IEEE Circuits and Systems Magazine - Q2 2018 - 24
16
Av (V)
12
8
4
0
-7
2
x2(t ) + iL (t )
Av (V) =
-6.5
-6
V
(a)
-5.5
Av (V) =
x2(t ) + iL (t )
-5
1.2
Av (V), Am (V)
0.96
2
Edge of Chaos Domain: - 9 V 1 V 1 - 3 V .
0.72
0.48
0.24
Am (V) = k [V + 7.0]1/2
0
-7
-6.95
V
(b)
-6.9
15
AL (L)
10
5
AL (L) =
0
0.04167
0.04192
2
x2(t ) + iL (t )
0.04217
L
(c)
0.04242
0.04267
Figure 10. (a). numerical verification of super-critical Hopf bifurcation of the 4-lobe chua corsage memristor oscillator as a parameter of V. (b). comparison between A V (V) = (x 2 (t) + i 2L (t)) 1/2
and the analytical formula A m (V) = k [V + 7.0] 1/2 over a small
neighborhood (- 7 V # V # 6.9 V) of V = 7 V. (c). Verification of
super-critical Hopf bifurcation of the 4-lobe chua corsage memristor oscillator as a parameter of L.
24
the open right-half plane where Re [s p] > 0, or Y (s) has
multiple poles on the imaginary axis, or Re [Y (i~)] < 0
for some ~ ! (- 3, 3) . The 4-lobe Chua corsage memristor is a locally active device because it satisfies the
condition of Re [Y (i~)] < 0 for some ~ ! (- 3, 3) which
is shown in Fig. 6. Moreover, the 4-lobe Chua corsage
memristor is locally active for -9 V 1 V 1 - 3 V as the
slope of the red DC V-I curve in the right inset of Fig. 4 is
negative over the range -9 V 1 V 1 - 3 V which satisfies the condition Re [Y (i~)] < 0 for ~ = 0 rad/s.
The Edge of Chaos regime is identified if all the poles
of the admittance Y (s) of a nonlinear system are located
in the open left-half plane, where Re [s p] < 0, as well as
Re [Y (i~)] < 0 for some ~ ! (- 3, 3) [11]. The 4-lobe
Chua corsage memristor has an edge of chaos regime over
the range -9 V 1 V 1 - 3 V because all poles of Y (s, V )
of the 1-st order generic memristor are located on open
left-half plane as shown in Fig. 5(c) and the Re [Y (i~)] < 0
for some ~ ! (- 3, 3) as shown in Fig. 6. In particular,
Re [Y (i~)] < 0 for ~ = 0 over an input voltage range
-9 V 1 V 1 - 3 V as shown in Fig. 4 right inset.
IEEE cIrcuIts and systEms magazInE
(32)
Hopf bifurcation is considered as a critical point
through which a system's stability switches and a periodic solution emerges. It is a local bifurcation where an
equilibrium point of a dynamical system loses its stability as a pair of complex conjugate eigenvalues, or poles,
cross the imaginary axis of the complex plane for some
critical parameter value ยต [9], [13]. The 4-lobe Chua corsage memristor oscillator has two critical Hopf bifurcation
points at V = - 7 V and V = - 5 V as the generic memristor oscillator circuit provides pairs of complex conjugate
poles on the imaginary axis at those voltages (as shown
in Fig. 8(a)). Over the range of - 7 V # V # - 5 V, the
dynamical system of the 4-lobe Chua corsage memristor
oscillator loses its stability as the poles cross the imaginary axis and migrate to the open right-half plane.
The 4-lobe Chua corsage memristor oscillator exhibits a super-critical Hopf bifurcation because
the amplitude A V (V ) = (x 2 (t ) + i 2L (t)) 1/2 (e. g. shown in
Fig. 10(a)) over a small neighborhood of Hopf bifurcation
point at V = - 7 V is very close to the curve computed
from the analy tical formula A m (V ) = k [V + 7.0] 1/2,
where k = (11 7 + V ) is chosen by trial and error [3],
as shown in Fig. 10(b). Another verification in terms of
A L (L) = (x 2 (t) + i 2L (t)) 1/2 is shown in Fig. 10(c).
V. Limit Cycle
In nonlinear dynamical system with two-dimensional
phase space, Hopf bifurcation gives birth to a limit cycle
from an equilibrium point which changes the system
sEcOnd quartEr 2018
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