基于混沌电路的新的忆阻器(英文)

designed to become an equivalent active device. K. Thamilmaran presented a hyperchaotic circuit system based on Chua’s circuit in 2004[8]. In the hyperchaotic circuit, we replace the Chua's diode with a flux-controlled active memristor mentioned above, and obtain a new menristor based circuit of Fig. 1.
w(ϕ ) =
dq(ϕ ) ⎧ ⎪c W (ϕ ) = ⎨ dϕ ⎪ ⎩d
ϕ < 1; ϕ > 1;
(5)
(6)
p ( t ) = W (ϕ ) v ( t ) 2 < 0 . Thus, the two-terminal circuit can be
If c < G
ϕ < 1; dq (ϕ ) ⎧ ⎪c - G (7) =⎨ dϕ d G ϕ > 1; ⎪ ⎩ or d < G , then, the instantaneous power
Now we study the behaviors of this circuit. Equation (8) can be transformed into the form = α z − α sx ⎧x ⎪y = β y − ζ (z + u) ⎪ ⎪ = θ ( y − x) − λ z (9) ⎨z ⎪u = μy ⎪ = x ⎪ ⎩w Where v1 = x , v2 = y , iL1 = z , iL = u , ϕ = w , w(ϕ ) = s ,
= ∫ ( w(ϕ )v − Gv )dτ = ∫ ( w(ϕ ) − G )vdτ = ∫ ( w(ϕ ) − G )dϕ = dϕ + 0.5(c − d )( ϕ + 1 − ϕ − 1) − Gϕ = (d -G )ϕ + 0.5(c-d )( ϕ + 1 − ϕ − 1) Memductance W (ϕ ) of the equivalent two-terminal circuit is given by
School of Electronics Information Hangzhou Dianzi University Hangzhou, China
E-mail: weiling_wang@163.com
Abstract—This paper represents a novel memristor based chaotic circuit, which is obtained by replacing the Chua's diode with a flux-controlled active memristor in a hyperchaotic circuit. Some characteristics of this proposed system are analyzed in detail. Analysis results show that the memristor circuit has an equilibrium set and the special phenomenon of transient chaos and state transfer, and exhibits a complex topological structure of the attractor. Moreover, a method for overcoming the transient chaos and generating continuous chaos is proposed for the engineering applications of memristor based chaotic circuits. Keywords-Chaos; memristor; transient chaos;
(8)
III. BASIC PROPERTY ANALYSIS OF THE SYSTEM The Divergence of the system is given by ⎧ ∂y ∂z ∂w ∂u ∂x ⎪−2.132< 0 w <1 ∇v = + + + + = −αs + β − λ = ⎨ (11) ∂x ∂y ∂z ∂u ∂w ⎪ ⎩−9.755 < 0 w > 1 Obviously the system is dissipative. The equilibria of Eq. (9) is given by A = {(x, y, z, u, w)|x = y = z = u = 0, w = constant} , which corresponds to the w-axis and is a equilibrium set. The Jacobian matrix at this equilibrium set is given by α 0 0⎤ ⎡−αs 0 ⎢0 β β β 0⎥ ⎢ ⎥ (12) J = ⎢−λ λ −λ 0 0 ⎥ . ⎢ ⎥ μ 0 0 0⎥ ⎢0
Applying Kirchoff's laws to the Figure 1, the state equations of describing the circuit is obtained as
⎧ dv1 ⎪ c1 dt = i L1 − v1 w (ϕ ) ⎪ ⎪ c dv 2 = G v − i − i L1 L2 1 2 ⎪ 2 dt ⎪ ⎪ diL 1 = v 2 − v1 − R iL 1 ⎨ L1 dt ⎪ ⎪ di L2 = v2 ⎪ L2 dt ⎪ ⎪ dϕ = v1 ⎪ ⎩ dt
(3)
or (4) q (ϕ) = d ϕ + 0.5( c − d )( ϕ + 1 − ϕ − 1 ) From equations (3) and (4), memductance W(ϕ) and memristance M (q) are given by
M (ϕ ) = ⎧a dϕ (q) ⎪ ⎨ dq ⎪ ⎩b q < 1; q > 1;
Where the two nonlinear functions M (q) and W (ϕ ) , called the memristance and memductance, respectively, are defined by
dq (ϕ ) ⎧ ⎪W (ϕ ) = d ϕ ⎪ . ⎨ ⎪ M (q) = dϕ (q) ⎪ dq ⎩
I. INTRODUCTION Memristor predicted in 1971 by Leon O Chua of UC Berkeley as the 4th passive circuit clement[1]. It does not appear in any circuit theory textbook or literature until May 1, 2008, when Stanley Williams and his team declared the milestone discovery in the Journal Nature [2]. Recently, application of nonlinear circuits for secure communication purposes has become an active area of theoretical and experimental investigations. Memristor based applications have attracted much attention. Itoh and Chua derived several oscillators from Chua's oscillators by replacing Chua's diodes with memristors described by a monotone-increasing and piecewise-linear nonlinearity [3]. Their works pave the way of constructing more oscillator circuits. Recently, some chaotic circuits are constructed from third-order or fourth-order Chua’s circuits by replacing Chua’s diodes with memristors characterized by a smooth continuous cubic nonlinearity [4-7]. In this paper, we construct a new chaotic oscillator circuit by replacing the Chua's diodes in a hyperchaotic circuit with the memristors described via a flux-controlled peace-wise linear characteristic[8]. Some novel properties including equilibrium set and its stability, transient chaos phenomenon and bifurcations, are investigated in detail. CONSTRUCTION OF MEMRISTOR BASED CHAOTIC CIRCUIT Memristor is a new passive two-terminal circuit element in which there is a functional relationship between charge q and magnetic flux linkage ϕ . The memristor is governed by the following relation II.
(2)
In this paper, we assume that the memristor is characterized by the monotone-increasing and piecewise-linear nonlinearity, which is defined by
ϕ ( q ) = bq + 0.5(a − b)( q + 1 − q − 1 )
⎧v = M (q )i ⎨ ⎩i = W (ϕ )v
978-0-7695-4560-8/11 $26.00 © 2011 IEEE DOI 10.1109/IWCFTA.2011.58
Consider a two-terminal circuit that consists of a negative conductance G and a passive memristor and is a parallel connection of the both elements. One has q(ϕ ) = ∫ i (τ )dτ = ∫ (i1 (τ ) + i2 (τ ))dτ
(1)
57
L1
R
iL1
+ +
iL2
-G1 L2
v
-
+
i2
2
C2
C1
v
-
1
M -
(c) x-y-w (d) x-u-w Fig. 2. Chaotic attractor of the memristor based circuit.
Fig. 1 The new circuit with a flux-controlled active m来自百度文库mristor
2011 Fourth International Workshop on Chaos-Fractals Theories and Applications
A New Memristor Based Chaotic Circuit
Weiling Wang, Guangyi Wang and De Tan