Multifolded torus chaotic attractors: Design and implementation

CHAOS 17, 013118 2007 

Multifolded torus chaotic attractors: Design and implementation 

Simin Yu 

College of Automation, Guangdong University of Technology, Guangzhou 510090, China 

Jinhu Lu a 

Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics 

and Systems Science, Chinese Academy of Sciences, Beijing 100080, China; State Key Laboratory 

of Software Engineering, Wuhan University, Wuhan 430072, China; and Department of Ecology 

and Evolutionary Biology, Princeton University, Princeton, New Jersey 08544 

Guanrong Chen b 

Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, China 

Received 31 October 2006; accepted 9 January 2007; published online 22 March 2007 

This paper proposes a systematic methodology for creating multifolded torus chaotic attractors from 

a simple three-dimensional piecewise-linear system. Theoretical analysis shows that the multifolded 

torus chaotic attractors can be generated via alternative switchings between two basic linear systems. 

The theoretical design principle and the underlying dynamic mechanism are then further 

investigated by analyzing the emerging bifurcation and the stable and unstable subspaces of the two 

basic linear systems. A novel block circuit diagram is also designed for hardware implementation of 

3-, 5-, 7-, 9-folded torus chaotic attractors via switching the corresponding switches. This is the first 

time a 9-folded torus chaotic attractor generated by an analog circuit has been verified experimentally. 

Furthermore, some recursive formulas of system parameters are rigorously derived, which is 

useful for improving hardware implementation. © 2007 American Institute of Physics. 

DOI: 10.1063/1.2559173 

Chua’s circuit, as the first chaotic circuit, has been intensively 

investigated as a platform for engineering applications 

over the past three decades. Since Chua’s circuit has 

a typical double-scroll chaotic attractor, a natural question 

to ask is, can we design various complex multiscroll 

chaotic attractors via some simple electronic circuits? After 

the rapid development over the past three decades, 

multiscroll chaos generation now has seen promising advances 

and has become an active research field in chaos. 

This is the case not only in deeper and wider theoretical 

studies but also in many newly found real-world applications. 

This paper introduces a systematic methodology 

for generating complex multifolded torus chaotic attractors 

from a simple three-dimensional piecewise-linear 

system. The theoretical design principle and the underlying 

dynamic mechanism are then studied further by analyzing 

the emerging bifurcation and the stable and unstable 

subspaces of the two basic generators. In addition, 

a block circuit diagram is designed for experimental verification 

of 3-, 5-, 7-, 9-folded torus chaotic attractors. It 

should be especially pointed out that this is the first time 

a 9-folded torus chaotic attractor has been realized physically 

by an analog circuit. 

I. INTRODUCTION 

Nowadays, theoretical design and circuit implementation 

of chaotic oscillators have been an increasingly interesting 

a Electronic mail: jhlu@iss.ac.cn 

b Electronic mail: eegchen@cityu.edu.hk 

subject for research due to their real applications in various 

information systems and chaos-based technologies. 1–27 Although 

there are many techniques reported for generating 

various multiscroll chaotic attractors, 2,9–27 to the best of our 

knowledge, there are very few publications on designing 

complex multifolded torus chaotic attractors. Torus breakdown 

was observed and confirmed in many real physical 

systems. 9,10 However, theoretical design and circuit implementation 

of multifolded torus chaotic attractors are still 

needed, which remains a technical challenge to both physicists 

and engineers today. 

Historically, Suykens and Vandewalle introduced a family 

of n-double-scroll chaotic attractors. 11 Aziz-Alaoui studied 

multispiral attractors in both autonomous and nonautonomous 

systems. 13 Yalcin et al. also proposed a family of 

hyperchaotic systems that have n-scroll attractors. 14 The essence 

of these methods lies in adding breakpoints into the 

piecewise-linear PWL characteristic function of the nonlinear 

resistor in Chua’s circuit. Later, Tang et al. developed a 

sine-function approach for creating n-scroll chaotic 

attractors. 15 Lu et al. introduced a switching manifold approach 

for creating chaotic attractors with multiple-merged 

basins of attraction. 1,17 Recently, Lu et al. developed hysteresis 

and saturated function series approaches for creating 

scroll-grid attractors, with rigorous mathematical proofs and 

physical circuit verifications for the chaotic behaviors. 20–23 

As a detailed survey, Lu and Chen reviewed the recent advances 

in multiscroll chaotic attractor generation, including 

theories, methods, and applications. 10 

1054-1500/2007/171/013118/12/$23.00 

17, 013118-1 

© 2007 American Institute of Physics 

Downloaded 22 Mar 2007 to 144.214.40.14. Redistribution subject to AIP license or copyright, see http://chaos.aip.org/chaos/copyright.jsp

013118-2 Yu, Lu, and Chen Chaos 17, 013118 2007 

Chua and his colleagues proposed a simple third-order 

autonomous circuit, 9,10,14 called a folded torus circuit, which 

can generate a double-folded torus chaotic attractor. Later, 

many researchers studied further the torus breakdown in a 

PWL forced van der Pol oscillator and a forced Rayleigh 

oscillator. As far as we know, previous works on torus breakdown 

only focused on laboratory measurement and numerical 

simulation for a torus or a folded torus. 9,10 Therefore, it is 

very interesting to ask whether the folded torus circuit can be 

slightly modified so as to generate multifolded torus chaotic 

attractors. This paper gives a positive answer to this 

question. 

More precisely, this paper reports our studies of constructing 

a general PWL characteristic function to replace the 

characteristic function used in the folded torus circuit, 

thereby generating multifolded torus chaotic attractors. In 

particular, our theoretical analysis reveals that these multifolded 

torus chaotic attractors can be generated via alternative 

switchings of two basic linear systems. The theoretical 

design principle and the underlying dynamic mechanism are 

then further investigated by analyzing the emerging bifurcation 

and the stable and unstable subspaces of the two linear 

systems. Moreover, a novel block circuit diagram is designed 

for hardware implementation of 3-, 5-, 7-, and 9-folded torus 

chaotic attractors. Recursive formulas for system parameters 

and for physical circuit parameters are also rigorously derived, 

useful for improving hardware implementation. It 

should be noted that this is the first experimental verification 

of a 9-folded torus chaotic attractor. 

The rest of this paper is organized as follows. In Sec. II, 

a systematic theoretical design approach for generating multifolded 

torus chaotic attractors is proposed, and some recursive 

formulas for system parameters are rigorously derived. 

The underlying dynamic mechanism and emerging bifurcation 

are then discussed in Sec. III. In Sec. IV, a simple circuit 

diagram is constructed for experimentally verifying the multifolded 

torus chaotic attractors. Conclusions are finally 

drawn in Sec. V. 

II. THEORETICAL DESIGN OF MULTIFOLDED TORUS 

CHAOTIC ATTRACTORS 

In this section, we construct a general PWL characteristic 

function to replace the characteristic function used in the 

fold torus circuit for generating multifolded torus chaotic 

attractors. 

A. Double-folded torus chaotic attractor 

Chua and his colleagues proposed a double-folded torus 

chaotic circuit, 9,10 called a folded torus circuit, described by 

where 

ẋ =−gy − x 

ẏ =−gy − x − z 

ż = y, 

1 

FIG. 1. Double-folded torus chaotic attractor. 

gy − x = m 1 y − x + m 0 − m 1 

y − x + x 1 − y − x − x 1 2 

2 

is a PWL odd function satisfying gx−y=−gy−x. When 

=15, =1, m 0 =0.1, m 1 =−0.07, and x 1 =1, system 1 has a 

double-folded torus chaotic attractor as shown in Fig. 1. The 

maximum Lyapunov exponent of this attractor is 0.0270. 

System 1 has three equilibrium points: O0,0,0 

and P ± 

± 

17 

7 ,0,0 . Linearizing system 1 at equilibrium point 

O gives the corresponding eigenvalues: O 1 1.4328 

and O 2,3 −0.0164±1.0231i, which have the corresponding 

eigenvectors 0.9985,0.0448,0.0312 T and 

−0.41940.2829i,−0.6169,0.0097±0.6028i T , respectively. 

Thus, system 1 has a one-dimensional unstable 

eigenspace E U x 

O: 

9985 = y 

448 = z 

312 corresponding to 1 O and a 

two-dimensional stable eigenspace E S O:37186732x 

O 

−25007019y+17452101z=0 corresponding to 2,3 at the 

neighboring region of O. 

Similarly, linearizing system 1 at the equilibrium 

points P ± gets the following eigenvalues: P 1 −1.0145 

and P 2,3 0.0172±1.0172i, with the corresponding 

eigenvectors 0.9989,0.0338,−0.0333 T and 

0.32780.3124i,0.6358,0.01060.6249i T , respectively. 

System 1 has a one-dimensional stable eigenspace 


013118-3 Multifolded torus chaotic attractors Chaos 17, 013118 2007 

x 

9989 = y 

338 = z 

E S P 

P ± : 

−333 

corresponding to 1 and twodimensional 

unstable eigenspace E S P ± :−19865571x 

P 

+10076539y+9931196z=0 corresponding to 2,3 in the 

neighboring region of P ± . 

B. Design principles of system parameters 

Definition 1: An n-folded torus attractor is an attractor 

that is created via an n-torus breakdown route, where the 

n-torus is described as a quotient of R n under integral shifts 

in any coordinate with a zero Lyapunov number and an irrational 

rotation number. 8–10 

To generate multifolded torus chaotic attractors from the 

folded circuit 1, we introduce a new PWL odd function to 

replace the characteristic function 2, which is described by 

gy − x = m N−1 y − x 

N−1 

mi−1 − m 

+ 

i 

y − x + x i − y − x − x i , 3 

i=1 2 

where gx−y=−gy−x and , are real parameters. 

Note that the above PWL characteristic function gy−x 

can be recast as follows: 

y − x x 1 , m 0 y − x 

if x i y − x x i+1 , 1 i N −2, 

i 

gy − x =if 

m i y − x + m j−1 − m j sgny − xx j 

j=1 

N−1 

if y − x x n−1 , m N−1 y − x + m j−1 − m j sgny − xx j , 

j=1 

4 

where m i 0iN−1 are the slopes of the segments, or 

radials in various piecewise subregions, and ±x i x i 0, 1 

iN−1 are the switching points. Since g· is an odd 

function, we only need to determine the positive switching 

points x i 1iN−1. 

Obviously, the equilibrium points of system 1 with 3 

are ±x i E ,0,0x i E 0,0iN−1, where ±x i E 0iN−1 

satisfies the following equation: 

gx i E =0, 0 i N −1. 5 

Substituting 4 into 5, and solving x E i 0iN−1 from 

5, yields the recursive formulas of x E i 0iN−1 as follows: 

= 0 i =0, 

x E i 

i j=1 m j − m j−1 x j 

6 

1 i N −1. 

m i 

Denote 

k i = x E 

i+1 − x i 

x E , 7 

i − x i 

E 

where 1iN−2. Let k i =1 for 1iN−2, thus x i 

= 1 2 x i+x i+1 for 1iN−2. That is, all internal equilibrium 

points x E i 0iN−2 are the midpoints of two neighboring 

switching points. According to 6, we have the recursive 

formulas of the positive switching points x i 2iN−1 as 

follows: 

x i+1 = 2 i 

j=1m j − m j−1 x j 

− x i , 8 

m i 

where 1iN−2. 

C. Multifolded torus chaotic attractors 

Numerical simulation shows that system 1 with 3 has 

a large parameter region for chaos generation. Here, assume 

that =14.5, =1.25, and x 1 =0.75. Then, we can calculate 

all other switching points x i 2iN−1 from the recursive 

formulas 8. 

In the following, two typical cases are first discussed: 

N=2 and N=5. When N=2, we have gy−x=m 1 y−x 

+ 1 2 m 0−m 1 y−x+x 1 −y−x−x 1 . Let m 0 =0.15 and m 1 

=−0.17. Figure 2a shows the PWL characteristic function 

gx with N=2. For the above set of parameters, system 1 

with 3 has a 3-folded torus chaotic attractor, as shown in 

Fig. 3a. The maximum Lyapunov exponent of this 3-folded 

torus chaotic attractor is 0.1018. It is clear that there are three 

tori folded in the chaotic attractor, as shown in Fig. 3a. 

Obviously, system 1 with 3 for N=2 has three equilibrium 

points: O0,0,0 and P ± ±1.4118,0,0 denoted by “,” 

as shown in Fig. 3a. Also, the two switching points are 

denoted by “,” as shown in Fig. 3a. 

When N=5, we have 

gy − x = m 4 y − x 

4 

+ 2 

1 m i−1 − m i y − x + x i − y − x − x i . 

i=1 

Let m 0 =−0.17, m 1 =0.15, m 2 =−0.17, m 3 =0.15, and m 4 = 

−0.17. From 8, we have x 2 =2.45, x 3 =3.95, and x 4 =5.65. 

Figure 2b shows the PWL characteristic function gx for 

N=5. Here, system 1 with 3 has a 9-folded torus chaotic 

attractor, as shown in Fig. 3b. The maximum Lyapunov 

exponent of this 9-folded torus chaotic attractor is 0.0730. It 

is clear that there are nine tori folded in the chaotic attractor, 



FIG. 2. PWL characteristic function gx. a Three segments; 

b nine segments. 

as shown in Fig. 3b. System 1 with 3 for N=5 has nine 

equilibrium points: O0,0,0, P ± 91 ±1.6,0,0, 

P ± 92 ±3.2,0,0, P ± 93 ±4.8,0,0, and P ± 94 ±6.4,0,0 denoted 

by “,” as shown in Fig. 3b. The eight switching points are 

denoted by “,” as shown in Fig. 3b. 

Similarly, system 1 with 3 has a 5-folded torus chaotic 

attractor with maximum Lyapunov exponent 0.1151 for 

parameters N=3, m 0 =−0.17, m 1 =0.15, m 2 =−0.17, and x 2 

=2.45. Also, system 1 with 3 has a 7-folded torus chaotic 

attractor with maximum Lyapunov exponent 0.0901 for parameters 

N=4, m 0 =0.15, m 1 =−0.17, m 2 =0.15, m 3 =−0.17, 

x 2 =2.0735, and x 3 =3.5735. Obviously, system 1 with 3 

can create a maximum 2N−1-folded torus chaotic attractor 

for N1, where every torus corresponds to a unique segment 

or radial of the PWL characteristic function gx. Furthermore, 

theoretical analysis and numerical simulation both 

show that the slopes of the two radials of the PWL characteristic 

function gx must be negative, as depicted in Fig. 2. 

III. DYNAMICAL BEHAVIORS OF MULTIFOLDED 

TORUS CHAOTIC SYSTEMS 

In this section, the dynamical behaviors of multifolded 

torus chaotic systems are further investigated, including 

symmetry, bifurcation, eigenspaces, approximation solutions, 

and system trajectory. 

A. Symmetry and bifurcation of 3-folded torus 

chaotic system 

Denote S + =x,y,zy=x−x 1 , S − =x,y,zy=x+x 1 , 

V 0 =x,y,zy−x x 1 , V + =x,y,zy−x−x 1 , and V − 

=x,y,zy−xx 1 . 



FIG. 3. Multifolded torus chaotic attractors. a 3-folded torus; b 9-folded 

torus. 

FIG. 4. Bifurcation diagram of 3-folded torus system 1 with 3 for N 

=2, m 0 =0.15, and m 1 =−0.17. a Parameter =1.25; b parameter 

=14.5. 

Note that system 1 is invariant under the transformation 

x,y,z→−x,−y,−z; that is, system 1 is symmetric 

about the origin. The symmetry persists for all values of the 

system parameters. Moreover, 

V = ẋ 

x + ẏ 

y + ż 

 

2 = m 0 −1 x,y,z V 0 

m 1 −1 x,y,z V ± , 

where V is the unit volume of the flow of system 1. Therefore, 

system 1 with 3 for N=2 is dissipative in V 0 for 

m 0 −10 and in V ± for m 1 −10. 

Assume that m 0 0 and m 1 0. Then V changes its 

sign at =1; 

= 

that is, 

0 x,y,z V 0 

1 

0 x,y,z V ± 

V 0 =1 

0 x,y,z V 0 

1. 

0 x,y,z V ± 

Theoretical analysis shows that system 1 with 3 for N 

=2 has a periodic repeller, which coexists with an attracting 

torus for 1. However, system 1 with 3 for N=2 has a 

periodic attractor, which coexists with a repelling torus for 

1. When =1, neither repeller nor attractor is observed 

as expected. Instead, a Hopf bifurcation occurs at =1. 8 

System 1 with 3 for N=2, m 0 =0.15, and m 1 =−0.17 

has a wide parameter region for a 3-folded torus chaotic 

attractor. Figures 4a and 4b show the bifurcation diagrams 

of parameters with =1.25 and with =14.5, 

respectively. According to the bifurcation diagrams in Fig. 4, 

it is clear that system 1 with 3 for N=2 has three different 

tori folded in a chaotic attractor. 

B. Dynamical behaviors of a 3-folded torus chaotic 

system 

From the discussion in Sec. II C, system 1 with 3 for 

N=2 has three equilibriums: O0,0,0 and 

P ± ±1.4118,0,0. Linearizing system 1 with 3 for N=2 

at equilibrium point O gives 



TABLE I. Parameters a ij , b ij ,andc ij 1i, j3. 

a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 

1.0299 −0.7954 0.3863 0.0549 −0.0424 0.0206 0.0333 −0.0257 0.0125 

b 11 b 12 b 13 b 21 b 22 b 23 b 31 b 32 b 33 

−0.0299 0.7954 −0.3863 −0.0549 1.0424 −0.0206 −0.0333 0.0257 0.9875 

c 11 c 12 c 13 c 21 c 22 c 23 c 31 c 32 c 33 

−0.0469 0.4557 0.6980 −0.0315 0.0391 0.9075 0.0602 −1.1345 0.0078 

Ẋ = m 0 − m 0 0 

m 0 − m 0 −1 

0 0 

x y 

z A 0X, 

where X=x,y,z T . The corresponding characteristic equation 

of O is 

3 + m 0 1− 2 + − m 0 =0. 

9 

10 

Let =14.5, =1.25, m 0 =0.15, and m 1 =−0.17. Solving 

10 gives 1 2.0592, 2,3 −0.0171±1.1489i, and the corresponding 

eigenvectors are v 1 0.9981,0.0532,0.0323 T 

and 

Denote 

0.2440 

v 2 = w 2 ± w 3 i 0.0086 

− 0.6328 − 0.4492 

± − 0.5816 i. 

0 

= m 1 − m 1 0 

x 

z 

Ẋ m 1 − m 1 −1 y A 1X, 

0 0 

11 

where X=x,y,z T , and its corresponding eigenvalues are 

¯ 1−2.3269, ¯ 2,30.0159±1.1506i. The corresponding 

eigenvectors are 

and 

Denote 

v¯1 0.9980,0.0559,− 0.03 

0.2258 

v¯2 = w¯ 2 ± w¯ 3i 0.0080 

0.6279 0.4696 

± 0.5780 i. 

0 

0.9981 0.2440 − 0.4492 

P = v 1 ,w 1 ,w 2 0.0532 0.0086 − 0.5816 

. 

0.0323 − 0.6328 0 

Thus, the solution of linear system 9 with initial value X 0 is 

XtP s 11 0 0 

 

0 s 22 − s 32 P−1 X 0 

0 s 32 s 22 

t 11 t 12 t 13 

t 21 t 22 t 23 

t 31 t 32 t 33 X 0, 

where s 11 =e 2.0592t , s 22 =e −0.0171t cos1.1489t, s 32 

=e −0.0171t sin1.1489t, and t ij =a ij e 2.0592t +b ij e −0.0171t 

cos1.1489t+c ij e −0.0171t sin1.1489t1i, j3, in 

which a ij ,b ij ,c ij 1i, j3 are given in Table I. 

Therefore, the unstable and stable subspaces of system 

9 are E U =Spanv 1 and E S =Spanw 2 ,w 3 , respectively. 

Figure 5a shows the stable and unstable subspaces of linear 

system 9. Note that system 9 is the linearized system of 

the 3-folded torus system 1 with 3 and N=2 at equilibrium 

point O. The unstable and stable eigenspaces of system 

x 

9981 = y 

532 = z 

323 

1 with 3 and N=2 corresponding to O are 

and 4600456x−3553172y+1725591z=0, respectively. 

Linearizing system 1 with 3 at equilibrium points P ± 

gives 

FIG. 5. Stable and unstable subspaces E S and E U of the PWL system 1 

with 3 for N=2. a In V 0 ; b in V ± . 



TABLE II. Parameters ā ij , b¯ij, and c¯ij 1i, j3. 

ā 11 ā 12 ā 13 ā 21 ā 22 ā 23 ā 31 ā 32 ā 33 

1.0363 −0.8419 −0.3619 0.0580 −0.0472 −0.0203 −0.0312 0.0253 0.0109 

b¯11 b¯12 b¯13 b¯21 b¯22 b¯23 b¯31 b¯32 b¯33 

−0.0363 0.8419 0.3619 −0.0580 1.0472 0.0203 0.03123 −0.0253 0.9891 

c¯11 c¯12 c¯13 c¯21 c¯22 c¯23 c¯31 c¯32 c¯33 

0.0461 −0.4281 0.7368 0.0295 −0.0378 0.9104 0.0635 −1.1379 −0.0083 

0.9980 0.2258 0.4696 

P¯ = v¯1,w¯ 1,w¯ 2 

0.0559 0.0080 0.5780 . 

− 0.03 0.6279 0 

The solution of linear system 11 with initial value X 0 is 

XtP¯ s¯11 0 0 

 

t¯12 t¯13 

0 s¯22 − s¯32 P¯ −1 X 0 t¯11 

t¯21 t¯22 t¯23 0 , 

0 s¯32 s¯22 t¯31 t¯32 t¯33X 

where s¯11 =e −2.3269t , s¯22 =e 0.0159t cos1.1506t, s¯32 

=e 0.0159t sin1.1506t, and t¯ij =ā ij e −2.3269t +b¯ije 0.0159t 

cos1.1506t+c¯ij e 0.0159t sin1.1506t1i, j3, in which 

ā ij ,b¯ij,c¯ij 1i, j3 are given in Table II. 

Moreover, the stable and unstable subspaces of system 

11 are E S =Spanv¯1 and E U =Spanw¯ 2,w¯ 3, respectively. 

Figure 5b shows the stable and unstable subspaces of linear 

system 11. Note that system 11 is the linearized system of 

the 3-folded torus system 1 with 3 and N=2 at equilibrium 

point P ± . The stable and unstable eigenspaces of system 

x 

1 with 3 and N=2 corresponding to P ± are 

= z 

−300 

9980 = y 

559 

and −9073155x+7371546y+3168890z=0, respectively. 

According to 10 and the Routh-Hurwitz theorem, the 

equilibrium point O0,0,0 is stable if and only if 

m 0 1− 0, m 0 0, m 0 0. 

x 

Similarly, the equilibrium point P ± 

 

1 m 0 −m 1 

 

m 1 

,0,0 is stable 

if and only if 

m 1 1− 0, m 1 0, m 1 0. 

Referring to 10, denote p=− m 0 2 1− 2 

3 

, q= 2m 0 3 1− 3 

27 

− m 01− 

3 

−m 0 , and =− m 0 4 1− 3 

27 

− m 0 2 2 1− 2 

108 

+ m 0 2 2 1− 

6 

+ 3 

27 + m 0 2 2 2 

4 

. Based on the classical formula of the solutions 

of cubic equations, solving 10 gives 1 =− m 01− 

+ 3 − q 2 + + 3 − q 2 − and 2,3 =− m 01− 

3 

− 1 2 3 − q 2 + 

+ 3 − q 2 − ± 

3 

2 i 3 − q 2 + − 3 − q 2 − 1 ± 2 i. Theoretical 

analysis and numerical simulations show that the PWL 

system 1 with 3 for N=2 can generate a 3-folded torus 

chaotic attractor under the conditions of 1 0, 1 0, and 

2 0. That is, the equilibrium point O is a one-dimensional 

unstable saddle point. 

Similarly, denote p¯ =− m 1 2 1− 2 

3 

, q¯ = 2m 1 3 1− 3 

27 

− m 11− 

3 

−m 1 , and ¯ =− m 1 4 1− 3 

27 

− m 1 2 2 1− 2 

108 

+ m 1 2 2 1− 

6 

+ 3 

27 

3 

+ m 1 2 2 2 

4 

, and the eigenvalues are ¯ 1=− m 11− 

3 

+ 3 − q¯2 + ¯ 

+ 3 − q¯2 − ¯ and ¯ 2,3=− m 11− 

3 

− 1 2 3 − q¯2 + ¯ 

+ 3 − q¯2 − ¯ ± 

3 

2 i 3 − q¯2 + ¯ − 3 − q¯2 − ¯ ¯ 1±¯ 2i. 

Theoretical 

analysis and numerical simulations show that the 

PWL system 1 with 3 and N=2 can create a 3-folded 

torus chaotic attractor under the conditions of ¯ 10, ¯ 1 

0, and ¯ 20. That is, the equilibrium points P ± are twodimensional 

unstable saddle points. 

Therefore, to generate a 3-folded torus chaotic attractor 

from 1 with 3 and N=2, one may assume that 

0, ¯ 0, 1 0, 

12 

¯ 1 0, 1 0, ¯ 1 0. 

Remark 1: The double-folded torus chaotic system 1 

with 2 and the 3-folded torus chaotic system 1 with 3 

have the same dynamical equation. The main difference lies 

in their different parameters , , m 0 , and m 1 . More importantly, 

different parameters correspond to different stable and 

unstable subspaces, which finally lead to different dynamical 

behaviors. Furthermore, different parameters may have different 

intrinsic dynamic mechanisms. It is clear that the 

double-folded torus chaotic attractor is nonsymmetrical 

about the origin O, as shown in Fig. 1; however, the 3-folded 

torus chaotic attractor is symmetrical about the origin O, as 

shown in Fig. 3a. 

C. Dynamical behaviors of multifolded torus 

chaotic systems 

The linear systems in the three domains V 0 and V ± of R 3 

are Ẋ=AX−E, where X=x,y,z T , A=A 0 , and E=O for X 

V 0 , and A=A 1 and E= P ± for XV ± . When XV ± , let 

X=X− P ± , so that Ẋ=A 1 X. Thus, the dynamical behaviors 

of the 3-folded torus system 1 with 3 and N=2 are completely 

determined by the two basic linear systems 9 and 

11. That is, the 3-folded torus chaotic attractor is generated 

via alternative switchings of the two basic linear systems 9 

and 11. 

Figure 5 shows the stable and unstable subspaces E S and 

E U of the PWL system 1 with 3 and N=2. Figure 5a 

shows the stable and unstable eigenspaces of system 1 with 

3 and N=2 corresponding to O in V 0 , where the arrows 

denote the directions of the flow. Figure 5b shows the 

stable and unstable eigenspaces of system 1 with 3 and 

N=2 corresponding to P ± in V ± , where the arrows denote the 



For N2, we can transform the multifolded torus system 

1 with 3 into two basic linear systems 9 and 11 by 

using a simple linear transformation. In the following, we 

only select N=5 as an example to explain how to construct 

the linear transformation. When N=5, construct the following 

linear transformation: 

FIG. 6. Equilibrium points and their corresponding eigenspaces of system 

1 with 3 and N=2. 

directions of the flow. Let t be the flow generated by 1 

with 3 and N=2, and X 0 be the initial value that is near O 

above T 0 but not on l 0 , where T 0 and l 0 are the stable and 

unstable eigenspaces of system 1 with 3 and N=2 corresponding 

to O, respectively. Because 1 0, t,X 0 moves 

forward with respect to the x axis while rotating clockwise 

around l 0 E U , as shown in Figs. 5a and 6. After long 

enough time, t,X 0 will hit S + and then enter V + . Due to 

the relative positions of P + and l + , t,X 0 continues to 

move forward while rotating around l + E S , as shown in Figs. 

5b and 6, where l + and T + are the stable and unstable 

eigenspaces of system 1 with 3 and N=2 corresponding 

to P + , respectively. Because ¯ 10, t,X 0 increases its 

magnitude of oscillation and eventually returns to V 0 . Then, 

t,X 0 moves backward around l 0 . After a long time, it 

eventually hits S − and then enters V − . Then, t,X 0 moves 

further backward while rotating around l − , which is the stable 

eigenspace of system 1 with 3 and N=2 corresponding to 

P − . Since ¯ 10, t,X 0 increases its magnitude of oscillation 

and finally returns to V 0 . Because 1 0, it decreases its 

magnitude of oscillation and returns to the original neighboring 

region of O. The trajectory t,X 0 is repeatedly 

stretched and folded for infinitely many times and finally 

forms a 3-folded torus chaotic attractor. Figure 6 shows the 

relative positions of all equilibrium points O, P ± and their 

eigenspaces l 0 , T 0 , l ± , and T ± . We can get the approximation 

formulas of the eigenspaces as follows: 

l 0 : 

x 

9981 = y 

532 = z 

323 , 

T 0 : 4600456x − 3553172y + 1725591z =0, 

l ± : x 1.4118 

9980 

= y 

559 = z 

− 300 , 

T ± : − 9073155x 1.4118 + 7371546y + 3168890z =0. 

Moreover, the flow t,X 0 is completely determined by 

the two basic linear systems 9 and 11. Therefore, the flow 

t,X 0 follows the approximation solution of 9 in V 0 and 

the approximation solution of 11, as verified by using a 

linear transformation X=X− P ± in V ± . In particular, the vector 

field of system 1 with 3 and N=2 is symmetrical about 

the origin. 

94 

− P − x − y − x 4 

93 

X − P − − x 4 x − y − x 3 

92 

X − P − − x 3 x − y − x 2 

91 

X − P − − x 2 x − y − x 1 

X =X X x − y x 1 

91 

X − P + x 1 x − y x 2 

92 

X − P + x 2 x − y x 3 

93 

X − P + x 3 x − y x 4 

94 

X − P + x − y x 4 . 

13 

Since m 0 =−0.17, m 1 =0.15, m 2 =−0.17, m 3 =0.15, and m 4 = 

−0.17, using the linear transformation 13, system 1 with 

3 and N=5 becomes 

= 

x − y x 1 

A 1 X or x 2 x − y x 3 

Ẋ or x − y x 4 

A 0 X x 1 x − y x 2 

or x 3 x − y x 4 . 

14 

Therefore, the 9-folded torus system 1 with 3 and N=5 is 

also generated via alternative switchings of the two basic 

linear systems 9 and 11. Similarly, the underlying dynamic 

mechanism of the 9-folded torus system 1 with 3 

and N=5 is the same as that of the 3-folded torus system 1 

with 3 and N=2. 

In the following, we compare the dynamical mechanisms 

of the 3-folded torus system 1 with 3 for N=2 with that of 

Chua’s double-scroll system. 8–10 

It is noticed that the magnitude of the real part 

Re 2,3 0.0171 or Re¯ 2,30.0159 of the pair of 

complex eigenvalues of system 9 or system 11 is less 

than that of the imaginary part Im 2,3 1.1489 or 

Im¯ 2,31.1506 of the pair of complex eigenvalues of 

system 9 or system 11 by two quantity levels O10 2 . 

Moreover, the magnitude of the real part characterizes the 

increasing or damping speed of the amplitude of the oscillator. 

However, the magnitude of the imaginary part characterizes 

the frequency of the oscillation. For the 3-folded torus 

system 1 with 3 and N=2, the increasing or damping 

speed of the amplitude of the oscillator is relatively much 

smaller than the frequency of the oscillator. Therefore, the 

3-folded torus system 1 with 3 and N=2 can easily form 

increased or damped oscillations in every PWL region. That 

is, every PWL region can generate a torus. 

The familiar Chua’s circuit is described by 8 



FIG. 7. Circuit diagram of multifolded 

torus chaotic attractors. 

ẋ = „y − fx… 

ẏ = x − y + z 

ż =−y, 

15 

where fx=m 1 x+ 1 2 m 0−m 1 x+x 1 −x−x 1 . When =10, 

=15, m 0 =− 1 7 , and m 1= 2 7 

, system 15 has a double-scroll 

attractor. 8 

Denote V¯ 0=x,y,zx x 1 , V¯ +=x,y,zxx 1 , and 

V¯ −=x,y,zx−x 1 . When XV¯ 0, system 15 has a 

unique equilibrium point O and the corresponding eigenvalues 

are 1 2.4777, 2,3 −1.0246±2.7566i. When XV¯ + 

or XV¯ −, system 15 has a unique equilibrium point 

P¯ + m 1−m 0 

m 1 

x 1 ,0,− m 1−m 0 

m 1 

x 1 

or P¯ −− m 1−m 0 

m 1 

x 1 ,0, m 1−m 0 

m 1 

x 1 

and the 

corresponding eigenvalues are ¯ 1−4.3290,¯ 2,3 

0.2359±3.1376i. Obviously, when XV¯ 0, the magnitudes 

of the real part Re 2,3 1.0246 and the imaginary part 

Im 2,3 2.7566 are in the same quantity level O1. 

Thus, the increasing or damping speed of the amplitude of 

the oscillator is relatively much bigger than the frequency of 

the oscillator. Hence, system 15 cannot create a scroll in 

V¯ 0. However, when XV¯ +or XV¯ −, the magnitude of the 

real part Re¯ 2,3 0.2359 is less than that of the imaginary 

part Im¯ 2,3 3.1376 by one quantity level O10. Thus, 

the increasing or damping speed of the amplitude of the oscillator 

is relatively much smaller than the frequency of the 

oscillator. Therefore, system 15 can generate a scroll in V¯ + 

or V¯ −. 

Remark 2: In conclusion, all multifolded torus chaotic 

systems have the same dynamical mechanism; that is, all 

multifolded torus chaotic attractors are generated via alternative 

switchings between two basic linear systems 9 and 

11. Moreover, all PWL regions of the characteristic function 

gx with a positive slope correspond to the linear system 

9, and the PWL regions of the characteristic function 

gx with a negative slope correspond to the linear system 

11. In particular, the two outer regions correspond to the 

negative slope. However, the dynamical mechanism of the 

multifolded torus chaotic attractors is very different from that 

of Chua’s double-scroll. 8 

IV. CIRCUIT IMPLEMENTATION FOR MULTI-FOLDED 

TORUS CHAOTIC ATTRACTORS 

This section designs a novel circuit to experimentally 

verify the multifolded torus chaotic attractors. 

A. Fundamental principle of circuit design 

Based on the above theoretical analysis, a block circuit 

diagram is designed for generating multifolded torus chaotic 

attractors, as shown in Fig. 7. This circuit is described by 



FIG. 8. Equivalent circuit of adjustable inductance and 

capacitance. 

where 

dv 

C C1 

1 =−fv C2 − v C1 

dt 

dv C2 

C 2 =−fv C2 − v C1 − i L 

dt 

L di L 

dt = v C2, 

N−1 

16 

fv C2 −v C1 =G N−1 v C2 −v C1 + 1 2 G i−1 −G i v C2 −v C1 

+E i −v C2 −v C1 −E i . 

According to 16, we have 

dv C1 

dt 

dv C2 

dt 

dRi L 

dt 

=− 1 

RC 2 

C 2 

C 1 

Rfv C2 − v C1 

i=1 

=− 1 

RC 2 

Rfv C2 − v C1 + Ri L 

= 1 

RC 2 

R 2 C 2 

L v C2. 

17 

Comparing 1 with 17, we get the following equivalent 

relationships: 

0 = RC 2 , = t , = C 2 

= 14.5, 

t 0 C 1 

= R2 C 2 

L 

= 1.25, x = v C1 

, y = v C2 

, z = Ri L 

, 

V BP V BP V BP 

x i = E i 

, G = 1 18 

V BP R , G i = m i G0 i N −1, 

fv C2 − v C1 = 1 gy − x, 

R 

where V BP =1 V, and 1 0 

= 1 

RC 2 

is the time-scale transformation 

factor. Let R=1 k. From 18, we have C 1 =1.29 nF, C 2 

=18.75 nF, and L=15 mH. 

The subcircuitry N S in Fig. 7 is the subtraction generator 

and its output is v C2 −v C1 . The subcircuitry N R in Fig. 7 is 

the generator of the PWL characteristic function fv C2 

−v C1 and its input and output satisfy the condition I N 

= fv C2 −v C1 . Moreover, we can rigorously calculate the theoretical 

values of all resistors in N R by using the recursive 

formulas. 25 The operational amplifier is selected as type 

TL082, and the supply voltage of electrical source is ±E C 

=±15V. Thus, the saturating voltage of the operational amplifier 

is E sat =14.3V. In the following, we calculate the theoretical 

values of all resistors in N R for the experimental 

confirmation of the 3-, 5-, 7-, and 9-folded torus chaotic 

attractors. 

To generate a 3-folded torus chaotic attractor, we rigorously 

calculate the theoretical values of the resistors in N R , 

based on the given parameters in Sec. II, as follows: 

G 0 = m 0 

R = 0.15 mS, G 1 = m 1 

= − 0.17 mS, 

R 

E 1 = x 1 V BP , r 1 = R 12 

=−G 1 R 2 = 0.34, 

R 11 

r 2 = R 22 

= E sat 

− 1 = 18.07, 

R 21 E 1 

r 3 = R 32 1+r 2 

=− 

R 31 R 2 G 1 − G 0 − 1 = 28.79. 

19 

Similarly, we can rigorously calculate the theoretical values 

of the resistors in N R for the 5-, 7-, and 9-folded torus 

chaotic attractors based on the given parameters in Sec. II, as 

follows: G 0 =−0.17 mS, G 1 =0.15 mS, G 2 =−0.17 mS, E i 

=x i V BP i=1,2, r 1 =0.34, r 2 =4.84, r 3 =8.12, r 4 =19.1, and 

r 5 =28.79 for a 5-folded torus chaotic attractor; G 0 

=0.15 mS, G 1 =−0.17 mS, G 2 =0.15 mS, G 3 =−0.17 mS, E i 

=x i V BP i=1,2,3, r 1 =0.34, r 2 =3.00, r 3 =5.25, r 4 =6.90, r 5 

=9.78, r 6 =18.07, and r 7 =28.79 for a 7-folded torus chaotic 

attractor; G 0 =−0.17 mS, G 1 =0.15 mS, G 2 =−0.17 mS, G 3 

=0.15 mS, G 4 =−0.17 mS, E i =x i V BP 1i4, r 1 =0.34, r 2 

TABLE III. Status of all switches, ratios of resistors r n = R n2 

R n1 

1n9, and number of folded tori. 

K 1 K 2 K 3 K 4 K 5 r 1 r 2 r 3 r 4 r 5 r 6 r 7 r 8 r 9 Number of tori 

on on off off off 0.34 18.1 28.8 3 

on on on off off 0.34 4.84 8.12 19.1 28.8 5 

on on on on off 0.34 3.00 5.25 6.90 9.78 18.1 28.8 7 

on on on on on 0.34 1.53 2.95 3.62 4.66 4.84 8.12 19.1 28.8 9 



TABLE IV. Status of switches, resistors R n2 =r n R n1 1n9, and number of folded tori. 

K 1 K 2 K 3 K 4 K 5 R 12 R 22 R 32 R 42 R 52 R 62 R 72 R 82 R 92 Number of tori 

on on off off off 3.4 181 28.8 3 

on on on off off 3.4 48.4 8.12 191 28.8 5 

on on on on off 3.4 30.0 5.25 69.0 9.78 181 28.8 7 

on on on on on 3.4 15.3 2.95 36.2 4.66 48.4 8.12 191 28.8 9 

=1.53, r 3 =2.95, r 4 =3.62, r 5 =4.66, r 6 =4.84, r 7 =8.12, r 8 

=19.1, and r 9 =28.79 for a 9-folded torus chaotic attractor. 

B. Circuit implementation 

In the circuit design, we select all operational amplifiers 

shown in Fig. 7 to be type TL082. The supply voltage of the 

electrical source is ±E C = ±15 V, and the saturating voltages 

of the operation amplifiers are E sat =14.3 V. Moreover, all 

resistors in Fig. 7 are exactly adjustable resistors or potentiometers. 

Note also that the capacitances C 1 , C 2 , and inductance 

L are adjustable, as shown in Fig. 8. Therefore, one can 

adjust the real parameter values of and by tuning the 

capacitances C 1 , C 2 , and inductance L. 

For the PWL function generator N R in Fig. 7, all 

unknown resistors R n2 1n9 can be rigorously calculated 

by using a formula similar to 19 as shown in Tables 

III and IV. 

FIG. 9. Experimental observations of the multifolded torus chaotic attractors. From left to right: a 3-folded torus, where x=0.86V/div and y=0.6V/div; b 

5-folded torus, where x=1.25V/div and y=0.64V/div; c 7-folded torus, where x=1.25V/div and y=0.64V/div; d 9-folded torus, where x=1.6V/div and 

y=0.9V/div. 



Let R 1 =100 k, R 2 =2 k, R 31 =R 51 =R 71 =R 91 =1 k, 

and R 11 =R 21 =R 41 =R 61 =R 81 =10 k. According to a formula 

similar to 19 and R n2 =r n R n1 1n9, we can rigorously 

calculate all resistors in N R . 

Thus, the circuit diagram Fig. 7 can be controlled based 

on Tables III and IV, to generate 3-, 5-, 7-, and 9-folded torus 

chaotic attractors via the switchings of the switches 

K i 1i5. Figure 9 shows the experimental observation 

results for 3-, 5-, 7-, and 9-folded torus chaotic attractors. 

Remark 3: The real measurement values of r n and R n2 

for 1n9 in the circuit experiment may have a small departure 

from the theoretically calculated values shown in 

Tables III and IV, due to the discrete nature of real circuit 

parameters and the measurement errors. These differences 

can be corrected via a small adjustment of the resistors R n2 

for 1n9 in the circuit experiment. Our experimental results 

show that it is technically very difficult to implement a 

chaotic attractor with more than nine tori by analog circuits 

because a circuit is highly sensitive to small variations of 

parameters with the increased number of tori. For example, 

the measurement precision of the switching points x 2 

=2.0735 and x 3 =3.5735 of the 7-folded torus attractor has to 

pinpoint to the fourth digit of the decimal. Here, our circuit 

can realize up to a maximum of 9-folded tori in the chaotic 

attractor under the real experimental conditions. 

V. CONCLUDING REMARKS 

This paper has developed a systematic methodology for 

generating multifolded torus chaotic attractors from a simple 

three-dimensional autonomous circuit. Recursive formulas 

for system parameters have been rigorously derived, useful 

for improving hardware implementation. Dynamical behaviors 

of the multifolded torus system, including symmetry, 

bifurcation, eigenspaces, and conditions for chaos generation, 

have also been investigated. Our theoretical analysis 

shows that multifolded torus chaotic attractors can be generated 

via alternative switchings between two linear systems. A 

simple circuit diagram has been designed for experimentally 

verifying 3-, 5-, 7-, and 9-folded torus chaotic attractors. 

This is the first time in the literature that an experimental 

realization of a 9-folded torus chaotic attractor has been reported. 

The circuit design method developed in this paper outperforms 

the existing methods, in the sense that all system 

design parameters and physical circuit parameters can be rigorously 

derived beforehand by our new techniques. Although, 

as is well known, there are many technical reasons 

that cause difficulties in hardware implementation of multifolded 

torus chaotic attractors, our approach has been physically 

implemented via circuitry, which can generate up to 

9-fold torus chaotic attractors visible on the oscilloscope, 

showing the effectiveness and realizability of the proposed 

methodology. 

ACKNOWLEDGMENTS 

This work was supported by the National Natural Science 

Foundation of China under Grants No. 60304017, No. 

20336040, and No. 60572073, the National Key Basic Research 

and Development 973 Program of China under Grant 

2006CB708202, the Scientific Research Startup Special 

Foundation on Excellent Ph.D. Thesis and Presidential 

Award of Chinese Academy of Sciences, the Natural Science 

Foundation of Guangdong Province under Grants No. 32469 

and No. 5001818, the Science and Technology Program of 

Guangzhou City under Grant No. 2004J1-C0291, and the 

Strategic Research Grants of the City University of Hong 

Kong under Grant No. 7001702/EE. 

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Multifolded torus chaotic attractors: Design and implementation

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