magazine - Q2 2018 - 104
IEEE Circuits and Systems Magazine - Q2 2018 - 105
IEEE Circuits and Systems Magazine - Q2 2018 - 106
IEEE Circuits and Systems Magazine - Q2 2018 - 107
IEEE Circuits and Systems Magazine - Q2 2018 - 108
IEEE Circuits and Systems Magazine - Q2 2018 - Cover3
IEEE Circuits and Systems Magazine - Q2 2018 - Cover4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2023Q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2023Q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2023Q1
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2022Q4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2022Q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2022Q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2022Q1
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2021Q4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2021q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2021q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2021q1
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2020q4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2020q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2020q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2020q1
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2019q4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2019q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2019q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2019q1
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2018q4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2018q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2018q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2018q1
https://www.nxtbookmedia.com