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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 438328, 12 pages
http://dx.doi.org/10.1155/2012/438328
Research Article

A New Four-Scroll Chaotic Attractor Consisted of Two-Scroll Transient Chaotic and Two-Scroll Ultimate Chaotic

1Department of Mathematics and Finance, Yunyang Teachers’ College, Hubei Shiyan 442000, China
2Computer School of Wuhan University, Wuhan 430079, China
3College of Information Science and Technology, Donghua University, Shanghai 201620, China
4NOSTA, The Ministry of Science and Technology of China, GPO Box 2143, Beijing 100045, Tianjin University, Tianjin 300072, China
5School of Management, Tianjin University, Tianjin 300072, China

Received 21 January 2012; Accepted 12 March 2012

Academic Editor: Ahmad M. Harb

Copyright © 2012 Yuhua Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A new four-scroll chaotic attractor is found by feedback controlling method in this paper. The novel chaotic system can generate four scrolls two of which are transient chaotic and the other two of which are ultimate chaotic. Of particular interest is that this novel chaotic system can generate one-scroll, two 2-scroll and four-scroll chaotic attractor with variation of a single parameter. We analyze the new system by means of phase portraits, Lyapunov exponents, fractional dimension, bifurcation diagram, and Poincaré map, respectively. The analysis results show clearly that this is a new chaotic system which deserves further detailed investigation.

1. Introduction

Since Lorenz found the first chaotic attractor, considerable research interests have been made in searching for new chaotic attractors. Particularly, research interests are turning in searching for new chaotic attractors in the three-dimensional (3D) autonomous ordinary differential equations. For example, Lorenz system [1], Rössler system [2], Chen and Ueta system [3], Lü and Chen system [4], and Liu system [5] were reported and analyzed.

In very recent years, creating complex multiscroll or multiwing chaotic attractors in 3D autonomous systems has been rapidly developed [6, 7]. It stimulates the current research interest in creating various complex multiscroll chaotic attractors by using simple electronic circuits and devices. After the rapid development in more than a decade, multiscroll chaotic attractors generation has become a relatively mature research direction [8]. In fact, most of the multiscroll attractors were generated by increasing the breakpoints in the nonlinearity. Recently, a four-wing or three-wing attractor was generated in some 3D systems by relying on two embedded state-controlled binary switches [9]. But these 3D systems are not usually smooth systems. Although a few 3D smooth autonomous chaotic systems have been reported to display two-, three-, and four-wing attractor, respectively [1017], how to generate multiwing chaotic attractors remains an open problem. Therefore, it is important and even necessary to investigate various possible chaotic behaviors such that we can establish a unified theory for a 3D system generating chaos.

Particularly, over the last two decades chaos in engineering systems, such as nonlinear circuits, has gradually been moved from simply being a scientific curiosity to a promising subject with practical significance and applications. Recently, it has been noticed that purposefully creating chaos can be a key issue in many technological applications such as communication, encryption, and information storage. An appropriate system for such applications could be chosen from a category of easier controlled chaotic system to optimize factors such as robustness to errors in the parameters or immunity to noise [18]. Therefore, it is obviously significant to create more complicated chaotic systems with simple expressions in three-dimensional for engineering applications such as secure communications.

In this paper, we propose a four-scroll chaotic attractor which consists of the two-scroll transient chaotic and the two-scroll chaotic. It is very desirable for engineering applications such as secure communications. For example, in terms of decryption, we can generate a certain degree of confusion by using chaos system of transient chaos feature to encrypt. Moreover, this novel system can generate one-scroll, left two-scroll, right two-scroll (according to the system’s geometric locations), and four-scroll chaotic attractors, respectively, with the variation of a single parameter. This paper is devoted to a more detailed analysis of this new chaotic attractor.

2. New Chaotic System

Based on the chaotification analysis [18, 19], the Liu-Chen system may be added with two terms, which leads to the finding of the new chaotic attractor.

Start with the controlled Liu-Chen system [10]:̇𝑥=𝑏𝑥+𝑐𝑦𝑧+𝑢1,̇𝑦=𝑑𝑦+𝑒𝑥𝑧+𝑢2,̇𝑧=𝑓𝑧+𝑔𝑥𝑦.(2.1)

By various trial tests, we find a simpler anticontroller, that is,𝑢1𝑢=𝑎𝑦,2=𝑧,(2.2) which yields the following new chaotic system:̇𝑥=𝑎𝑦+𝑏𝑥+𝑐𝑦𝑧,̇𝑦=𝑑𝑦𝑧+𝑒𝑥𝑧,̇𝑧=𝑓𝑧+𝑔𝑥𝑦,(2.3) where [𝑥(𝑡),𝑦(𝑡),𝑧(𝑡)]𝑇𝑅3 is the state vector and 𝑎,𝑏,𝑐,𝑑,𝑒,𝑓, and 𝑔 are real constants.

This new system (2.3) is found to be chaotic in a wide parameter range and possesses many interesting complex dynamical behaviors. For example, the system (2.3) can generate four-scroll for the parameters 𝑎=2.4, 𝑏=3, 𝑐=14, 𝑑=11, 𝑒=4,𝑓=5.85, 𝑔=1 and the initial conditions [1, 3, 5] (see Figure 1(a)).

fig1
Figure 1: (a) The new chaotic attractor. (b) 𝑥-𝑧 phase plane of the new chaotic attractor. (c) When time 0<𝑡55, the new system generate two-scroll transient chaotic. (d) When time 𝑡>55, the new system generate two-scroll chaotic attractor.

Sprott [17] has suggested that the long calculation time helps ensure that the solutions are steady states. Notably, in comparison with those of existing two-scroll or four-scroll chaotic attractors in 3D autonomous systems, the novel chaotic attractor can generate four-scroll consisted of the two-scroll transient chaotic and the two-scroll chaotic (see Figures 1(b)1(d)). After time 𝑡>55, simulation results show that the attractor is no longer a four-scroll attractor and becomes a two-scroll attractor.

3. Some Basic Properties of the New System

In this section, we will investigate some basic properties of the new system (2.3).

3.1. The Four-Scroll Chaotic Attractor
3.1.1. Equilibria

It is known that the number of equilibrium points of the system and the stabilities at the equilibrium points are very important for the emergence of chaos. In the sequel, we consider the equilibrium points of system (2.3).

Let 𝑎𝑦+𝑏𝑥+𝑐𝑦𝑧=0,𝑑𝑦𝑧+𝑒𝑥𝑧=0,𝑓𝑧+𝑔𝑥𝑦=0.(3.1)

Let 𝑎=2.4, 𝑏=3, 𝑐=14, 𝑑=11, 𝑒=4, 𝑓=5.85, 𝑔=1. Equation (3.1) has five equilibrium points as follows:𝐸1𝐸(0,0,0),2𝐸(4.1379,1.0050,0.7109),3𝐸(3.8879,1.2560,0.8347),4𝐸(3.88790.9981,0.6633),5(4.1379,1.2473,0.8823).(3.2)

As shown in Figure 2, the equilibria points of system (2.3), 𝐸2, 𝐸3, 𝐸4, and 𝐸5 are located at the center of the four wings of the attractor, respectively, and the origin (𝐸1) is located at the center of whole chaotic attractor. Moreover, it can be seen from Figure 2 that the equilibria 𝐸2, 𝐸5 play an important role in generating the two-scroll transients chaotic, while the equilibria 𝐸3 and 𝐸4 play an important role in ultimate generating two-scroll chaotic attractor.

438328.fig.002
Figure 2: Five equilibria of the new four-scroll chaotic attractor.

For equilibrium 𝐸1, the system (2.3) is linearized, and the Jacobian matrix at 𝐸1 is as follows:=𝐽=𝑏𝑎+𝑐𝑧𝑐𝑦𝑒𝑧𝑑𝑒𝑥1𝑔𝑦𝑔𝑥𝑓32.400111005.85.(3.3)

To gain its eigenvalues, we let |𝜆𝐼𝐽|=0.

So the corresponding eigenvalues at 𝐸1 are𝜆1=3,𝜆2=11,𝜆3=5.85.(3.4)

Similarly, the corresponding eigenvalues at 𝐸2 are𝜆1=12.9021,𝜆2=2.3761+7.0804𝑖,𝜆3=2.37617.0804𝑖.(3.5)

The corresponding eigenvalues at 𝐸3 are𝜆1=12.8005,𝜆2=2.3252+7.7881𝑖,𝜆3=2.32527.7881𝑖.(3.6)

The corresponding eigenvalues at 𝐸4 are𝜆1=12.5469,𝜆2=2.1984+6.9804𝑖,𝜆3=2.19846.9804𝑖.(3.7)

The corresponding eigenvalues at 𝐸5 are𝜆1=13.1525,𝜆2=2.5013+7.8516𝑖,𝜆3=2.50137.8516𝑖.(3.8) From (3.4)–(3.8), we know that 𝐸1, 𝐸2, 𝐸3, 𝐸4, and 𝐸5 are all unstable saddle points.

3.1.2. Dissipativity and the Existence of Attractor

For dynamical system (2.3), we can obtain𝑉=𝜕̇𝑥+𝜕𝑥𝜕̇𝑦+𝜕𝑦𝜕̇𝑧𝜕𝑧=𝑏+𝑑+𝑓,(3.9) where 𝑏+𝑑+𝑓=8.15 is a negative value. Dynamical system (2.3) is a dissipative system, and an exponential contraction of the system (2.3) is𝑑𝑉𝑑𝑡=𝑒8.15.(3.10)

In the dynamical system (2.3), a volume element 𝑉0 is apparently contracted by the flow into a volume element 𝑉0𝑒8.15𝑡 in time 𝑡. It means that each volume containing the trajectory of this dynamical system shrinks to zero as 𝑡 at an exponential rate −8.15. So, all this dynamical system orbits are eventually confined to a specific subset that have zero volume, and the asymptotic motion settles onto an attractor of the system (2.3).

3.1.3. Lyapunov Exponents and Spectrum Map

Any system containing at least one positive Lyapunov exponents is defined to be chaotic [20]. The Lyapunov exponent spectrum of the system (2.3) is found to be 𝐿1=2.8063, 𝐿2=0.0050, 𝐿3=2.8020 (see Figure 3). In addition, the Lyapunov dimension of this system is𝐷𝐿1=𝑗+||𝐿𝑗+1||𝑗𝑖=1𝐿𝑖𝐿=2+1+𝐿2||𝐿3||=2+2.8013||||2.8020=2.9998,(3.11) which means that the system (2.3) is really a dissipative system, and the Lyapunov dimension of this system is fractional. The fractal nature of an attractor does not merely mean this system has nonperiodic orbits; it also causes nearby trajectories to diverge.

438328.fig.003
Figure 3: Lyapunov exponents.

We can further find that the spectrum of system (2.3) exhibits a continuous broadband feature as shown in Figure 4.

438328.fig.004
Figure 4: An apparently continuous broadband frequency spectrum Log|𝑥|.
3.1.4. Forming Mechanism of the Four-Scroll Chaotic Attractor Structure

In order to reveal the forming mechanism of the four-scroll chaotic attractor structure, a controlled system is proposed. The autonomous differential equations of this controlled system are expressed aṡ𝑥=𝑎𝑦+𝑏𝑥+𝑐𝑦𝑧,̇𝑦=𝑑𝑦𝑧+𝑒𝑥𝑧,̇𝑧=𝑓𝑧+𝑔𝑥𝑦+𝑢.(3.12)

In this system, 𝑢 is a parameter of control and the value of 𝑢 can be changed within a certain range.

When the parameter 𝑢 is changed, the chaos behavior of this system can effectively be controlled. So it is a controller. In the numerical simulation, the initial values of the system (3.12) is (1, 3, 5).

For 𝑢=9, the attractor evolves into the limit cycles; the limit cycles are shown in Figure 5(a).

fig5
Figure 5: Phase portraits of the system (3.5) at (a) 𝑢=9, (b) 𝑢=5.11, (c) 𝑢=1.94, (d) 𝑢=0.45, (e) 𝑢=0.45, (f) 𝑢=1.94, (g) 𝑢=5.11, (h) 𝑢=9.

For 𝑢=5.11, the attractor evolves into the period-doubling bifurcations; period-doubling bifurcations are shown in Figure 5(b).

For 𝑢=1.94, the corresponding strange attractors are shown in Figure 5(c). Moreover the attractors are evolved into the one lower-scroll attractor.

For 𝑢=0.45, the strange attractors are shown in Figure 5(d), the attractor evolves into the single left two-scroll attractor.

For 𝑢=0.45, the corresponding strange attractors are shown in Figure 5(e); the attractor evolves also into the single right two-scroll attractor.

For 𝑢=1.94, the corresponding strange attractors are shown in Figure 5(f). Moreover, the attractors are evolved into the one upper-scroll attractor.

For 𝑢=5.11, the attractor evolves into the period-doubling bifurcations; period-doubling bifurcations are shown in Figure 5(g).

For 𝑢=9, the attractor evolves into the limit cycles; the limit cycles are shown in Figure 5(h).

In the controller, one can see that when |𝑢| is large enough, chaos attractor disappears; when |𝑢| is small enough, a complete chaos attractor appears. So |𝑢| is an important parameter to control chaos in the nonlinear system.

For 9𝑢9 the bifurcation diagram of system (3.12) shows the complicated bifurcation phenomena (see Figure 6). It is clear that the bifurcation phenomenon well coincides with the forming mechanism.

438328.fig.006
Figure 6: Bifurcation diagram of system states versus parameter 9𝑢9.
3.2. The One-Scroll Chaotic Attractor

The system (2.3) has been found to generate a one-scroll chaotic attractor by only varying a single parameter. Here, the parameter 𝑓 is selected to be varied. For example, if we let 𝑎=2.4, 𝑏=3, 𝑐=14, 𝑑=11, 𝑒=4, 𝑓=7, 𝑔=1 then a one-scroll chaotic attractor can be observed, as depicted in Figure 7.

438328.fig.007
Figure 7: The new one-scroll chaotic attractor.
3.3. The Left and Right Two-Scroll Chaotic Attractor

With parameters 𝑎=2.4, 𝑏=3, 𝑐=14, 𝑑=11, 𝑒=4, 𝑓=6, 𝑔=1, the system (2.3) can exhibit a right two-scroll chaotic attractor (see Figure 8), and with parameters 𝑎=2.4, 𝑏=3, 𝑐=14, 𝑑=11, 𝑒=4, 𝑓=5.5, 𝑔=1, the system (2.3) can exhibit a left two-scroll chaotic attractor (see Figure 9).

438328.fig.008
Figure 8: The new right two-scroll chaotic attractor.
438328.fig.009
Figure 9: The new left two-scroll chaotic attractor.

4. Poincaré Map, Bifurcation Diagram, and the Maximum Lyapunov Exponent Spectrum of the New Chaotic System

Varying the parameter 𝑓, Poincaré mapping of the chaotic attractors of the systems (2.3) are shown in Figures 10(a), 10(b), 10(c), and 10(d), respectively. Several sheets of the attractors are displayed. It is noticeable that the Poincaré map of many chaotic systems such as the generalized Lorenz system [19] only shows a branch with several twigs. The Poincaré map in Figures 10(a), 10(b), 10(c) and 10(d), however, consists of virtually symmetrical branches and a number of nearly symmetrical twigs.

fig10
Figure 10: Poincaré maps of 𝑥-𝑦 plane for 𝑧=0, (a) 𝑓=7, (b) 𝑓=6, (c) 𝑓=5.58, (d) 𝑓=5.5.

The bifurcation diagram would be far better to summarize all of the possible behaviors as the parameter varies on one diagram. For 1𝑓9.4 the bifurcation diagram of system (2.3) shows the complicated bifurcation phenomena (see Figure 11).

438328.fig.0011
Figure 11: Bifurcation diagram of system states versus parameter 1𝑓9.4.

5. Conclusions

In this paper, a new four-scroll chaotic attractor in 3D autonomous system has been reported and confirmed analytically and numerically. In comparison with that of existing two-scroll or four-scroll chaotic attractors in 3D autonomous system, the novel chaotic attractor can generate four-scroll that consisted of the two-scroll transient chaotic and the two-scroll chaotic. The particular interest is that this novel system can generate different scroll chaotic attractors with variation of a single parameter. The topological structure of the new system should be completely and thoroughly investigated. It is expected that more detailed theory analyses and simulation investigations will be provided in a forthcoming paper.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (61075060), the Innovation Program of Shanghai Municipal Education Commission (12zz064), the Key Basic Research Program of Shanghai City (09JC1400700), the Science and Technology Research Key Program for the Education Department of Hubei Province of China (D20105001, D20126004) and China Postdoctoral Science Foundation (2012M511259).

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