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

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1 Mathematical Problems in Engineering Volume, Article ID 88, pages doi:.//88 Research Article A New Four-Scroll Chaotic Attractor Consisted of Two-Scroll Transient Chaotic and Two-Scroll Ultimate Chaotic Yuhua Xu,,, Bing Li, Yuling Wang, Wuneng Zhou, and Jian-an Fang Department of Mathematics and Finance, Yunang Teachers College, Hubei Shian, China Computer School of Wuhan Universit, Wuhan 79, China College of Information Science and Technolog, Donghua Universit, Shanghai, China NOSTA, The Ministr of Science and Technolog of China, GPO Bo, Beijing, Tianjin Universit, Tianjin 7, China School of Management, Tianjin Universit, Tianjin 7, China Correspondence should be addressed to Yuhua Xu, uhuau@.com Received Januar ; Accepted March Academic Editor: Ahmad M. Harb Copright q 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 an medium, provided the original work is properl cited. A new four-scroll chaotic attractor is found b feedback controlling method in this paper. The novel chaotic sstem 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 sstem can generate one-scroll, two -scroll and four-scroll chaotic attractor with variation of a single parameter. We anale the new sstem b means of phase portraits, Lapunov eponents, fractional dimension, bifurcation diagram, and Poincaré map, respectivel. The analsis results show clearl that this is a new chaotic sstem which deserves further detailed investigation.. Introduction Since Loren found the first chaotic attractor, considerable research interests have been made in searching for new chaotic attractors. Particularl, research interests are turning in searching for new chaotic attractors in the three-dimensional D autonomous ordinar differential equations. For eample, Loren sstem,rössler sstem, Chen and Ueta sstem,lü and Chen sstem,andliusstem were reported and analed. In ver recent ears, creating comple multiscroll or multiwing chaotic attractors in D autonomous sstems has been rapidl developed, 7. It stimulates the current research interest in creating various comple multiscroll chaotic attractors b using simple electronic circuits and devices. After the rapid development in more than a decade, multiscroll chaotic

2 Mathematical Problems in Engineering attractors generation has become a relativel mature research direction 8. In fact, most of the multiscroll attractors were generated b increasing the breakpoints in the nonlinearit. Recentl, a four-wing or three-wing attractor was generated in some D sstems b reling on two embedded state-controlled binar switches 9. But these D sstems are not usuall smooth sstems. Although a few D smooth autonomous chaotic sstems have been reported to displa two-, three-, and four-wing attractor, respectivel 7, how to generate multiwing chaotic attractors remains an open problem. Therefore, it is important and even necessar to investigate various possible chaotic behaviors such that we can establish a unified theor for a D sstem generating chaos. Particularl, over the last two decades chaos in engineering sstems, such as nonlinear circuits, has graduall been moved from simpl being a scientific curiosit to a promising subject with practical significance and applications. Recentl, it has been noticed that purposefull creating chaos can be a ke issue in man technological applications such as communication, encrption, and information storage. An appropriate sstem for such applications could be chosen from a categor of easier controlled chaotic sstem to optimie factors such as robustness to errors in the parameters or immunit to noise 8. Therefore, it is obviousl significant to create more complicated chaotic sstems with simple epressions in three-dimensional for engineering applications such as secure communications. In this paper, we propose a four-scroll chaotic attractor which consists of the twoscroll transient chaotic and the two-scroll chaotic. It is ver desirable for engineering applications such as secure communications. For eample, in terms of decrption, we can generate a certain degree of confusion b using chaos sstem of transient chaos feature to encrpt. Moreover, this novel sstem can generate one-scroll, left two-scroll, right twoscroll according to the sstem s geometric locations, and four-scroll chaotic attractors, respectivel, with the variation of a single parameter. This paper is devoted to a more detailed analsis of this new chaotic attractor.. New Chaotic Sstem Based on the chaotification analsis 8, 9, the Liu-Chen sstem ma be added with two terms, which leads to the finding of the new chaotic attractor. Start with the controlled Liu-Chen sstem : ẋ b c u, ẏ d e u, ż f g.. B various trial tests, we find a simpler anticontroller, that is, u a, u,. which ields the following new chaotic sstem: ẋ a b c, ẏ d e, ż f g,. where t, t, t T R is the state vector and a, b, c, d, e, f,andg are real constants.

3 Mathematical Problems in Engineering This new sstem. is found to be chaotic in a wide parameter range and possesses man interesting comple dnamical behaviors. For eample, the sstem. can generate four-scroll for the parameters a., b, c, d, e,f.8, g andthe initial conditions,, see Figure a. Sprott 7 has suggested that the long calculation time helps ensure that the solutions are stead states. Notabl, in comparison with those of eisting two-scroll or four-scroll chaotic attractors in D autonomous sstems, the novel chaotic attractor can generate fourscroll consisted of the two-scroll transient chaotic and the two-scroll chaotic see Figures b d. Aftertimet>, simulation results show that the attractor is no longer a four-scroll attractor and becomes a two-scroll attractor.. Some Basic Properties of the New Sstem In this section, we will investigate some basic properties of the new sstem.... The Four-Scroll Chaotic Attractor... Equilibria It is known that the number of equilibrium points of the sstem and the stabilities at the equilibrium points are ver important for the emergence of chaos. In the sequel, we consider the equilibrium points of sstem.. Let a b c, d e,. f g. Let a., b, c, d, e, f.8, g. Equation. has five equilibrium points as follows: E,,, E.79,.,.79, E.8879,.,.87,. E ,., E.79,.7,.88. As shown in Figure, the equilibria points of sstem., E, E, E,andE are located at the center of the four wings of the attractor, respectivel, and the origin E is located at the center of whole chaotic attractor. Moreover, it can be seen from Figure that the equilibria E, E pla an important role in generating the two-scroll transients chaotic, while the equilibria E and E pla an important role in ultimate generating two-scroll chaotic attractor.

4 Mathematical Problems in Engineering 8 8 a c b d Figure : a The new chaotic attractor. b - phase plane of the new chaotic attractor. c When time <t, the new sstem generate two-scroll transient chaotic. d When time t>, the new sstem generate two-scroll chaotic attractor. 8 E E E 8 E E Figure : Five equilibria of the new four-scroll chaotic attractor.

5 Mathematical Problems in Engineering For equilibrium E, the sstem. is linearied, and the Jacobian matri at E is as follows: b a c c. J e d e.. g g f.8 To gain its eigenvalues, we let λi J. So the corresponding eigenvalues at E are λ, λ, λ.8.. Similarl, the corresponding eigenvalues at E are λ.9, λ.7 7.8i, λ.7 7.8i.. The corresponding eigenvalues at E are λ.8, λ i, λ i.. The corresponding eigenvalues at E are λ.9, λ.98.98i, λ.98.98i..7 The corresponding eigenvalues at E are λ., λ. 7.8i, λ. 7.8i..8 From..8, we know that E, E, E, E,andE are all unstable saddle points.... Dissipativit and the Eistence of Attractor For dnamical sstem., we can obtain V ẋ ẏ ż b d f,.9 where b d f 8. is a negative value. Dnamical sstem. is a dissipative sstem, and an eponential contraction of the sstem. is dv dt e 8... In the dnamical sstem., a volume element V is apparentl contracted b the flow into a volume element V e 8.t in time t. It means that each volume containing the trajector of this dnamical sstem shrinks to ero as t at an eponential rate 8.. So, all this dnamical sstem orbits are eventuall confined to a specific subset that have ero volume, and the asmptotic motion settles onto an attractor of the sstem..

6 Mathematical Problems in Engineering Lapunov eponent 8 t Figure : Lapunov eponents.... Lapunov Eponents and Spectrum Map An sstem containing at least one positive Lapunov eponents is defined to be chaotic. The Lapunov eponent spectrum of the sstem. is found to be L.8, L., L.8 see Figure. In addition, the Lapunov dimension of this sstem is D L j L j j i L i L L L ,.8. which means that the sstem. is reall a dissipative sstem, and the Lapunov dimension of this sstem is fractional. The fractal nature of an attractor does not merel mean this sstem has nonperiodic orbits; it also causes nearb trajectories to diverge. We can further find that the spectrum of sstem. ehibits a continuous broadband feature as shown in Figure.... 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 sstem is proposed. The autonomous differential equations of this controlled sstem are epressed as ẋ a b c, ẏ d e,. ż f g u. In this sstem, u is a parameter of control and the value of u can be changed within a certain range.

7 Mathematical Problems in Engineering 7 Log Frequenc (rad/s) Figure : An apparentl continuous broadband frequenc spectrum Log. When the parameter u is changed, the chaos behavior of this sstem can effectivel be controlled. So it is a controller. In the numerical simulation, the initial values of the sstem. is,,. For u 9, the attractor evolves into the limit ccles; the limit ccles are shown in Figure a. For u., the attractor evolves into the period-doubling bifurcations; perioddoubling bifurcations are shown in Figure b. For u.9, the corresponding strange attractors are shown in Figure c. Moreover the attractors are evolved into the one lower-scroll attractor. For u., the strange attractors are shown in Figure d, the attractor evolves into the single left two-scroll attractor. For u., the corresponding strange attractors are shown in Figure e ; the attractor evolves also into the single right two-scroll attractor. For u.9, the corresponding strange attractors are shown in Figure f. Moreover, the attractors are evolved into the one upper-scroll attractor. For u., the attractor evolves into the period-doubling bifurcations; perioddoubling bifurcations are shown in Figure g. For u 9, the attractor evolves into the limit ccles; the limit ccles are shown in Figure 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 sstem. For 9 u 9 the bifurcation diagram of sstem. shows the complicated bifurcation phenomena see Figure. It is clear that the bifurcation phenomenon well coincides with the forming mechanism... The One-Scroll Chaotic Attractor The sstem. has been found to generate a one-scroll chaotic attractor b onl varing a single parameter. Here, the parameter f is selected to be varied. For eample, if we let a., b, c, d, e, f 7, g then a one-scroll chaotic attractor can be observed, as depicted in Figure 7.

8 8 Mathematical Problems in Engineering a c 8 e. g..... b d f 8 h Figure : Phase portraits of the sstem. at a u 9, b u., c u.9, d u., e u., f u.9, g u., h u 9.

9 Mathematical Problems in Engineering 9 The bifurcation diagram 8 8 u Figure : Bifurcation diagram of sstem states versus parameter 9 u 9. Figure 7: The new one-scroll chaotic attractor... The Left and Right Two-Scroll Chaotic Attractor With parameters a., b, c, d, e, f, g, the sstem. can ehibit a right two-scroll chaotic attractor see Figure 8, and with parameters a., b, c, d, e, f., g, the sstem. can ehibit a left two-scroll chaotic attractor see Figure 9.. Poincaré Map, Bifurcation Diagram, and the Maimum Lapunov Eponent Spectrum of the New Chaotic Sstem Varing the parameter f, Poincaré mapping of the chaotic attractors of the sstems. are shown in Figures a, b, c, and d, respectivel. Several sheets of the attractors are displaed. It is noticeable that the Poincaré map of man chaotic sstems such as the generalied Loren sstem 9 onl shows a branch with several twigs. The Poincaré map

10 Mathematical Problems in Engineering 8 8 Figure 8: The new right two-scroll chaotic attractor. Figure 9: The new left two-scroll chaotic attractor. in Figures a, b, c and d, however, consists of virtuall smmetrical branches and a number of nearl smmetrical twigs. The bifurcation diagram would be far better to summarie all of the possible behaviors as the parameter varies on one diagram. For f 9. the bifurcation diagram of sstem. shows the complicated bifurcation phenomena see Figure.. Conclusions In this paper, a new four-scroll chaotic attractor in D autonomous sstem has been reported and confirmed analticall and numericall. In comparison with that of eisting two-scroll or four-scroll chaotic attractors in D autonomous sstem, 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 sstem can generate different scroll chaotic

11 Mathematical Problems in Engineering a b 8 8 c d Figure : Poincare maps of - plane for, a f 7, b f, c f.8, d f.. The bifurcation diagram f Figure : Bifurcation diagram of sstem states versus parameter f 9..

12 Mathematical Problems in Engineering attractors with variation of a single parameter. The topological structure of the new sstem should be completel and thoroughl investigated. It is epected that more detailed theor analses and simulation investigations will be provided in a forthcoming paper. Acknowledgments This research is supported b the National Natural Science Foundation of China 7, the Innovation Program of Shanghai Municipal Education Commission, the Ke Basic Research Program of Shanghai Cit 9JC7, the Science and Technolog Research Ke Program for the Education Department of Hubei Province of China D, D and China Postdoctoral Science Foundation M9. References E. N. Loren, Deterministic non-periodic flows, the Atmospheric Sciences, vol., pp. 8, 9. O. E. Rössler, An equation for continuous chaos, Phsics Letters A, vol. 7, no., pp , 97. G. Chen and T. Ueta, Yet another chaotic attractor, International Bifurcation and Chaos in Applied Sciences and Engineering, vol. 9, no. 7, pp., 999. J. Lü and G. Chen, A new chaotic attractor coined, International Bifurcation and Chaos in Applied Sciences and Engineering, vol., no., pp. 9,. C. Liu, T. Liu, L. Liu, and K. Liu, A new chaotic attractor, Chaos, Solitons and Fractals, vol.,no., pp. 8,. J. Lü, T. Zhou, G. Chen, and X. Yang, Generating chaos with a switching piecewise-linear controller, Chaos, vol., no., pp. 9,. 7 W. Liu, W. K. S. Tang, and G. Chen, -scroll attractors generated in a three-dimensional smooth autonomous sstem, International Bifurcation and Chaos, vol. 7, no., pp. 7, 7. 8 J. Lü and G. Chen, Generating multiscroll chaotic attractors: theories, methods and applications, International Bifurcation and Chaos in Applied Sciences and Engineering, vol., no., pp ,. 9 A. S. Elwakil, S. Öoǧu, and M. P. Kenned, A four-wing butterfl attractor from a full autonomous sstem, International Bifurcation and Chaos in Applied Sciences and Engineering, vol., no., pp. 9 98,. W. Liu and G. Chen, A new chaotic sstem and its generation, International Bifurcation and Chaos in Applied Sciences and Engineering, vol., no., pp. 7,. G. Qi, G. Chen, S. Li, and Y. Zhang, Four-wing attractors: from pseudo to real, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol., no., pp ,. J. Lü, G. Chen, and D. Cheng, A new chaotic sstem and beond: the generalied Loren-like sstem, International Bifurcation and Chaos in Applied Sciences and Engineering, vol., no., pp. 7 7,. S. Dadras and H. R. Momeni, A novel three-dimensional autonomous chaotic sstem generating two, three and four-scroll attractors, Phsics Letters A, vol. 7, no., pp. 7, 9. A. Chen, J. Lu, J. Lü, and S. Yu, Generating hperchaotic Lü attractor via state feedback control, Phsica A, vol., pp.,. G. Qi, G. Chen, S. Du, Z. Chen, and Z. Yuan, Analsis of a new chaotic sstem, Phsica A, vol., no. -, pp. 9 8,. G. Qi, G. Chen, M. A. van Wk, B. J. van Wk, and Y. Zhang, A four-wing chaotic attractor generated from a new -D quadratic autonomous sstem, Chaos, Solitons and Fractals, vol. 8, no., pp. 7 7, 8. 7 J. C. Sprott, Some simple chaotic flows, Phsical Review E, vol., no., pp. R7 R, T. Ueta and G. Chen, Bifurcation analsis of Chen s equation, International Bifurcation and Chaos, vol., pp. 97 9, 9. 9 S. Čelikovský and G. Chen, On the generalied Loren canonical form, Chaos, Solitons and Fractals, vol., no., pp. 7 7,. A. Wolf, J. B. Swift, H. L. Swinne, and J. A. Vastano, Determining Lapunov eponents from a time series, Phsica D, vol., no., pp. 8 7, 98.

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