INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONSInt. J. Circ. Theor. Appl. (2013)Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/cta.1912
Hyperchaos and horseshoe in a 4D memristive system with a line ofequilibria and its implementation
Qingdu Li1,2,*,†, Shiyi Hu1, Song Tang2 and Guang Zeng1
1Institute for Nonlinear Circuits and Systems, Chongqing University of Posts and Telecommunications, Chongqing,400065, China
2Key Laboratory of Industrial Internet of Things and Networked Control of Ministry of Education, Chongqing Universityof Posts and Telecommunications, Chongqing 400065, China
Hyperchaos, first introduced by Rössler in 1979[1], is defined as a chaotic attractor with more than onepositive Lyapunov exponents (LE). Compared with ‘common’ chaos with only one positive LE,hyperchaos can exhibit multi-directional expansion which leads to much more complex dynamicalbehaviors. Therefore, hyperchaos is often considered better than common chaos in many engineeringfields, especially in secure communication for its capability of improving the security of chaoticcommunication systems. Many efforts have been paid in recent decades to develop efficient methodsand techniques for the generation and circuit implementation of hyperchaos [2–9].
The memristor predicted by Leon O. Chua [10] is recognized as the fourth fundamental electronicelement besides the well-known resistor, capacitor and inductor. Since its first prototype wasimplemented in a nanoscale system with coupled ionic and electronic transport by Stan Williamset al. in 2008 [11], many potential unique applications have been reported in spintronic devices,ultra-dense information storage, neuromorphic circuits and programmable electronics [11–14]. It isexpected to play a significant role in electronics in the near future. Since the memristor is normally anonlinear element [10, 11, 15, 16], which often leads to complex phenomena, there is growing interestin studying nonlinear dynamics in memristive circuits.
Due to the extraordinary function of the memristor [16], the state equation of an ideal memristive circuitis very different from a classical circuit with only inductors and capacitors in general. The equilibria
*Correspondence to: Qingdu Li, Institute for Nonlinear Circuits and Systems, Chongqing University of Posts andTelecommunications, Chongqing, 400065, China.†Email: [email protected]
usually do not consist of isolated points but make up a non-zero dimensional sets, e.g. lines, planes, etc.[17, 18] Hence, the memristive system has a different state-space structure from classical dynamicalsystems, which often leads to unusual behaviors. For example, the dynamics may not only be sensitiveto the parameters, but also closely depend on the initial conditions [19]. All these facts make it muchharder to study nonlinear dynamics of the memristive system than the classical systems.
A thorough investigation of such nonlinear dynamics is a key to design memristor-based circuits,which probably results in more efficient applications and more potential fields. As a typical nonlinearbehavior, chaos has been observed in many excellent memristive circuits, such as, a one-dimensionalnonautonomous circuit proposed by Driscoll et al. [20], a three-dimensional (3D) circuit invented byMuthuswamy and Chua [21], a number of four-dimensional (4D) circuits studied by Itoh, Chua,Muthuswamy, Bao, Buscarino et al. [22–26] and higher dimensional systems presented by Chua andBuscarino, et al. [27, 28], etc. We notice there is only one positive LE observed in the above papers.For hyperchaos, it has been observed in three five-dimensional circuits with two positive LEs, suchas the circuit with two memristors by Bao et al. [17] and the two circuits modified from the canonicalChua’s circuit by Fitch, Qi, et al. [5,29]
Hyperchaos has been found in many 4D circuits with capacitors or inductors. However, up to now,the 4D hyperchaotic memristive system is still absent in literature according to the best of ourknowledge. Thus, a natural question would be if such a system exists, which is not only crucialfrom our standpoint on the functionality and flexibility of the memristor as a fundamental circuitelement, but also important for our understanding of a dynamical system with a line of equilibria.Hence, the motivation of this paper is to give a positive answer to the above question. By adding amemristor into a 3D chaotic circuit, we will show that a 4D memristive circuit can not onlydemonstrate hyperchaos, but also exhibit many interesting phenomena that have not yet been seen inclassical systems, e.g. transient hyperchaos appears on the bifurcation route from a long period-1limit cycles to hyperchaos, and hyperchaos is frequently interrupted by non-attractive behaviors.
The rest of this paper is organized as follows. In Section 2, we review the memristor and give therelationship between a common system and an ideal memristive system. In Section 3, we show richnonlinear dynamics in a 4D memristive system. Then, hyperchaos is studied by a topologicalhorseshoe with two-directional expansion in Section 4. In Section 5, we present our circuitimplementation and experimental results. Finally, the conclusions are drawn in Section 6.
2. THE MEMRISTORS AND MEMRISTIVE SYSTEMS
According to Chua’s definition [10], the memristor is a two-terminal electronic element, where themagnetic flux ’ between the terminals is a function of the electric charge q that passes through thedevice. The memristor used in this work is a flux-controlled memristor. The nonlinear constitutiverelation between the device terminal voltage v and terminal current i is given by the differential form
i ¼ W ’ð Þv; _’ ¼ v; (1)
where W(’) is an incremental memductance function, describing the flux-dependent rate of change ofcharge:
W ’ð Þ ¼ dq ’ð Þd’
;
where q(’) is generally a nonlinear function [16]. In many papers [5,23,30–32], a cubic nonlinearity ischosen for the q�’ function
q ’ð Þ ¼ a’þ b’3;
where a and b are two positive constant parameters. The intuition is that smooth nonlinearities shouldbe easier to analyze and implement. Since Chua’s circuit with a cubic nonlinearity does give rise tochaos, many researchers use this form for its potential of introducing complex dynamics. Therefore,the memductance function in the following paper is given by:
It is easy to see from (1) and (2) that the function of the memristor are not alike any inductor orcapacitor. Hence, a memristive circuit is very different from a classical circuit with only inductorsand capacitors. Apparently, the first-order differential equations of an ideal memristive system mustcontain an equation like (1). If the system has an equilibrium, v should be zero according to (1).Hence, i must be zero, too, which means ’ has nothing to do with the rest state equations.Therefore, the equilibria of the whole system only rely to the state variables except ’. As a result, ’in the equilibria could be any constant. Thus, the equilibrium set does not consist of isolated points;it will be either nothing or a non-zero-dimensional set, where the dimension is not less than thenumber of ideal memristors. Consequently, the memristive system is generally harder to study thanthe classical dynamical system with only a limit number of equilibria.
The aim of this paper is to investigate hyperchaotic dynamics of a 4D ideal memristive system. Thesystem will be created by adding a memristor into a common chaotic circuit. We will show that amemristor makes the circuit hyperchaotic.
3. COMPLEX DYNAMICS IN A MEMRISTIVE SYSTEM
Now, we consider a 3D electronic circuit with its dynamics described by the following ordinarydifferential equations,
_x ¼ �a 2xþ yþ yzð Þ_y ¼ �b x� xzð Þ_z ¼ �z� gxy
;
8<: (3)
where the a, b and g are positive parameters. It is easy to see that system (3) is invariant under thetransformation
x; y; zð Þ↔ �x;�y; zð Þ:
Therefore, if (x,y,z) is a solution of (3), then (�x,� y, z) is also a solution. It is noticed that thesystem is always dissipative since there exists an exponential contraction rate
r�V ¼ @ _x
@xþ @ _y
@yþ @ _z
@z¼ �2a� 1 < 0:
System (3) has three equilibria. The first one is origin O, and the eigenvalues of the Jacobian matrixof (3) at O are
When we choose a = 5 and b= g = 6, they are both stable. However, our computer simulation froman initial point around O suggests (3) has a strange attractor, as shown in Figure 1. The correspondingLyapunov exponents are [0.197, 0.000, �11.2]. Clearly, the first one is positive, which implies that (3)is chaotic.
Figure 1. The phase portrait of the 3D system (3).
Q. LI ET AL.
This system can be easily realized as a chaotic circuit with op-amps and multipliers. A surprisingphenomenon here is that when we add one dimension by introducing in a memristor with themethod in Section 2, the new system can exhibit hyperchaos. According to (1), if we consider y asthe state voltage v inputting to the memristor and let k be a positive parameter indicating thestrength of the memristor, then (3) can be modified as follows:
where W(w) is the same as (2). Obviously, when k =0, the first three equations are the same as (3).Here, we are interested in the nonlinear dynamics affected by the memristor, so we considerk> 0 in the following paper. Since the memristor is the only difference here, it is easy for us to findout its function and effect by comparing the dynamical difference between the above two systems.
Comparing with (3), system (4) has the similar symmetry, i.e.,
x; y; z;wð Þ↔ �x;�y; z;�wð Þ
The two stable equilibria, O� and O+, disappear, and only the unstable equilibrium O is left. Asmentioned in Section 2, the equilibrium states only relies to x, y and z, and are independent from w.Supposing c is a real constant, the equilibria of (4) is
�O ¼ x; y; z;wð Þ x ¼ y ¼ z ¼ 0;w ¼ cj g;fwhich is actually a line. The Jacobian matrix of (4) on this line is
J�O ¼�2a �a 0 0�b kW cð Þ 0 00 0 �1 00 1 0 0
0BB@
1CCA:
The four eigenvalues and eigenvectors of J�O are listed in Table I. Since (3) and (4) are the samewhen k= 0, their first three eigenvalues and eigenvectors are also the same. Because all parametersare positive, we have
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2W cð Þ þ a� �2 þ ab
qe2 ¼ �a� 2a
b 2aþ l3ð Þ l2 0 1h iT
l3 ¼ k2W cð Þ � a�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2W cð Þ þ a� �2 þ ab
qe3 ¼ �a� 2a
b 2aþ l2ð Þ l3 0 1h iT
l4 = 0 e4 = [0 0 0 1]T
HYPERCHAOS IN A 4D MEMRISTIVE CIRCUIT
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2W cð Þ þ a
h i2þ ab
r> and
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2W cð Þ þ a
h i2þ ab
r>
k2W cð Þ:
Then, it is not hard to see that
l2 > 0 and l3 < 0:
Therefore, there are one positive eigenvalue, one zero eigenvalue and two negative eigenvalues.Hence, the system is unstable. Apparently, the eigenvector for l1 =� 1 is on the z-axis. Theeigenvector for l4 = 0 is on the w-axis, which is also the line of equilibria.
From the equations of the new system, we have
r�V ¼ @ _x
@xþ @ _y
@yþ @ _z
@zþ @ _w
@w¼ kW wð Þ � 2a� 1:
If the system has a chaotic attractor, it must be dissipative, i.e. rV< 0. Hence, k must be smallenough, i.e.
k < 2aþ 1ð Þ=W wð Þ:
Now, we choose a= 1 and b = 2. To explore the nonlinear dynamics caused by the memristor, wecompute the Lyapunov spectrum of (4) as we increase k from 0 to 0.3 with a stepsize 0.002.
In the computation, we choose the Jacobian method with QR-decomposition for calculating LEs andthe Dormand–Prince method (a fourth- and fifth-order Runge–Kutta method) for solving the ODEs.The error tolerance of each step is 10�9. The initial condition is x0 = (x, y, z, w) = (0, 0.01, 0.01, 0).The final time tmax takes 50000, which is long enough for a good accuracy, as well as excludingtransient dynamics. We consider the system non-attractive (toward infinity) if the distance betweenthe current state and the origin is greater than 100. In this case, we do not output LEs. Thenumerical results are shown in Figure 2, where only the first three LEs are presented, the last one isomitted because it is a big negative number between �11.2 and �10.5. Moreover, the LEs fork> 0.2 are also not presented because the system is not attractive. This figure shows that the systemexhibits complex nonlinear dynamics, such as, limit cycles, chaos, hyperchaos, etc.
In order to have a detailed view of the complex dynamics, we also compute the bifurcation diagramwhile k varies from 0 to 0.2 with a much smaller stepsize 0.0001. In the computation, we first tracealong the trajectory until t= 3000, then take a section hyperplane y = 0 with _y < 0 , and save allintersection points of the plane and the trajectory with t from 3000 to 4000. The bifurcation diagramis shown in Figure 3, which matches the LEs in Figure 2 very well.
From Figure 2, it is clear to see that when k= 0, the system is common chaotic. The LEs are 0.1969,0.0000, 0.0000 and �11.20, and they are the same as (3) except an extra zero. As k increase, this zeroLE monotonously rises up, and becomes a positive number, which implies the system turns
Figure 2. The first three Lyapunov exponents of (4).
Figure 3. Bifurcation diagram of (4) while adjusting k.
Q. LI ET AL.
hyperchaotic. This interesting phenomenon caused by the memristor is agreed by Figure 3, where thedynamical behaviors become more complex and the range of x gets larger. A typical phase portrait isshown in Figure 4, where k= 0.05 and the four LEs are 0.1615, 0.0054, 0.0000 and �10.985,respectively. This figure may give a false impression that there are four cycles in the x-w and y-w planeprojections of this hyperchaotic attractor, which probably remind the readers some similar hyperchaoticattractors with multi-scrolls discussed in the highly cited papers [2] and [33]. However, the truth is thatthere is actually only one big cycle, i.e. x and y always turn back as they get close to zero while w isbetween �1.2 and 1.2.
When 0.06< k< 0.078, the first LE turns to zero, the second and third LEs become negative, thatimplies the existence of limit cycles. For example, when k= 0.07, the LE are 0.0001, �0.0251,�0.0253 and �10.59, and the limit cycle is shown in Figure 5. Obviously, it is a very long period-1orbit, from which there is a bifurcation route to hyperchaos according to Figure 3.
However, if we compare the bifurcation diagram in Figure 3 with the LEs’ plot in Figure 2 carefully,we may discover the mismatch in many places, which imply the existence of transient dynamics. Tofind what dynamics it is, we recompute the LEs with a smaller stepsize 0.0001 and a shorter timetmax =5000. The results are shown in Figure 6, where two positive LEs occur on many parametervalues, e.g. 0.0655, 0.075, 0.076, 0.077, etc. All this evidence suggests a surprising phenomenon:there exists transient hyperchaos on the bifurcation route from the period orbit to hyperchaos.
When 0.078≤ k≤ 0.124, the first two LEs become positive, and the third LE becomes zero. Andthen, the system transits back to hyperchaos again. From Figure 2, we can also see that the second
LE is much larger than the previous hyperchaotic range 0<k< 0.06, which implies the dynamicsbecomes more complex. This is also supported by the bifurcation diagram. A typical phase portraitof the hyperchaotic attractor is shown in Figure 7, where k= 0.1 and the LE are 0.1267, 0.0145,�0.0001 and �10.82. Since the second LE is much greater than the one at k= 0.05, thehyperchaotic attractor in Figure 7 is more complicated than the attractor in Figure 4.
As we continuously increase k from 0.124, the hyperchaotic dynamics transits to limit cycles again,and then transmits back to hyperchaos at k= 0.134. The bifurcation diagram shows there exists aperiod-3 limit cycle, from which there is another route bifurcating to hyperchaos.
When k> 0.134, the dynamics becomes hyperchaotic. However, it is much different from the previoustwo hyperchaotic ranges of k. The hyperchaos is frequently interrupted by non-attractive behaviors.The interruption can also be seen in Figure 3. It is very interesting that the system suddenly becomesnon-attractive and then suddenly comes back to hyperchaos, which is another surprising phenomenonthat has rarely been seen in classical hyperchaotic systems according to our knowledge.
In order to see what happened when the system becomes non-attractive, we numerically simulate thesystem at k= 0.2, and the phase portrait is shown in Figure 8, where the red color indicates a trajectory
Figure 8. A non-attractive trajectory of (4) at k = 0.2.
Figure 7. The hyperchaotic attractor of (4) at k = 0.1.
with its time from 0 to 350, and the blue color indicates the same trajectory with its time from 350 to400. At the beginning, the trajectory evolves like chaos. When the oscillation of w decreases, states xand y slowly spiral out, meanwhile state z oscillates toward infinity. However, longer time simulationshows minimal value of w does not exceed �3, so r �V< 0, which implies the system is stilldissipative. This is perhaps the reason why x, y and z head to infinity so slowly.
All this evidence suggests that a memristor may make a common chaotic system demonstrateabundant interesting nonlinear phenomena, which are very different from the classical dynamicalsystem we used to study. We believe that the memristive system featured by non-zero-dimensionalequilibria could be a special type of dynamical systems, and it is ripe for intensive study.
4. COMPUTER-ASSISTED VERIFICATION OF HYPERCHAOS
We notice that the above studies with LEs and bifurcations are not reliable enough. Hence, in thissection, we will present a rigorous study on the hyperchaotic dynamics by combining thetopological horseshoe theory with a computer-assisted approach of Poincaré maps. The parametersare taken as follows:
a ¼ 5; b ¼ g ¼ 6; k ¼ 0:1; a ¼ 1 and b ¼ 2: (5)
And the corresponding attractor is shown in Figure 7. Before the detailed study, we first introducesome criteria for the existence of topological horseshoes by codimension-one crossing in terms‘separation’ [34, 35].
Let D be a compact region of Rn, and Di, i= 1,2,3,. . ., m, be compact subsets of D. For each Di, letD1
i and D2i be its two fixed disjointed connected nonempty compact subsets contained in the boundary
of Di. Let f :Di!Rn be a piecewise continuous map which is continuous on each Di.
Definition 1A connected subset S of Di is said to be a separation of D1
i and D2i , if for any connected subset l ⊂Di
with l ∩D1i 6¼ ∅ and l ∩D2
i 6¼ ∅, we have l ∩ S 6¼∅.
Definition 2
We say that f :Di↦Dj is codimension-one crossing with respect to two pairs D1i ; D2
i
� �and D1
j ; D2j
� �,
if f (S) ∩Dj is also a separation of D1j ; D2
j
� �for each separation S of D1
j ; D2j
� �.
Theorem 1If the codimension-one crossing relation f :Di↦Dj, holds for 1≤ i, j≤m, then there exists a compactinvariant set K ⊂D, such that f | K is semi-conjugate to the m-shift.
Here, the m-shift is usually denoted by s|Σm, which is also called the Bernoulli m-shift. Thesymbolic series space Σm is compact, totally disconnected and perfect. A set having these threeproperties is often defined as a Cantor set, and such a Cantor set frequently appears in thecharacterization of complex structures of chaotic invariant sets.
Proposition 1[36]. Let X be a compact metric space and f :X!X be a continuous map. If there exists an invariant setK ⊂X such that f | K is semiconjugate to the m-shift s|Σm, then the entropy of f,
When m> 1, the shift map s has a positive topological entropy and, therefore, is sensitive to initialconditions, i.e. chaotic. Then, it follows that f must be chaotic, too.
According to the above two definitions, the codimension-one crossing in terms of separationactually characterizes codimension-one expansion, i.e. map f is expanding in n�1 directions [34,35]. In our case, the Poincaré map will have a three-dimensional state space, i.e. n= 3. Hence, ahorseshoe satisfying the above theorem should expand in two directions, which implies that f shouldhave two positive Lyapunov exponents. Then, f must be hyperchaotic on the invariant set K. In thisway, we may provide a computer assisted verification of hyperchaos. The key to this study isfinding a topological horseshoe with two-directional expansion.
Now, let Π ¼ x � x; y; z;wð Þ w ¼ 0; _w < 0j gf be a Poincaré cross-section hyperplane. Then, thecorresponding Poincaré map P :Π!Π of (4) can be defined as follows: For each x2Π, P(x) istaken to be the first return point in Π under the flow of the dynamical system with the initialcondition x.
Unlike many other studies on topological horseshoes for two-dimensional chaotic maps [37–39], itis too hard to find a horseshoe directly due to the higher dimensionality. Hence, we use the followingtechnique to make it possible. First, we deduct the dimension along the direction of contraction to get a2D projective map; then, we detect a projective horseshoe with two-directional expansion; at last, weconstruct a 3D horseshoe back to the original map P.
Since the last LE is a large negative number, the dynamics in this direction contracts very quickly.Computer simulation shows the strange attractor of P in the state space looks like a curved surface.Since most points in the attractor are very close to the surface, the function of the surface x = s(y,z)can be easily fitted by these points. Since s can be computed numerically in practice, e.g., the built-in function ‘griddata’ in MATLAB, we do not need to know its explicit formula. This way, we havethe following 2D projective system [40]:
yz
nþ1
¼ jyz
n
¼ 0
010
01
P
s y; zð Þyz
0@
1A
n
0@
1A: (6)
Since the dimension in the direction of the negative LE has been deducted, we then only need todetect a 2D horseshoe of the projective map (6) on the yoz plane. We use a similar method proposedin [38] with an efficient and powerful tool called ‘A toolbox for finding horseshoes in 2D map’(downloadable from http://www.mathworks.com/matlabcentral/ fileexchange/14075), which has beensuccessfully applied in many systems [38, 39, 41]. The difference here is that the horseshoe shouldhave two-directional expansion, which is the same as the hyperchaotic spacecraft circuit studied in [4].
By several attempts, we find two quadrilaterals in the yoz plane. The first one is Ayoz with its fourvertices in terms of (y, z) as follows:
In order to construct a 3D horseshoe of the original Poincaré map P, we first project Ayoz and Byoz
back to the curved surface by adding the x coordinate with the numerical function x = s(y, z). Then,we have two curved ‘quadrilaterals’ Ac and Bc, respectively. At last, we move Ac and Bc up anddown for a small distance, d= 0.001, along the contractive direction of P. Where Ac and Bc havepassed through are block A and block B in the 3D state space, respectively. To approximate thisdirection, we conveniently use the average normal vector of Ac and Bc
v ¼ -0:306223817697261; 0:841717808314292; 0:44467752882454½ �T :
For clarity, we rotate the coordinates via the following Householder transform.
x ¼ x1; x2; x3½ �T ¼ H x; y; z½ �T ;where the Householder matrix:
H ¼ I� 2hhT ; h ¼ v� uv� uj jj j ;
which rotates the vector v to u= [0,0,1]T. After the rotation, blocks A and B and their images under P2 areshown in Figure 9, where At, Ab and As indicate the top surface, the bottom surface and the other four sidesurfaces of A, respectively, and Bt, Bb and Bs indicate the similar surfaces of B, respectively. Numericalcomputation suggests thatP2 is continuous onA and B. Then, it is not hard to draw the following conclusion.Theorem 2For the Poincaré map P :Π!Π, there exists a closed invariant set Λ ⊂A ∪B on which P2|Λ issemiconjugate to the 2-shift, and ent Pð Þ≥1
2 log2.
ProofTo this end, we need to show that the following relations of the codimension-one crossing,
P2 : A↦A; P2 : A↦B; P2 : B↦A and P2 : B↦B;
hold true with respect to two pairs (At, Ab) and (Bt, Bb).
(a) (b)
(c) (d)
Figure 9. The blocks A and B and their images under P2.
It is obvious from Figure 9(a) that P2(A) expands in two directions, then transversely intersectsblock A between its top surface At and its bottom surface Ab, and also transversely intersects block Bbetween its top surface Bt and its bottom surface Bb. It is also clear from Figure 9(c) that the imageof the side of A, i.e. P2(As), is outside of A and B. Hence, for each separation S of (At, Ab), f (S) ∩Amustbe a separation of (At, Ab), and f (S) ∩B must be a separation of (Bt, Bb). Then, we have P
2 :A↦A andP2 :A↦B.
Similarly, we can have P2 :B↦A and P2 :B↦B from Figure 9 (b) and (d). Then, let D=A ∪B, it fol-lows from Theorem 1 that there exists a compact invariant set Λ ⊂D, such that P2 | Λ is semiconjugateto the 2- shift. At last, according to Proposition 1, the topological entropy of P is not less than 1
2 log2.□The global picture of Figure 9 shows that P2 | A and P2 | B both expand in two directions. To verify
this fact, we compute their local expansions by numerically studying the Jacobian matrix of P2 atrandomly chosen 105 points in A and B, respectively. We numerically find that the two eigenvaluesof the matrix are both located outside of the unit circle. For A, the minimal norms of the twoeigenvalues are 1.09 and 7.17, respectively. For B, they are 1.25 and 17.5 respectively. Hence, P2 oninvariant set Λ ⊂A ∪B also expands two directionally, which implies that for any trajectory of P2 in Λ,it should have two positive LEs, therefore is hyperchaotic. This way, we have proposed a computer-assisted verification of hyperchaos. Here, we use the word ‘verification’ rather than ‘proof’ becauseP2(A) and P2(B) cannot be computed rigorously with interval arithmetic in an acceptable time. SinceA and B are both 3D blocks, the number of intervals needed to be computed is incredibly large.
5. CIRCUIT IMPLEMENTATION
In this section, we will realize the memristive system (4) with an electronic circuit. Generally speaking,the simplest way to do this is to use integrators. Hence, we discuss a memristor-based integrator and itscircuit implementation at first, then design the schematic of the whole circuit and verify thehyperchaotic attractor shown in Figure 7 by experiments at last.
A typical integrator with resistors and memristors is illustrated in Figure 10 (a), where u1, u2,⋯, unare voltage sources inputting to resistors R1,R2,⋯,Rn, and v1, v2,⋯, vm are voltage sources inputtingto memristors W1,W2,⋯,Wm. Since the change rate of the electric charges on capacitor C is thenegative sum of the electric currents through all resistors and memristors, the mathematical model ofthe integrator is given by the following differential form
C _vc ¼ �Xni¼1
ui=Ri �Xmj¼1
Wj ’j
� �vj
_’1 ¼ v1; _’2 ¼ v2;⋯; _’m ¼ vm
: (7)
Since the left-hand side of the capacitor is held to a zero voltage, due to the ‘virtual ground’ effect,the memristors become voltage-controlled current sources, which give great convenience forfabricating the memristor with classical components. This is important for circuit simulations andexperiments, because at present, the memristor is physically unavailable.
C
R1
vc
vm
u1
opamp
Rnun
Wm
v1W1
A B
.
.
.
.
..
IB
A BRa
Y
X
RbY
X
CwRw
Memristors
w
(a) (b)
Figure 10. A typical integrator with resistors and memristors.
The key to realizing this memristor is the nonlinear current on its right-hand side IB=W(’)v, wherev is the input voltage source on its left-hand side. Because
IB ¼ W ’ð Þv ¼ v aþ 3b’2ð Þ ¼ avþ 3bvZ
vdt
2
¼ avþ 3bvZ
� vdt
2 :
The schematic of our memristor is shown in Figure 10 (b), where the voltage of Cw indicates themagnetic flux ’ and the two multipliers are used for multiplication. The circuit is simpler than theone proposed in [23]. Since the voltage at terminals A and B are v and 0, respectively, the outputcurrent at B is given by
IB ¼ v
Raþ v
Rb
Zv
CwRwdt
2
:
In this case, we have
a ¼ 1Ra
; b ¼ 13RbC2
wR2w
:
Now, we apply the same group of parameters (5) in Section 4, then from (2) and (4), we have
_x ¼ �10x� 5y� 5yz_y ¼ �6xþ 6xzþ y 0:1þ 0:6w2
� �_z ¼ �z� 6xy_w ¼ y
;
8>><>>:
(8)
Before we design the schematic of the circuit, we first transform the above equations into the stateequations associated with (7). To realize the multiplication in the above equations, we choose the lowcost multiplier AD633, which has laser-trimmed accuracy and stability between �10V and 10V.However, Figure 7 shows that the values of all state variables are between �2 and 2. For a betteraccuracy, we rescale them by 5 V, i.e.
vx ¼ 5xV; vy ¼ 5yV; vz ¼ 5zVvw ¼ 5wV:
Since AD633 provides an overall scale factor of 1/10V. We need to put a factor 0.1/V on eachmultiplication. Then, we take the real time t = t �RC, where t is the dimensionless time in (8), R is areference for choosing the values of resistors and C is a reference capacitor. Finally, we rewrite theabove equations as follows:
If we take R= 18 kΩ, C = 10 nF, the circuit can be implemented as Figure 11, where the op-amps areLF347 and LF353 types and the supply voltages are �15V and +15V.
According to the schematic in Figure 11, the experimental circuit setup is shown in Figure 12(a),where the part on the left side is the topside circuit in Figure 11, and the part on the right side is thedownside circuit in Figure 11. We should note here that the original 3D circuit can be gotten bysimply removing the memristor. The experimental results are shown in Figure 12(b)–(d), which arethe screenshots of the oscilloscope, and the input signals are as follows: (b) vx vs. vy, (c) vy vs. vzand (d) vz vs. vw. Obviously, they look similar with the attractor shown in Figure 7. The slightdifference comes from the offset voltage of op-amps, as well as the inaccuracy and nonlinearity ofthe multipliers (which have a total accuracy of 2%). The experimental results suggest that the mainidea about the hyperchaotic memristive circuit of this paper is effective.
6. CONCLUSIONS
We have complex dynamics, especially hyperchaos, in a 4D memristive system modified from a 3Dcommon chaotic system by adding a memristor. Our finding and contributions are as follows:
Figure 12. The experimental circuit (a) and the screenshots of the oscilloscope: (b) vx vs. vy, x-axis 2V/div,y-axis 2V/div; (c) vy vs. vz, x-axis 2V/div, y-axis 5V/div; (d) vz vs. vw, x-axis 5V/div, y-axis 2V/div.
HYPERCHAOS IN A 4D MEMRISTIVE CIRCUIT
1. The system has a line of infinitely many equilibria, which is the main difference from other 4Dhyperchaotic systems we have ever known, such as the hyperchaotic Rössler system [1], thehyperchaotic Lorenz system [7,42, 43], the hyperchaotic Chen’s system [8], the hyperchaoticLü’s system [6], etc. [9,44], which at most have just a limited number of isolated equilibria.We also discover some new interesting phenomena that have not yet been seen in classicalsystems, such as, transient hyperchaos appears on the bifurcation route from a long period-1 limitcycle to hyperchaos, and hyperchaos is frequently interrupted by non-attractive behaviors.
2. To verify the existence of hyperchaos and reveal its mechanism, we study a topologicalhorseshoe with two-directional expansion, which implies that each trajectory in the invariantset Λ should have two positive Lyapunov exponents. Comparing other studies on horseshoesof 3D hyperchaotic maps [22,34], this paper demonstrates a detailed procedure on how to detectthem, whose best advantage is that it can be implemented with the MATLAB toolbox. In thisway, finding a horseshoe in the 3D space becomes applicable.
3. At last, we provide a way to implement the memristive system with an electronic circuit by a typicalintegrator with resistors and memristors. In this way, the memristors can be fabricated simply forcircuit simulations and experiments. Our experiments show that the screenshots of the oscilloscopewell match the phase portraits of the simulated attractor, which verifies the whole idea about a 4Dhyperchaotic memristive circuit of the paper. This circuit can be used as hyperchaotic signal sourcesin engineering applications, such as random numbers generator, chaos communications, etc.
We believe that the memristive system featured by non-zero dimensional equilibria could be aspecial type of dynamical system, which is a key to understanding how the memristor works.However, there are a lot of questions unresolved, such as what these interesting phenomena are,why they happen and so on. Hence, the memristive system is worth of more intensive studies due tothe promising applications of the memristor.
In addition, most real-world engineering systems can be described by complex networks composed ofmany different subsystems with connections in and among them [45]. The memristor is a naturalelectronic element to implement these connections, especially for neural networks. Thus, it is importantto further explore the networked memristive systems, i.e. many memristive systems merge into acomplex network via various coupling. It is very interesting to ask such a question: How can we extendthe current results to the real-world networked memristive systems? Due to the significance of scientificand engineering background of collective dynamics, one could start with studying the synchronizationof networked memristive systems, some very valuable methods [46] and criteria can be found in [47].
The authors are deeply grateful to the referees for the very valuable and detailed comments. This work issupported in part by the National Natural Science Foundation of China (61104150), Natural ScienceFoundation Project of Chongqing (cstcjjA40044) and Doctoral Fund of CQUPT (A2009-12).
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