CHIN. PHYS. LETT. Vol. 27, No. 7 (2010) 070504
Initial State Dependent Dynamical Behaviors in a Memristor Based Chaotic
Circuit *
BAO Bo-Cheng(包伯成)1** , XU Jian-Ping(许建平)2 , LIU Zhong(刘中)3
1
School of Electrical and Information Engineering, Jiangsu Teachers University of Technology, Changzhou 213001
2
School of Electrical Engineering, Southwest Jiaotong University, Chengdu 610031
3
Department of Electronic Engineering, Nanjing University of Science and Technology, Nanjing 210094
(Received 29 December 2009)
Unlike conventional chaotic systems, a memristor based chaotic circuit has an equilibrium set, whose stability is
dependent on the initial state of the memristor. The initial state dependent dynamical behaviors of the memristor
based chaotic circuit are investigated both theoretically and numerically.
PACS: 05. 45. −a, 05. 45. Ac, 05. 45. Pq
DOI: 10.1088/0256-307X/27/7/070504
The “memristor”, an abbreviation for memory resistor studied by Chua in 1971,[1,2] is the fourth fundamental electronic component along with resistors,
capacitors and inductors. A memristor, either a fluxcontrolled memristor or a charge-controlled memristor, is a two-terminal element characterized by a relation of the type 𝑓 (𝜑, 𝑞) = 0. Since the successful realization of the memristor by Stan Williams’s
group at HP Labs in 2008,[3,4] memristor based applications have attracted much attention.[5,6] Itoh and
Chua[5] derived several oscillators from Chua’s oscillators by replacing Chua’s diodes with memristors
characterized by a monotone-increasing piecewiselinear function. Muthuswamy[6] proposed memristor based chaotic circuits with the memductance
𝑊 (𝜑) = 𝑑𝑞(𝜑)/𝑑𝜑 mathematically defined as a discontinuous function. These memristor based chaotic circuits can generate deformed chaotic attractors. However, the constitutive relations of these memristors
in Refs. [5,6] are non-smooth piecewise-linear functions, which make the corresponding memristance
𝑀 (𝑞) and memductance 𝑊 (𝜑) discontinuous nonlinear functions, and thus make the physical realization
of these memristors difficult.
In this Letter, we assume that an active twoterminal flux-controlled memristor is characterized by
smooth continuous nonlinearity as
2
𝑞(𝜑) = −𝑎𝜑 + 0.5𝑏𝜑 sgn(𝜑),
𝑑𝑞(𝜑)
= −𝑎 + 𝑏|𝜑|.
𝑑𝜑
𝑥˙ = 𝛼(𝑦 − 𝑥 − 𝑊 (𝑤)𝑥),
𝑧˙ = −𝛽𝑦 − 𝛾𝑧,
(2)
Then, by replacing the Chua’s diode in Chua’s chaotic
circuit with this active flux-controlled memristor, a
𝑦˙ = 𝑥 − 𝑦 + 𝑧,
𝑤˙ = 𝑥,
(3)
where 𝑥 = 𝑣1 , 𝑦 = 𝑣2 , 𝑧 = 𝑖3 , 𝑤 = 𝜑, 𝛼 = 1/𝐶1 ,
𝛽 = 1/𝐿, 𝛾 = 𝑟/𝐿, 𝜉 = 𝐺, 𝐶2 = 1, 𝑅 = 1, and define
the nonlinear function 𝑊 (𝑤) as
𝑊 (𝑤) =
𝑑𝑞(𝑤)
= −𝑎 + 𝑏|𝑤|.
𝑑𝑤
(4)
To find the equilibria of system (3), let 𝑥˙ = 𝑦˙ = 𝑧˙ =
𝑤˙ = 0. It is obvious that the equilibrium state of
system (3) is given by an equilibrium set
𝐴 = {(𝑥, 𝑦, 𝑧, 𝑤)|𝑥 = 𝑦 = 𝑧 = 0, 𝑤 = 𝑐.},
(5)
which corresponds to the 𝑤-axis. Here 𝑐 is a real constant.
i3
R
r
v2
–
i
+
+
+
L
C2
v1
C1
v
–
–
(1)
where 𝑎, 𝑏 > 0. In this case, the continuous memductance 𝑊 (𝜑) is given by
𝑊 (𝜑) =
new memristor based chaotic circuit as shown in Fig. 1
is designed.
From Fig. 1, we can obtain a set of four first-order
differential equations in dimensionless form[5]
Fig. 1. Memristor based chaotic circuit.
The Jacobian matrix 𝐽𝐴 at the equilibrium set is
given by
⎡
⎤⃒
𝛼(−1 − 𝑊 (𝑤)) 𝛼
0 −𝛼𝑏𝑥sgn(𝑤) ⃒⃒
⎢
⎥⃒
1
−1 1
0
⎥⃒ .
𝐽𝐴 = ⎢
⎣
⎦⃒
0
−𝛽 −𝛾
0
⃒
⃒
1
0
0
0
𝐴
(6)
* Supported by the National Natural Science Foundation of China under Grant No 60971090, and the Natural Science Foundations
of Jiangsu Province under Grant No BK2009105.
** To whom correspondence should be addressed. Email: [email protected]
c 2010 Chinese Physical Society and IOP Publishing Ltd
○
070504-1
CHIN. PHYS. LETT. Vol. 27, No. 7 (2010) 070504
The coefficients of the cubic polynomial equation in
brackets are all nonzero. Then according to the
Routh–Hurwitz condition, the real parts of the roots
of Eq. (7), except the zero eigenvalue, are negative if
and only if
7|𝑐| − 2.5 > 0,
7|𝑐| − 0.5 > 0,
a positive Lyapunov exponent, two zero Lyapunov exponents along with a negative Lyapunov exponent observed from Fig. 2(a), and with three different undergoing chaotic routes observed from Fig. 2(b). As 𝑐
increases in the parameter variation range, Hopf and
period-doubling bifurcation routes to chaos exist in
the memristor based chaotic circuit. Consequently,
the numerical simulation results of the initial state depending dynamical behaviors are consistent with the
theoretical analysis.
Lyapunov exponents
When 𝛼 = 7, 𝛽 = 10, 𝛾 = 0.0, 𝑎 = 1.5, 𝑏 = 1, and
𝑊 (𝑐) = −𝑎+𝑏|𝑐|, the characteristic equation of Eq. (6)
is given by
[︀
𝜆 𝜆3 + (7|𝑐| − 2.5)𝜆2
]︀
+ (7|𝑐| − 0.5)𝜆 + 70|𝑐| − 35 = 0.
(7)
70|𝑐| − 35 > 0,
(7|𝑐| − 2.5)(7|𝑐| − 0.5) − (70|𝑐| − 35) > 0,
which further give rise to
0.5 < |𝑐| < 0.5786,
|𝑐| > 1.2785.
(8)
L1
0.0
L2
−0.5
L3
L4
−1.0
−1.5
(a)
−2.0
1.5
1.0
To make the equilibrium set 𝐴 unstable, thereby yielding the possibility for chaos to occur, the constant 𝑐
in the cubic polynomial equation needs to be chosen
to satisfy the following conditions, which are obtained
by the Routh–Hurwitz criterion,
0.5
w
0.0
−0.5
−1.0
−1.5
|𝑐| < 0.5 and 0.5786 < |𝑐| < 1.2785.
(9)
(b)
−2.0
−1.5
−1.0
0
−0.5
0.5
1.0
1.5
c
Fig. 2. Lyapunov exponent spectra and bifurcation diagram of 𝑤 with increasing 𝑐 of system (3): (a) Lyapunov
exponents, (b) bifurcation diagram.
1.5
0.5
0.60
0.55
−1.0
−1.5
−4
1.0
0.50
−2
0
2
4
−0.2 −0.1
0
0.1
0.2
0.3
1.28
(d)
(c)
0.5
𝜆2,3
−0.9558 ± 𝑗2.6495
−0.5 ± 𝑗1.6583
−0.1676 ± 𝑗1.7133
±𝑗1.8843
0.1005 ± 𝑗2.7267
±𝑗2.9068
−0.0696 ± 𝑗2.9833
1.26
0.0
w
𝜆1
4.4117
0
−0.9449
−1.5506
−4.701
−6.4495
−7.8607
(b)
0.65
0.0
−0.5
Table 1. Three eigenvalues 𝜆𝑖 (𝑖 = 1, 2, 3) of the equilibrium set
𝐴.
|𝑐|
0
0.5
0.54
0.5786
1
1.2785
1.5
(a)
1.0
w
The three non-zero eigenvalues 𝜆𝑖 (𝑖 = 1,2,3) (except
an eigenvalue of zero) of the equilibrium set 𝐴 for some
typical values of the constant 𝑐 are listed in Table 1,
which are characterized by an unstable or stable saddle focus. Thus, the dynamical behavior of the memristor based chaotic circuit is closely dependent on the
initial state of the state variable 𝑤. Such characteristics well reflect the unique properties of the memristor
element.
1.24
−0.5
−1.0
1.22
−1.5
1.20
−2.0
−5
When 𝛼 = 7, 𝛽 = 10, 𝛾 = 0.0, 𝑎 = 1.5, 𝑏 = 1,
and initial states 𝑥(0) = 0, 𝑦(0) = 10−10 , 𝑧(0) = 0
are fixed while the initial state 𝑤(0) = 𝑐 is varied,
the Lyapunov exponent spectra of system (3) and the
corresponding bifurcation diagram of the state variable 𝑤 are shown in Figs. 2(a) and 2(b). For clarity,
only part of the fourth Lyapunov exponent is presented. Observed from Fig. 2, system (3) has abundant and complex dynamical behaviors. In three regions (−1.23, −0.58), (−0.5, 0.42) and (0.58, 1.23) of
the initial state 𝑤(0) = 𝑐, system (3) is chaotic with
0
z
5
1.18
−0.4
−0.2
0
0.2
0.4
z
Fig. 3. Typical orbits in the memristor based chaotic
circuit: (a) double-scroll attractor (𝑐 = 0), (b) period1 (𝑐 = 0.4), (c) 1-scroll attractor (𝑐 = 0.7), (d) sink
(𝑐 = 1.23).
Some typical orbits of system (3) are obtained from
numerical simulations, as shown in Fig. 3. Figure 3(a)
shows a double-scroll chaotic attractor. Figure 3(b)
depicts the limit cycle with period 1. Figure 3(c) illustrates a single-scroll chaotic attractor. Figure 3(d)
exhibits a sink. Observed from Fig. 3, there are great
variations for the orbits of system (3) under different
initial values.
070504-2
CHIN. PHYS. LETT. Vol. 27, No. 7 (2010) 070504
Therefore, it is remarkable that compared with the
conventional chaotic system,[7−9] the memristor based
chaotic circuit has more complicated routes to chaos
depending on the initial states.
In conclusion, we have presented and studied a
memristor based chaotic circuit, which is directly extended from Chua’s oscillator by replacing Chua’s
diode with an active two-terminal memristive circuit.
The novel chaotic circuit can generate a chaotic attractor and has complex nonlinear dynamics under
different initial states. The results demonstrate that
the introduction of a memristor makes the dynamical
behavior more complicated, completely different from
the existing chaotic system. A more detailed analysis
will be provided elsewhere in the near future.
References
[1] Chua L O 1971 IEEE Trans. Circuit Theor. 18 507
[2] Chua L O and Kang S M 1976 Proc. IEEE 64 209
[3] Strukov D B, Snider G S, Stewart D R and Williams R S
2008 Nature 453 80
[4] Tour J M and He T 2008 Nature 453 42
[5] Itoh M and Chua L O 2008 Int. J. Bifur. Chaos 18 3183
[6] Muthuswamy B 2009 IETE Tech. Rev. 26 415
[7] Jin Y F and Xu M 2010 Chin. Phys. Lett. 27 040501
[8] Hu G S 2009 Chin. Phys. Lett. 26 120501
[9] Bao B C and Liu Z 2008 Chin. Phys. Lett. 25 2396
070504-3