IEEE Circuits and Systems Magazine - Q2 2018 - 17

The dynamic route is a powerful tool for analyzing the
dynamics of nonlinear ordinary differential equations.

n a m e l y, Q 0 (x = 9), Q 1 (x = 31), Q 2 (x = 49), Q 3 (x = 81),
and Q 4 (x = 109) . These points are equilibrium points
when the 4-lobe Chua corsage memristor is short-circuited
(v = 0) . Among these equilibrium points Q 0, Q 2, and Q 4
are stable equilibrium points, whereas Q 1 and Q 3 are
unstable because the state variable x (t) diverges away
from Q 1 and Q 3 . Any initial state x Q 1 (0) = 31 + dx Q 1, diverges from Q 1 and converges to the stable equilibrium
point Q 2, whereas any initial state x Q 1 (0) = 31 - dx Q 1, diverges from Q 1 and converges to the stable equilibrium
point Q 0 . Similarly, any initial state x Q 3 (0) = 81 - dx Q 3,
diverges from Q 3 and converges to the stable equilibrium
point Q 2, whereas any x Q 3 (0) = 81 + dx Q 3, diverges from
Q 3 and converges to the stable equilibrium point Q 4 .
Let us plot the Lissajous figure in the i - v plane
with a sinusoidal input of v (t) = A sin (~t) . As expected,
Fig. 3 shows the 4-lobe Chua corsage memristor exhibits a frequency-dependent pinched hysteresis loop for
initial state x (0) = 8, amplitude A = 5 and, frequencies, ~ = 0.1 rad/s, 1 rad/s, 10 rad/s, and 100 rad/s. All
pinched hysteresis loops in Fig. 3 pass through the origin (0, 0) and the lobe area of the pinched hysteresis
loops shrinks as the frequency increases and tends to
a straight line at ~ = 100 rad/s, thereby confirming the
fingerprints of a generic memristor [8], [9].

DC V-I curve of the 4-lobe Chua corsage memristor is a
unique feature, because most published highly-nonlinear DC V-I curves of nonlinear resistors have several disconnected branches.
The red, green, blue, magenta, and purple branches
of the DC V-I curve represents the equilibrium points
of Q 0, Q 1, Q 2, Q 3, and Q 4, respectively. Observe that
at v = 0 the state variables are x = 9 (Q 0), x = 31 (Q 1),
x = 49 (Q 2), x = 81 (Q 3), and x = 109 (Q 4), respectively.
Hence, the slopes (memductance value) at v = 0 are
also different at each of the five branches shown in the
left inset of Fig. 4. It follows from the dynamic route in
Fig. 2, which is plotted at v = 0, namely, on the poweroff-plot (POP), only 3 of the 5 small-signal conductance
at v = 0 are stable; namely, the red, blue, and purple
branch. Hence the 4-lobe Chua corsage memristor is a
Tri-stable memory device at v = 0 V. Moreover, the right
inset of Fig. 4 shows that the red DC V-I curve contains
a negative slope over the range - 9 V < V < - 3 V with a
slope = - 24 Siemens at V = - 7 V. The locally-active negative slope region of the 4-lobe Chua corsage memristor
could give rise to oscillation and chaos which provides
the opportunity to use the 4-lobe memristor as an oscillator by imbedding it in locally-passive circuit, powered
by a battery.

B. DC V-I Curve
The DC V-I curve is used to determine the basic parameters of a device or a system when operating at DC. In
particular, a generic memristor [8] behaves as a nonlinear resistor in the DC regime [4]. The explicit formula of
the DC V-I curve of the 4-lobe Chua corsage memristor
is obtained by equating (3) to zero and by computing
x = X as a function of v = V, i.e.

C. Small-signal Model and Frequency Response
Small-signal circuit modeling is a standard nonlinear
circuit analysis techniques through which the behavior

x = X = xt (V)

1

ω = 1 rad/s

(6)

ω = 10 rad/s

0.5

and
I = G (xt (V )) V.

ω = 0.1 rad/s

(7)

The corresponding DC V-I curve computed using (6)
and (7) is shown in Fig. 4 over the input voltage range
- 20 V # V # 20 V. The DC V-I curve in Fig. 4 contains 4
contiguous DC V-I lobes and henceforth the 1-st order
generic corsage memristor defined by (1)-(4) is dubbed
4-lobe Chua corsage memristor. The contiguousness of
sEcOnd quartEr 2018

i (KA)

-6

-4

-2

0

2

ω = 100 rad/s
v (V )
4
6

-0.5
Figure 3. Frequency-dependent pinched hysteresis loops of
the 4-lobe chua corsage memristor.

IEEE cIrcuIts and systEms magazInE

17



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