A New Three-Scroll Uniﬁed Chaotic System...

ISSN 1749-3889 (print), 1749-3897 (online)International Journal of Nonlinear Science

Vol.10(2010) No.4,pp.462-474

A New Three-Scroll Unified Chaotic System Coined

Lin Pan1,2 ∗, Wuneng Zhou1, Jian’an Fang1 , Dequan Li3,41College of Information Science and Technology, Donghua University, 201620 Shanghai, P.R. China

2Embedded Systems Group, Fernuniversitat in Hagen, 58084 Hagen, Germany3Department of Mathematics and Physics, Anhui University of Science and Technology, Huainan, 232001 Anhui

Province, P.R. China4Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, P.R. China

(Received 10 September 2009 , accepted 10 March 2010)

Abstract: This paper introduces a new Three-Scroll Unified Chaotic System (TSUCS) that contains theLorenz-like subsystem and the Chen-like subsystem as two extremes of its parameter spectrum. The newsystem represents the continued transition from the Lorenz-like subsystem to the Chen-like subsystem andis chaotic over the entire spectrum of the key system parameter. The basic dynamical properties, such asLyapunov exponents, fractal dimension, Poincare map and chaotic dynamical behaviors of the new unifiedchaotic system are studied, either numerically or analytically. Simulation results clearly show that this is anovel unified chaotic system and deserves further detailed investigation.

Keywords: Three-Scroll Unified Chaotic System (TSUCS); TS fuzzy model; Lorenz-like attractor; Chen-like attractor; Lu-like attractor

1 IntroductionSince Lorenz found the first chaotic attractor in a smooth three-dimensional autonomous system [1], more and morechaotic attractors were developed, such as the Rossler system [2], the Sprott system [3], the Chen system [4], the Lu sys-tem [5, 6], the generalized Lorenz system family [7], and the hyperbolic type of the generalized Lorenz canonical form [8].The Lorenz system, as the first chaotic model, revealed the complex and fundamental behaviors of the nonlinear dynam-ical systems. Notably, during the development history of chaos theory, considerable research interests have been madein searching for new chaotic attractors in simple three-dimensional autonomous systems. In this pursuit, two categoriesof research directions were developed. One direction is to find the simplest three-dimensional chaotic flow, such as thethree-dimensional autonomous smooth chaotic system found by Rossler [2]. The other is to develop autonomous chaoticflows that are algebraically simpler. It is notable that the Rossler attractor and the Sprott’s attractor are both algebraicallysimpler than the celebrated two-scroll Chen attractor, because they have a single nonlinear term while the Chen systemhas two [9–12]. Due to theoretical interests and important potential applications in various technologies and informationsystems based on chaos, a new research direction has been investigated which aims to generate three-dimensional smoothautonomous chaotic systems with complicated topological structures. In this effort, some new chaotic systems and chaoscontrols were recently developed [13–22].

However, a complete classification of all possible types for the known three-dimensional chaotic attractors withtheir corresponding topological properties is still lacking. In this pursuit, a new Three-Scroll Unified Chaotic System(TSUCS) is discussed in this paper, which is derived from the generalized Lorenz system and Rossler system via chaos-controlling/anti-controlling from its continuous-time TS fuzzy model and with the non-resonant parametric perturbationapproach. The rest of the paper is organized as follows. Some system descriptions are introduced in Section 2. A novelTSUCS is put forward and the basic properties of the proposed chaotic system are discussed in Section 3 and Section 4,respectively. Finally, conclusions are drawn in Section 5.

∗Corresponding author. E-mail address: Tel. : +49-2331-987-1731; Fax: +49-2331-987-375.E-mail address: [email protected](L. Pan), [email protected](w. Zhou), [email protected]

Copyright c⃝World Academic Press, World Academic UnionIJNS.2010.12.30/429

L. Pan,W. Zhou,J. Fang,D. Li: A New Three-Scroll Unified Chaotic System Coined 463

2 System DescriptionAbout forty years ago, Lorenz found the first classical chaotic system called Lorenz system, and then in 2002, Lu andChen found the Lu system which is described by the following simple three-dimensional autonomous system [7, 8]:⎧⎨⎩

�� = 𝑎(𝑦 − 𝑥),

�� = −𝑥𝑧 + 𝑐𝑦,

�� = 𝑥𝑦 − 𝑏𝑧.

(1)

where 𝑎 = 36, 𝑏 = 3, and 𝑐 = 20. System (1) is chaotic and exhibits the attractor shown in Fig. 1.

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z(d)

Figure 1: Phase portraits of Lu attractor in (a) the 𝑥− 𝑦 − 𝑧 space; and projected on (b) the 𝑥− 𝑦 plane, (c) the 𝑥− 𝑧 plane, and (d)the 𝑦 − 𝑧 plane.

The from the topological point of view simplest chaotic system with the symmetry property is the Rossler system asdescribed by the following equations [2, 3]: ⎧⎨⎩

�� = −(𝑦 + 𝑧),

�� = 𝑥+ 𝑎𝑦,

�� = 𝑧(𝑥− 𝑐) + 𝑏.

(2)

where 𝑎 = 0.2, 𝑏 = 0.2, and 𝑐 = 5.7. System (2) is chaotic and exhibits an attractor with only one scroll as shown inFig. 2.

To explore forming mechanism and compound structure of the new chaotic system, a hybrid or non-resonant paramet-ric perturbed Takagi-Sugeno fuzzy model for the Lu system is proposed as follows [14, 17, 18]:

Rule 1: IF 𝑥(𝑡) is about 𝐹𝑚𝑖𝑛 , THEN

�� = (𝐴1 +1

2𝑇1𝐴

21). (3)

Rule 2: IF 𝑥(𝑡) is about 𝐹𝑚𝑎𝑥 , THEN

�� = (𝐴2 +1

2𝑇2𝐴

22). (3′)

Here, 𝑇1 and 𝑇2 are two controller parameters,

𝐴1 =

⎛⎝ −𝜎 𝜎 0𝜌 𝛾 −𝑀1

0 𝑀1 −𝛽

⎞⎠ ,

𝐴2 =

⎛⎝ −𝜎 𝜎 0𝜌 𝛾 −𝑀2

0 𝑀2 −𝛽

⎞⎠ ,

IJNS homepage: http://www.nonlinearscience.org.uk/

464 International Journal of Nonlinear Science, Vol.10(2010), No.4, pp. 462-474

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Figure 2: Phase portraits of Rossler attractor in (a) the 𝑥− 𝑦 − 𝑧 space; and projected on (b) the 𝑥− 𝑦 plane, (c) the 𝑥− 𝑧 plane, and(d) the 𝑦 − 𝑧 plane.

𝑋 = (𝑥, 𝑦, 𝑧)𝑇 , with membership functions 𝐹𝑚𝑖𝑛 and 𝐹𝑚𝑎𝑥 given as 𝐹𝑚𝑖𝑛 = −𝑥+𝑀2

𝑀2−𝑀1and 𝐹𝑚𝑎𝑥 = 𝑥−𝑀1

𝑀2−𝑀1, and the

[𝑀1,𝑀2] chosen as [−30, 30].A new forming mechanism, named compression and pull forming mechanism [14], consists of varying the non-

resonant controller parameters 𝑇1 and 𝑇2 of the hybrid Takagi-Sugeno fuzzy model-based Lu system (1), and results inthe Lu attractor to assume more abundant and complex dynamical behavior. By simply varying the controller parameter,when 𝑇1 = 𝑇2, the Lu attractor can be compressed and distorted along the double scrolls, or the connect parts of thedouble scrolls can gradually become dominant, and pull the double scrolls high into the vertical direction (or alongthe diagonal direction of the two scrolls). Hence, the question arises whether a three-dimensional smooth autonomouschaotic system exists which produces a chaotic attractor. Under the simplifying condition 𝑀1 = 𝑀2 and by defusingthe hybrid Takagi-Sugeno fuzzy Lu model (3), we obtain after some tedious algebraic manipulations a three-dimensionaltime-varying system of the following form: ⎧⎨⎩

�� = 𝐴𝑥+𝐵𝑦 + 𝐶𝑥𝑧,

�� = 𝐷𝑥+ 𝐸𝑦 + 𝐹𝑥𝑧,

�� = 𝐺𝑥2 +𝐻𝑥𝑦 + 𝐼𝑧.

(4)

where the parameters of the system (4) are given as 𝐴 = −𝜎 + 0.5𝑇 (𝜎2 + 𝜎𝜌), 𝐵 = 𝜎 + 0.5𝑇 (−𝜎2 + 𝜎𝛾), 𝐶 =−0.5𝑇𝜎, 𝐷 = 𝜌 + 0.5𝑇 (−𝜎𝜌 + 𝛾𝜌), 𝐸 = 𝛾 + 0.5𝑇 (𝛾2 + 𝜎𝜌 + 𝑀1𝑀2), 𝐹 = −1 + 0.5𝑇 (−𝛾 + 𝛽), 𝐺 = 0.5𝑇𝜌,𝐻 = 1 + 0.5𝑇 (𝛾 − 𝛽), 𝐼 = −𝛽 + 0.5𝑇 (𝛽2 +𝑀1𝑀2), with 𝑇 = 𝑇1 = 𝑇2 and 𝑎 = 36, 𝑏 = 3, 𝑐 = 20. One can easilysee that (4) is essentially a time-varying system when the variable 𝑇 varies, and Lu system (1) is just a special case of(4) for 𝑇 = 0, 𝜎 = 𝑎, 𝜌 = 0, 𝛾 = 𝑐, and 𝛽 = 𝑏. Hence, by varying the non-resonant control parameter 𝑇 , abundantand complex dynamical behavior of the compound structures of the chaotic Lu attractor can be observed [13], which maycontribute to better understand the essence of the Lu system and its generalization.

To obtain the aforementioned three-scroll chaotic attractor from a three-dimensional smooth autonomous system hav-ing a similar basic algebraic structure as the Lu system, we let all parameter values of (4) be constant and 𝐴 = −𝐵 = −𝑎,𝐶 = 𝑑, 𝐷 = 0, 𝐸 = 𝑓 , 𝐹 = −1, 𝐺 = −𝑒, 𝐻 = 1, 𝐼 = 𝑐. The Lu attractor can be compressed and distorted alongthe double scrolls. Then, with trial-and-error numerical simulations, we found a new chaotic attractor from the followingthree-dimensional quadratic autonomous system with four smooth quadratic terms:⎧⎨⎩

�� = 𝑎(𝑦 − 𝑥) + 𝑑𝑥𝑧,

�� = −𝑥𝑧 + 𝑓𝑦,

�� = −𝑒𝑥2 + 𝑥𝑦 + 𝑐𝑧,

(5)

which has the strange attractor shown in Fig. 3. The initial values of the system are selected as (0.1, 1.0,−0.1). Usingthe MATLAB software, the numerical simulation was carried out with 𝑎 = 40, 𝑐 = 5

6 , 𝑑 = 0.50, 𝑒 = 0.65, and 𝑓 = 20.This new strange attractor has recently been called Lu-like attractor.

IJNS email for contribution: [email protected]


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Figure 3: Phase portraits of the Lu-like attractor in (a) the 𝑥− 𝑦 − 𝑧 space; and projected on (b) the 𝑥− 𝑦 plane, (c) the 𝑥− 𝑧 plane,and (d) the 𝑦 − 𝑧 plane.

Continue to use the above method, we can find the other two attractors: Lorenz-like attractor and Chen-like attractor.In order to obtain the aforementioned three-scroll chaotic attractor from a three-dimensional smooth autonomous systemwhich has a similar basic algebraic structure as the Lorenz system, we let all parameter values of (4) be constant and𝐴 = −𝐵 = −𝑎, 𝐶 = 𝑑, 𝐷 = 𝑏, 𝐸 = 𝑓 , 𝐹 = −1, 𝐺 = −𝑒, 𝐻 = 1, 𝐼 = 𝑐. The Lorenz attractor can be compressedand distorted along the double scrolls. Then, with trial-and-error numerical simulations, we found a new chaotic attractorfrom the following three-dimensional quadratic autonomous system with four smooth quadratic terms:⎧⎨⎩

�� = 𝑎(𝑦 − 𝑥) + 𝑑𝑥𝑧,

�� = 𝑏𝑥− 𝑥𝑧 + 𝑓𝑦,

�� = −𝑒𝑥2 + 𝑥𝑦 + 𝑐𝑧,

(6)

which has the strange attractor shown in Fig. 4. The initial values of the system are selected as (0.1, 1.0,−0.1). Usingthe MATLAB software, the numerical simulation was carried out with 𝑎 = 40, 𝑏 = 55, 𝑐 = 11

6 , 𝑑 = 0.16, 𝑒 = 0.65, and𝑓 = 20. This new strange attractor has recently been called Lorenz-like attractor.

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Figure 4: Phase portraits of the Lorenz-like attractor in (a) the 𝑥 − 𝑦 − 𝑧 space; and projected on (b) the 𝑥 − 𝑦 plane, (c) the 𝑥 − 𝑧

plane, and (d) the 𝑦 − 𝑧 plane.

At last, in order to obtain the aforementioned three-scroll chaotic attractor from a three-dimensional smooth au-tonomous system which has a similar basic algebraic structure as the Chen system, we let all parameter values of (4) beconstant and 𝐴 = −𝐵 = −𝑎, 𝐶 = 𝑑, 𝐷 = 𝑓 − 𝑎, 𝐸 = 𝑓 , 𝐹 = −1, 𝐺 = −𝑒, 𝐻 = 1, 𝐼 = 𝑐. The Chen attractor can



be compressed and distorted along the double scrolls. Then, with trial-and-error numerical simulations, we found a newchaotic attractor from the following three-dimensional quadratic autonomous system with four smooth quadratic terms:⎧⎨⎩

�� = 𝑎(𝑦 − 𝑥) + 𝑑𝑥𝑧,

�� = (𝑓 − 𝑎)𝑥− 𝑥𝑧 + 𝑓𝑦,

�� = −𝑒𝑥2 + 𝑥𝑦 + 𝑐𝑧,

(7)

which has the strange attractor shown in Fig. 5. The initial values of the system are selected as (0.1, 1.0,−0.1). Using theMATLAB software, the numerical simulation was carried out with 𝑎 = 60, 𝑐 = 5

6 , 𝑑 = 0.40, 𝑒 = 0.65, and 𝑓 = 25. Thisnew strange attractor has recently been called Chen-like attractor.

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Figure 5: Phase portraits of the Chen-like attractor in (a) the 𝑥− 𝑦− 𝑧 space; and projected on (b) the 𝑥− 𝑦 plane, (c) the 𝑥− 𝑧 plane,and (d) the 𝑦 − 𝑧 plane.

3 A New Three-Scroll Unified Chaotic System (TSUCS)Vanecek and Celikovsky [1996] classified a generalized Lorenz system family of the form, 𝑌 = 𝐴𝑋 + 𝑓(𝑋), where 𝐴is a instant coefficient matrix and 𝑓(𝑋) represents the nonlinear term, in terms of its linear part 𝐴 = [𝑎𝑖𝑗 ], 𝑖, 𝑗 = 1, 2, 3,with conditions, 𝑎12𝑎21 > 0, 𝑎12𝑎21 = 0 and 𝑎12𝑎21 < 0.

A novel Three-Scroll Unified Chaotic System (TSUCS) is constructed and described by⎧⎨⎩�� = (20𝛼+ 40)(𝑦 − 𝑥) +

6𝛼+ 4

25𝑥𝑧,

�� = (55− 90𝛼)𝑥+ (5𝛼+ 20)𝑦 − 𝑥𝑧,

�� = −13

20𝑥2 + 𝑥𝑦 +

11− 6𝛼

6𝑧.

(8)

where 𝛼 ∈ [0, 1]. System (8) is chaotic for 𝛼 ∈ [0, 1], which is shown in Fig. 6 and the system can be re-written asSystem (8) can be rewritten as

�� = 𝐴 ⋅𝑋𝑇 + 𝑥 ⋅𝐵 ⋅𝑋𝑇 ,

where 𝑋 = (𝑥, 𝑦, 𝑧),

𝐴 =

⎛⎝ 𝑎11 𝑎12 𝑎13𝑎21 𝑎22 𝑎23𝑎31 𝑎32 𝑎33

⎞⎠ =

⎛⎝ −(20𝛼+ 40) 20𝛼+ 40 055− 90𝛼 5𝛼+ 20 0

0 0 11−6𝛼6

⎞⎠ ,

𝐵 =

⎛⎝ 𝑏11 𝑏12 𝑏13𝑏21 𝑏22 𝑏23𝑏31 𝑏32 𝑏33

⎞⎠ =

⎛⎝ 0 0 6𝛼+425

0 0 −1− 13

20 1 0

⎞⎠ .

The critical value is decided by 𝑎12𝑎21, by which the new TSUCS (8) can be classified into three subsystems.



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Figure 6: Phase portraits of the new TSUCS attractor in (a) the 𝑥 − 𝑦 − 𝑧 space; and projected on (b) the 𝑥 − 𝑦 plane, (c) the 𝑥 − 𝑧

plane, and (d) the 𝑦 − 𝑧 plane.

1. Lorenz-like subsystem: When 𝛼 ∈ [0, 5590 ), one has 𝑎12𝑎21 < 0, and the system has two positive Lyapunov

exponents, 𝐿1 = 0.242627 and 𝐿2 = 0.030076, and one negative Lyapunov exponent, 𝐿3 = −2.055068 when𝛼 = 0. Thus, subsystem 𝑆− exhibits chaos as shown in Fig. 4 [17, 18].

2. Lu-like subsystem: When 𝛼 = 5590 , it holds 𝑎12𝑎21 = 0, and the system has a positive Lyapunov exponent,

𝐿1 = 0.0289, and two negative Lyapunov exponents, 𝐿2 = −0.0312, 𝐿3 = −1.4812. Thus, the second systemexhibits chaos, whose corresponding chaotic attractor is shwon in Fig. 3.

3. Chen-like subsystem: When 𝛼 ∈ ( 5590 , 1], it holds 𝑎12𝑎21 > 0, and the system has a positive Lyapunov exponent𝐿1 = 0.6971, and two negative Lyapunov exponents, 𝐿2 = −0.0392 and 𝐿3 = −3.7549 when 𝛼 = 1. Thus,system 𝑆+ exhibits chaos shown in Fig. 5.

All the Lyapunov exponents of the new TSUCS are shown in Fig. 7. The largest Lyapunov exponent of the system ispositive, Thus, the new united system exhibits chaos.

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Lyap

unov

exp

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L3

Figure 7: Lyapunov exponents for the new TSUCS.

From a control engineering point of view, the new chaotic attractor provides another interesting test platform for ad-vanced control techniques, because it has higher complexity than the known ordinary unified chaotic system and Rossler’ssystem [8, 11]. Moreover, it is generally more difficult to control the new system due to its rapid change in velocity in the𝑧-direction [17, 18].



4 Properties of the Proposed System

In this section, the basic properties of the proposed system (8) are given.

4.1 Symmetry and invariance

Firstly, it is easy to see the invariance of system term under the coordinate transformation (𝑥, 𝑦, 𝑧) → (−𝑥,−𝑦, 𝑧), i.e.,the TSUCS has rotation symmetry around the z-axis as for the ordinary unified chaotic system [8]. But unlike the unifiedchaotic system, where the chaotic dynamics only remain confined to positive half-space with respect to the z state variable(see Fig. 8 (c)), the proposed system’s chaotic dynamics can extend to the negative half-space with respect to the z statevariable (see Fig. 9 (c)). The time response of the unified chaotic system and the proposed system–TSUCS are shown inFig. 8 and Fig. 9, respectively.

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Figure 8: The Time response state of the unified chaotic system: (a) the Time response state 𝑥, (b) the Time response state 𝑦, (a) theTime response state 𝑧.

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Figure 9: The Time response state of the new TSUCS: (a) the Time response state 𝑥, (b) the Time response state 𝑦, (a) the Timeresponse state 𝑧.



4.2 Dissipativity and the existence of attractorFor dynamical system (8), we can obtain

∇𝑉 =∂��

∂𝑥+

∂��

∂𝑦+

∂��

∂𝑧

= −(20𝛼+ 40) +6𝛼+ 4

25𝑧 + (5𝛼+ 20) +

11− 6𝛼

6

= −16𝛼− 20 +11

6+

6𝛼+ 4

25𝑧

= 𝑔

(9)

as long as ∇𝑉 < 0 at 𝑧 < 25(96𝛼+109)12(3𝛼+2) , that is 𝑧 < 85.4167 then 𝑔 < 0, where 𝑔 is a negative value. Dynamical system

(8) is one dissipative system, and an exponential contraction of the system (8) is 𝑑𝑉𝑑𝑡 = 𝑒𝑔 = 𝑒−2, where 𝑔 is around −2.

In the dynamical system (8), a volume element 𝑉0 is apparently contracted by the flow into a volume element 𝑉0𝑒𝑔𝑡 =

𝑉0𝑒−2𝑡 at time 𝑡. It means that each volume containing the trajectory of this dynamical system shrinks to zero as 𝑡 → ∞

at an exponential rate 𝑔. So, all this dynamical system orbits are eventually confined to a specific subset that have zerovolume, the asymptotic motion settles onto an attractor of the system (8).

4.3 Lyapunov exponentThe Lyapunov exponent spectrum of system (8) is found to be 𝐿1 = 0.1566, 𝐿2 = −0.1864, 𝐿3 = −3.7889 forinitial value (0.1, 1.0,−0.1) when 𝛼 = 1, while, the maximal Lyapunov exponents are found to be 𝐿𝑚𝑎𝑥1 = 0.6971,𝐿𝑚𝑎𝑥2 = −0.0392, 𝐿𝑚𝑎𝑥3 = −3.7549 (see Fig. 7). Therefore, the Lyapunov dimension of this system is

𝐷𝐿 = 𝑗 +1

𝐿𝑗+1

𝑗∑𝑖=1

𝐿𝑖 = 2 +𝐿1 + 𝐿2

∣𝐿3∣ = 2 +0.1566− 0.1864

∣ − 3.7889∣ = 1.9921,

which means the system (8) is really a dissipative system, and the Lyapunov dimension of this system is fractional. Thefractal nature of an attractor does not merely mean this system has non-periodic orbits; it also causes nearby trajectoriesto diverge. As all strange attractors, orbits that are initiated from different initial conditions soon reach the attracting set,but two nearby orbits does not stay close to each other, they soon diverge and follow utterly different trajectories in theattractor. Therefore, there is really strange chaos in this system. More interestingly, the third Lyapunov exponent 𝐿3 of thenew TSUCS (see Fig. 10) is about −1.4812 and that of the Chen system is about −12.0003 (see Fig. 11) [6, 7], thereforethe new chaotic system has a slower contracting rate in phase space than the Chen system’s, which means the proposedsystem has a lower degree of orbital disorder and randomness.

4.4 EquilibriaFirstly, Let ⎧⎨⎩

(20𝛼+ 40)(𝑦 − 𝑥) +6𝛼+ 4

25𝑥𝑧 = 0,

(55− 90𝛼)𝑥+ (5𝛼+ 20)𝑦 − 𝑥𝑧 = 0,

− 13

20𝑥2 + 𝑥𝑦 +

11− 6𝛼

6𝑧 = 0,

(10)

The system has three equilibria points. For the convenience, here we take a special case to detailed describe the TSUCSwhen 𝛼 = 1. The TSUCS become Chen-like system and has three equilibria points which are described as follows,respectively: 𝑆0 = (0, 0, 0), 𝑆1 = (−𝑥0,−𝑦0, 𝑧0), 𝑆2 = (𝑥0, 𝑦0, 𝑧0). After mathematical manipulations, we obtain

𝑆0 = (0, 0, 0), 𝑆1 = (−𝑘,−𝑎+𝑎𝑑−𝑑𝑓𝑎+𝑑𝑓 𝑘, 2𝑎𝑓−𝑎2

𝑎+𝑑𝑓 ), 𝑆2 = (𝑘, 𝑎+𝑎𝑑−𝑑𝑓𝑎+𝑑𝑓 𝑘, 2𝑎𝑓−𝑎2

𝑎+𝑑𝑓 ), where 𝑘 =√

2𝑎𝑐𝑑𝑓−𝑎2𝑐𝑑𝑎𝑑𝑒+𝑑2𝑒𝑓−𝑎𝑑−𝑎𝑑2+𝑑2𝑓 .

When 𝑎 = 60, 𝑐 = 56 , 𝑑 = 0.40, 𝑒 = 0.65, 𝑓 = 25, that is 𝑆0 = (0, 0, 0), 𝑆1 = (−4.1885,−4.4279,−8.5714),

𝑆2 = (4.1885, 4.4279,−8.5714).For equilibrium 𝑆0 = (0, 0, 0), the corresponding Jacobian matrix is

𝐽(𝑆0) =

⎛⎝ −𝑎+ 𝑑𝑧 𝑎 𝑑𝑥𝑓 − 𝑎− 𝑧 𝑓 −𝑥−2𝑒𝑥+ 𝑦 𝑥 𝑐

⎞⎠𝑆0

=

⎛⎝ −60.0000 60.0000 0−35.0000 25.0000 0

0 0 0.8333

⎞⎠ . (11)



0 50 100 150 200−30

−25

−20

−15

−10

−5

0

5Dynamics of Lyapunov exponents

Time

Lyap

unov

exp

onen

ts

Figure 10: The Lyapunov exponent of the new system.

0 50 100 150 200−14

−12

−10

−8

−6

−4

−2

0

2

4Dynamics of Lyapunov exponents

Time

Lyap

unov

exp

onen

ts

Figure 11: The Lyapunov exponent of the Chen system.

To gain its eigenvalues, we let ∣𝜆𝐼 − 𝐽(𝑆0)∣ = 0, then the eigenvalues corresponding to equilibrium 𝑆0 are obtained as:

𝜆1 = −17.5000 + 17.1391𝑖, 𝜆2 = −17.5000− 17.1391𝑖, 𝜆3 = 0.8333.

Because 𝜆1 and 𝜆2 have negative real parts and 𝜆3 is a positive real number, the equilibrium 𝑆0 is a saddle point and thusthis equilibrium 𝑆0 is unstable.

Secondly, for the equilibrium point 𝑆1, the Jacobian matrix is

𝐽(𝑆1) =


⎞⎠𝑆1

=

⎛⎝ −63.4286 60.0000 −1.6754−26.4286 25.0000 4.18851.0172 −4.1885 0.8333

⎞⎠ . (12)

The eigenvalues corresponding to the equilibrium point 𝑆1 are

𝜆1 = −38.5975, 𝜆2 = 0.5011 + 5.0653𝑖, 𝜆3 = 0.5011− 5.0653𝑖,

where 𝜆1 is a negative real number, 𝜆2 and 𝜆3 are a pair of complex conjugate eigenvalues with positive real parts. Theequilibrium 𝑆1 is a saddle-focus point and thus this equilibrium point is also unstable.

Finally, for the equilibrium 𝑆2, the corresponding Jacobian matrix is

𝐽(𝑆2) =


⎞⎠𝑆2

=

⎛⎝ −33.3333 40.0000 3.4816𝑖−16.6667 20.0000 −8.7039𝑖−18.5683𝑖 8.7039𝑖 5/6

⎞⎠ . (13)

And the corresponding eigenvalues of 𝑆2 are

𝜆1 = −38.5975, 𝜆2 = 0.5011 + 5.0653𝑖, 𝜆3 = 0.5011− 5.0653𝑖,

Therefore, similar to 𝑆1, 𝑆2 is also a saddle-focus point and this equilibrium is also unstable. The above brief analysisshow that the three equilibria of the proposed system are all saddle focus-nodes. It is known that the Chen system hasthree real fixed points and displays a double-scroll attractor, while the new chaotic system exhibits a three-scroll attractor.

4.5 Poincare map, Poincare-section, spectrum map and bifurcation diagramWe can further find that the section of the attractor looks like three circles from Poincare map of system (8) when 𝛼 = 1as Fig. 12(a), (b), (c). The spectrum of system (5) exhibits a continuous broadband feature as shown in Fig. 13. Thebifurcation diagram of system (8) when 𝛼 = 1 shows the basic bifurcation phenomena in Fig. 14.

According to the results obtained above, the new chaotic attractor has three scrolls, which can be further clearly seenfrom the Poincare section as( Fig. 15(a), (b), (c)), that is, from the different section of the attractor looking like threecircles.



−200 −100 0 100 200−400

−200

0

200

400

x axis δ/rad

y ax

is ω

/(ra

d/s)

(a)

−200 −100 0 100 200−200

0

200

400

x axis δ/rad

z ax

is ω

/(ra

d/s)

(b)

−400 −200 0 200 400−200

0

200

400

y axis δ/rad

z ax

is ω

/(ra

d/s)

(c)

Figure 12: The Poincare map of the new attractor in y-z plane. (a) The x-y phase plane the new attractor, (b) x-z phase plane the newattractor, (c) y-z phase plane the new attractor.

0 50 100 150−250

−200

−150

−100

−50

0

50

log(feg)/Hz

the

pow

er(1

0 ∗

log(

pow

er))

/ ω

Figure 13: Spectrum of 𝑥.

−0.5 0 0.5 1−20

0

20

40

60

80

100

α

x m

ax

bifurcation diagram

Figure 14: Bifurcation diagram of system (5).



−200 −100 0 100 200−400

−200

0

200

400(a)

x

y

−200 −100 0 100 2000

100

200

300(b)

x

z

−400 −200 0 200 400−200

0

200

400(c)

y

z

Figure 15: The Poincare-section of the new chaotic attractor. (a) The x-y phase plane the new attractor, (b) x-z phase plane the newattractor, (c) y-z phase plane the new attractor.

4.6 Forming mechanism of this new chaotic attractor structureIn order to reveal the forming mechanism of this new chaotic attractor structure, a controlled system is proposed. Theautonomous differential equations of this controlled system are expressed as⎧⎨⎩

�� = (20𝛼+ 40)(𝑦 − 𝑥) +6𝛼+ 4

25𝑥𝑧,

�� = (55− 90𝛼)𝑥+ (5𝛼+ 20)𝑦 − 𝑥𝑧,

�� = −13

20𝑥2 + 𝑥𝑦 +

11− 6𝛼

6𝑧 +𝑚.

(14)

In this system, 𝑚 is a parameter of control and the value of 𝑚 can be changed within a certain range. When theparameter 𝑚 is changed, the chaos behavior of this system can effectively be controlled. So it is a controller. Here, weselect the initial values of the system as (0.1, 1.0,−0.1).

−100 0 100−100

0

100

x

z

(a)

−50 0 50−50

0

50

x

z

(b)

−50 0 50−50

0

50

x

z

(c)

−200 0 200−500

0

500

x

z

(d)

−200 0 200−500

0

500

x

z

(e)

−200 0 200−500

0

500

x

z

(f)

−500 0 500−500

0

500

x

z

(g)

−200 0 200−500

0

500

x

z

(h)

−200 0 200−500

0

500

x

z

(i)

Figure 16: 𝑥 − 𝑧 phase plane of the new TSUCS when (a) 𝑚 = −220, (b) 𝑚 = −80, (c) 𝑚 = −30, (d) 𝑚 = −8, (e) 𝑚 = 0, (f)𝑚 = 8, (g) 𝑚 = 30, (h) 𝑚 = 80, (i) 𝑚 = 220.



When 𝑚 = −220, the attractor evolves into the butterfly attractor. The butterfly attractor is shown in Fig. 16(a). When𝑚 = −80, the corresponding strange attractor is shown in Fig. 16(b). Moreover the attractor evolves into two big scrollsand a small above scroll attractor in this time. When 𝑚 = −30, the strange attractor is shown in Fig. 16(c), the attractorevolves into the period-doubling bifurcations. The attractor is still bounded in this time. When 𝑚 = −8, the strangeattractor is shown in Fig. 16(d), the attractor evolves also into the new attractor, but is still bounded in this time. When𝑚 = 0, the strange attractor is shown in Fig. 16(e), the attractor evolves also into the new TSUCS, but is bounded in thistime. When 𝑚 = 8, the strange attractor is shown in Fig. 16(f), the attractor evolves also into the new attractor, but isstill bounded in this time. When 𝑚 = 30, the strange attractors are shown in Fig. 16(g), the attractor evolves also intosmall new attractor, but is still bounded in this time. When 𝑚 = 80, the corresponding strange attractors are shown inFig. 16(h). Moreover the attractor evolves into a small three-scroll attractor in this time. When 𝑚 = 220, the attractorevolves into the globosity attractor, the globosity attractor is shown in Fig. 16(i).

In the controller, one can see that when ∣𝑚∣ is large enough, chaos attractor varies; when ∣𝑚∣ is small enough, acomplete new chaos attractor appears. So 𝑚 is an important parameter to control chaos in the nonlinear system.

This means that the new attractor reported is a compound structure obtained by merging together three simple attractorsafter performing some mirror operations.

Fig. 17 displays the maximum Lyapunov exponent spectrum of system for −600 ≤ 𝑚 ≤ 600. It is clear that there aresome periodic windows in Fig. 16, which are 𝑊1(−516,−492), 𝑊2(−482,−462), 𝑊3(−410,−394), 𝑊4(−352,−324),𝑊5(−298,−272), 𝑊6(−192,−164), 𝑊7(−32,−12), 𝑊8(52, 86), 𝑊9(102, 124), 𝑊10(192, 202). However, these peri-odic windows are very important in the evolution of the complex dynamics system (8).

−600 −400 −200 0 200 400 600−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

parameter m

Max

imum

Lya

puno

v ex

pone

nt

Figure 17: The maximum Lyapunov exponent spectrum.

5 ConclusionIn this paper, we have constructed a fully new three-scroll unified chaotic system. This new chaos attractor is differentfrom the other existing attractors. In this new chaotic system, there are abundant and complex dynamical behaviors. Thetopological structure of the new system should be completely and thoroughly investigated, and the generation mechanismof the system (8) needs further studying and exploration. It is expected that more detailed theory analysis and simulationinvestigations will be provided in a forthcoming paper.

AcknowledgementsThis work was supported by the National “863” Key Programme of China (2008AA042902), the Doctor Base Foundationof Colleges and Universities by the Ministry of Education of China (200802550007), the Key Scientific Research andInnovation Program of Shanghai Education Committee (09zz66), the open project of National Laboratory of Industrial



Control Technology and the National Excellent Ph.D. Innovation Foundation of China (104060019137). The authorsthank the reviewer for all the help comments.

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