IEEE Circuits and Systems Magazine - Q2 2018 - 26

L = 40.67 mH, x (0) = 3, iL(0) = -63
0

iL

-20

Q0(2, -28)

-40
-60
(x (0), iL(0)) = (3, -63)
-80

0

1

2

3

x
(a)
L = 41.67 mH, x (0) = 3, iL(0) = -63
0

iL

-20

Q0(2, -28)

-40
-60
(x (0), iL(0)) = (3, -63)
-80

0

2

1

3

x
(b)
L = 42.67 mH, (x1 , iL1) = (3, -63),
(x2 , iL2 ) = (2.15, -32.36)
0
Q0 (2, -28)

iL

-20
-40

(x2 , iL2) =
(2.15, -32.36)

-60
(x1 , iL1) = (3, -63)
-80

0

1

2

3

x
(c)
Figure 12. simulation results show the relationship between
the oscillation generated from the oscillator circuit and the
external inductance L for an input voltage V = - 7 V. (a).
transient waveform converges to the stable equilibrium
point Q 0 (2, -28) for L = 40.67 mH with an initial condition x (0) = 3 and i L (0) = 63. (b). transient waveform converges to a stable limit cycle for L = 41.67 mH for an initial
condition x (0) = 3 and i L (0) = 63. (c). transient waveforms
from two different initial states (x 1 (0), i L1 (0)) = (3, - 63) and
(x 2 (0), i L2 (0)) = (2.15, - 32.36) converge to the small green
stable limit cycle for L = 42.67 mH.

26

IEEE cIrcuIts and systEms magazInE

stability via a pair of complex-conjugate imaginary eigenvalues or poles of the admittance function [13]. Among
the two types of Hopf bifurcation phenomena; supercritical Hopf bifurcation results in a stable limit cycle [14]. As
the 4-lobe Chua corsage memristor oscillator exhibits a
supercritical Hopf bifurcation, it will generate a small stable oscillation at least over the range - 7 V # V # - 6.9 V
which is shown in Fig. 11. However, for any input voltage
V < - 7 V and V > - 5 V, the state variables of the oscillator circuit will converge to the stable equilibrium point
Q 10 (1.9, - 25.63 h a nd Q 20 (4.1, - 82.37 h, re spectively.
Fig. 11(a) shows the limit cycle of the 4-lobe Chua corsage
memristor oscillator for an input voltage V = - 7.1 V
which is near but slightly to the left of the Hopf bifurcation point V = - 7 V (see Fig. 8(a)). The transient waveforms converge to the asymptotical stable equilibrium
point Q 10 (1.9, - 25.63 h when V = - 7.1 V. Similar phenomenon is also observed in Fig. 11(c) for an input voltage V = - 4.9 V, where the DC input voltage V = - 4.9 V
is chosen near but slightly to the left of the Hopf bifurcation point V = - 5 V (see Fig. 8(a)). For V = - 4.9 V,
the transient waveforms converge to the asymptotical
stable equilibrium point Q 20 (4.1, - 82.37 h, However, for
an input voltage V = - 6.95 V which is near but slightly
right to the bifurcation point V = - 7 V (see Fig. 8(a)),
the transient waveforms converge to a stable limit cycle
as shown in Fig. 11(b). The numerical simulation results
in Fig. 11 confirm that the 4-lobe Chua corsage memristor oscillator has a stable limit cycle for the bifurcation
parameter V chosen between the two Hopf bifurcation
points V = - 7 V and V = - 5 V as predicted by the super-critical Hopf bifurcation theorem.
Numerical simulations are performed to observe the
relationship between the external inductance L and the
oscillation generated from the oscillator circuit which
is shown in Fig. 12. Fig. 8(b) shows that Re p 1 = 0 and
Re p 2 = 0 whereas Im p 1 = 2.4495 and Im p 2 = - 2.4495
at L = L) = 41.67 mH for an input voltage V = - 7 V. It follows that L $ L) generates oscillation whereas L < L)
converges to a stable equilibrium point. Fig. 12(a) shows
that the transient waveform converges to a stable equilibrium point for an inductance L = 40.67 mH, where
L < L). However, the transient waveform in Fig. 12(b) converges to a stable limit cycle because L = L) = 41.67 mH.
Fig. 12(c) shows that the transient waveforms generated
from two different initial states (x 1 (0), i L1 (0)) = (3, - 63 )
and (x 2 (0), i L2 (0)) = (2.15, - 32.36 ) converges to the
stable green limit cycle for an inductance L = 42.67 mH
where L > L). It follows from the numerical results in
Fig. 12 that the oscillator circuit in Fig. 7(a) has a stable
limit cycle for the bifurcation parameter L chosen such
that L > L) where L = L) is a supercritical Hopf bifurcation point.
sEcOnd quartEr 2018



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