MULTI-SCROLL CHAOTIC AND HYPERCHAOTIC ATTRACTORS GENERATED FROM CHEN SYSTEM

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1 International Journal of Bifurcation and Chaos, Vol. 22, No. 2 (212) 133 ( pages) c World Scientific Publishing Compan DOI: /S MULTI-SCROLL CHAOTIC AND HYPERCHAOTIC ATTRACTORS GENERATED FROM CHEN SYSTEM XINZHI LIU Department of Applied Mathematics, Universit of Waterloo, Waterloo, Ontario N2L 3G1, Canada liu@math.uwaterloo.ca XUEMIN (SHERMAN) SHEN and HONGTAO ZHANG Department of Electric and Computer Engineering, Universit of Waterloo, Waterloo, Ontario N2L 3G1, Canada shen@bbcr.uwaterloo.ca hhang@bbcr.uwaterloo.ca Received December, 211 In this paper, we create a multi-scroll chaotic attractor from Chen sstem b a nonlinear feedback control. The dnamic behavior of the new chaotic attractor is analed. Speciall, the Lapunov spectrum and Lapunov dimension are calculated and the bifurcation diagram is sketched. Furthermore, via changing the value of the control parameters, we can increase the number of equilibrium points and obtain a famil of more comple chaotic attractors with different topological structures. B introducing time dela to the feedback control, we then generalie the multi-scroll attractor to a set of hperchaotic attractors. Computer simulations are given to illustrate the phase portraits with different sstem parameters. Kewords: Chaos; Chen sstem; Lapunov eponent; Hopf bifurcation. 1. Introduction Over the past two decades, the generation of chaotic or hperchaotic sstems and their applications have attracted a great deal of attention [Ballinger & Liu, 1997; Chen & Ueta, 1999; Cuomo & Oppenheim, 1993; Khadra et al., 23; Kocarev et al., 1992; Liu, 26; Pecora & Carroll, 199; Yang & Chua, 1996]. Especiall, multi-scroll attractors and hperchaos sstems have been intensivel studied [Goedgebuer et al., 1998; Grassi & Mascolo, 1999; Li & Chen, 24; Li et al., ; Lu et al., 24; Peng et al., 1996; Rossler, 1979; Yalcin, 27; Yan & Yu, 28]. Recentl, a nonautonomous technique to generate multi-scroll attractors and hperchaos has been introduced [Elwakil & Oogu, 26; Li et al., ]. Using switching sstems, some new chaotic attractors have been achieved in [Liu et al., 26; Lu et al., 24]. Some fractional differential sstems have also been presented to generate chaos [Ahmad & Sprott, 23; Hartle et al., 199; Li & Chen, 24], and some simple circuits have been developed to realie chaos [Chua et al., 1993; Elwakil & Kenned, 21; Lu et al., 24; Yalcin, 27; Yalcin et al., 22]. Research supported b NSERC Canada

2 X. Liu et al Fig. 1. The phase portrait of Chen sstem with a = 3, b =3andc = 28, starting with [1, 1, 1]. It is well known that a first-order dela differential equation can generate chaotic and even hperchatic attractors [Farmer, 1982; Ikeda & Matsumoto, 1987]. Since Macke Glass sstem, which is a phsiological model, was found to possess chaotic behaviors, several modified chaotic sstems have been reported [Lu & He, 1996; Namajunas et al., 199; Tamasevicius et al., 26; Wang & Yang, 26]. Some eperimental observations of multiscroll attractors have been confirmed [Tamasevicius et al., 26; Wang & Yang, 26; Yalcin & Oogu, 27]. In this paper, we use feedback function to control Chen sstem to multi-scroll attractors, via increasing its equilibrium points. Furthermore, we introduce time dela to the control function to generalie it to more comple chaos and hperchaos. The remainder of this paper is organied as follows. In Sec. 2, we present a control function to generate a 6-scroll attractor from Chen sstem. In Sec. 3, we anale the dnamical behavior of the new attractor. In Sec. 4, b controlling the values of sstem parameter, we generalie the sstem to different multi-scroll chaotic attractors. In Sec., b introducing time dela to the control function, we achieve more comple chaos and hperchaos. Finall, conclusions are given in Sec New Chaotic Attractor Chen sstem was introduced in 1999 when he studied how to control the Loren sstem. It has been proved that Chen sstem is not topologicall equivalent to the Loren sstem. The mathematical model is given as ẋ = a( ), ẏ =(c a) + c, (1) ż = b. Figure 1 shows the phase portrait of Chen sstem with a = 3, b =3andc = 28. We consider the following controlled sstem, ẋ = a( ) ẏ =(c a) u + c (2) ż = b where the control function u(t) = d 1 (t) d 2 sin((t)) with a = 3,b = 3,c = 28, d 1 = 1 and d 2 =8.Figure2 shows the phase portraits of the state variables of the controlled Chen sstem, which is a 6-scroll attractor Fig. 2. The phase portraits of the controlled Chen sstem. ; ; ;

3 Multi-Scroll Chaotic and Hperchaotic Attractors Generated from Chen Sstem Fig. 2. (Continued) Dnamical Analsis Sstem (2) has eleven equilibrium points (,, ), (±6.999, ±6.999, ), (±7.472, ±7.472, 18.36), (±8.12, ±8.12, ), (±8.7929, ±8.7929,.7719) and (±9.92, ±9.92, ). At the equilibrium point (,, ), the characteristic equation of the linearied sstem is a λ a c a (d 1 d 2 sin ) c λ d 1 + d 2 cos = (3) b λ Table 1. List of equilibrium points and corresponding roots of characteristic equations. Roots of Characteristic Roots of Characteristic No. Equilibrium Point Equation No. Equilibrium Point Equation 1 (,, ) λ 1 = (8.12, 8.12, ) λ 1 = λ 2 = 3. λ 2 = i λ 3 = λ 3 = i 2 ( 6.999, 6.999, ) λ 1 = ( , ,.7719) λ 1 = i λ 2 = i λ 2 = i λ 3 = i λ 3 = (6.999, 6.999, ) λ 1 = (8.7929, ,.7719) λ 1 = i λ 2 = i λ 2 = i λ 3 = i λ 3 = ( 7.472, 7.472, 18.36) λ 1 = i 1 ( 9.92, 9.92, ) λ 1 = λ 2 = i λ 2 = i λ 3 =3.44 λ 3 = i (7.472, 7.472, 18.36) λ 1 = i 11 (9.92, 9.92, ) λ 1 = λ 2 = i λ 2 = i λ 3 =3.44 λ 3 = i 6 ( 8.12, 8.12, ) λ 1 = λ 2 = i λ 3 = i 133-3

4 I X. Liu et al Lapunov Eponents R Time Fig. 3. The roots of the characteristic equations of sstem (2) corresponding to equilibrium points (±6.999, ±6.999, ). Fig. 4. The Lapunov spectra of sstem (2) witht = 1, starting from (1, 1, 3). 4. Generalied Multi-Scroll Chaos All equilibrium points and corresponding roots are shownintable1. From Fig. 2, we can see that the above new chaos is From Table 1, we can see that all equilibrium a 6-scroll Chen attractor. Furthermore, b choosing points of sstem (2) are unstable. Figure 3 shows different values of the parameter d 2,wecancontrol the responding eigenvalues of points (±6.999, the number of equilibrium points of sstem (2). Subsequentl, we can design comple chaotic sstems ±6.999, ). A pair of comple conjugate eigenvalues of the lineariation around these equilibrium points have crossed the imaginar ais of the number of Hopf bifurcation points. with different topological structures b controlling the comple plane. It can be seen that Hopf bifurcation will occur when suitable parameters are chosen. rium points (,, ), (±7.847, ±7.847, ), When d 2 =, sstem (2) has seven equilib- Similarl, at points (±8.12, ±8.12, ) (±7.439, ±7.439, ) and (±8.917, ±8.917, and (±9.92, ±9.92, ), there eist 21.83). At points (±7.847, ±7.847, ) Hopf bifurcations when suitable parameters are and (±8.917, ±8.917, 21.83), there eist Hopf chosen. bifurcations when suitable parameters are chosen. Furthermore, we calculate the maimum Lapunov eponent LE ma =4.789 and Lapunov of sstem (2) with d 2 =, which is a 4-scroll Figure shows the phase portraits of state variables dimension d = using the Matlab LET toolbo. The Lapunov spectra are shown in Fig. 4. LE ma =.4 and Lapunov dimension d = attractor. The maimum Lapunov eponent is When d 2 = 14, sstem (2) has 19 equilibrium points At points (,, ), (±.7, ±.7, 1.292), (±.963, ±.963, 11.84), (±6.9429, ±6.9429, 16.68), (±7.4867, ±7.4867, ), (±8.112, ±8.112, 21.9), (±8.739, ±8.739,.467), (±9.1274, ±9.1274, ), (±9.834, ±9.834, ) and (±1.19, ±1.19, ). (±.7, ±.7, 1.292), (±6.9429, ±6.9429, 16.68), (±8.112, ±8.112, 21.9), (±9.1274, ±9.1274, ) and (±1.19, ±1.19, ), there eist Hopf bifurcations when suitable parameters are chosen. Figure 6 shows the phase portraits of state variables of sstem (2) withd 2 = 14, which is a 1-scroll attractor. The maimum Lapunov eponent is LE ma =7.722 and Lapunov dimension d =

5 Multi-Scroll Chaotic and Hperchaotic Attractors Generated from Chen Sstem Fig.. The phase portraits of sstem (2) withd 2 =, starting from (1, 1, 3). ; ; ;. When d 2 = 22, sstem (2) has 27 equilibrium points (,, ), (±3.4741, ±3.4741, 4.231), (±4.626, ±4.626,.), (±.4638, ±.4638, 9.91), (±6.38, ±6.38, 12.), (±6.92, ±6.92,.94), (±7.4994, ±7.4994, ), (±8.1144, ±8.1144, ), (±8.7174, ±8.7174,.339), (±9.74, ±9.74, ), (±9.7882, ±9.7882, ), (±1.89, ±1.89, ), (±1.76, ±1.76, ) and (±1.9291, ±1.9291, ). At points (±3.4741, ±3.4741, 4.231), (±.4638, ±.4638, 9.91), (±6.92, ±6.92,.94), (±8.1144, ±8.1144, ), (±9.74, ±9.74, ), (±1.89, ±1.89, ) and (±1.9291, ±1.9291, ), 133-

6 X. Liu et al Fig. 6. The phase portraits of sstem (2) with d 2 = 14, starting from (1, 1, 3). ; ; ;. there eist Hopf bifurcations when suitable parameters are chosen. Figure 7 shows the phase portraits of state variables of sstem (2) withd 2 = 22, which is a 14-scroll attractor. The maimum Lapunov eponent is LE ma = and Lapunov dimension d = When d 2 = 28, sstem (2) has 31 equilibrium points (,, ), (±3.3777, ±3.3777, 3.829), (±4.1371, ±4.1371,.73), (±.4318, ±.4318, ), (±6.618, ±6.618, ), (±6.947, ±6.947,.8914), (±7.4, ±7.4, ), (±8.1161, ±8.1161, 21.97), (±8.797, ±8.797,.2864), (±9.1686, ±9.1686, 28.29), (±9.7692, ±9.7692, ), (±1.112, ±1.112, ), (±1.7287, ±1.7287, ), (±1.9668, ±1.9668, 4.9), (±11.626, ±11.626, 4.129) and (± , ± , )

7 Multi-Scroll Chaotic and Hperchaotic Attractors Generated from Chen Sstem Fig. 7. The phase portraits of sstem (2) with d 2 = 22, starting from (1, 1, 3). ; ; ;. At points (±3.3777, ±3.3777, 3.829), (±.4318, ±.4318, ), (±6.947, ±6.947,.8914), (±8.1161, ±8.1161, 21.97), (±9.1686, ±9.1686, 28.29), (±1.112, ±1.112, ), (±1.9668, ±1.9668, 4.9) and (± , ± , ), there eist Hopf bifurcations when suitable parameters are chosen. Figure 8 shows the phase portraits of state variables of sstem (2) withd 2 = 28, which is a 16-scroll attractor. The maimum Lapunov eponent is LE ma = and Lapunov dimension d = Figure 9 shows that all three Lapunov eponents change as d 2 ranges in [, 2] with fied parameters a =3,b=3,c =28andd 1 =1. Remark. When parameter d 2 increases, the number of equilibrium points increases correspondingl

8 X. Liu et al Fig. 8. The phase portraits of sstem (2) withd 2 = 28, starting from (1, 1, 3). ; ; ;. Lapunov Eponents d2 Fig. 9. All Lapunov eponents versus parameter d 2 in [, 2] with a =3,b=3,c=28andd 1 =

9 Multi-Scroll Chaotic and Hperchaotic Attractors Generated from Chen Sstem With Hopf bifurcation occurring over and over ẋ = a( ) again, sstem (2) generates new attractors with ẏ =(c a) (d more scrolls. (t)+d 1 (t τ) (4) d 2 sin((t τ))) + c ż = b. Generalied Comple Chaos and Hperchaos It is well known that first-order dela differential equation can generate hperchaotic dnamic behavior. In order to generalie sstem (2) tomore comple chaos and hperchaos, we introduce time dela to control function, u(t) =d (t) +d 1 (t τ) d 2 sin((t τ)). Sstem (2) transforms to the following. where d,d 1,d 2 are constants and τ is the time dela. When suitable parameters are chosen, sstem (4) can generate comple chaotic and hperchaotic attractors as follows..1. Special case I : d = d 1 (i) When d =1,d 1 =1,d 2 =andτ =.2, sstem (4) generates comple chaotic dnamic behaviors(showninfig.1) Fig. 1. The phase portraits of sstem (4) withd =1,d 1 =1,d 2 =andτ =

10 X. Liu et al Fig. 11. The phase portraits of sstem (4) withd =1,d 1 =1,d 2 =andτ = Fig. 12. The phase portraits of sstem (4) withd =1,d 1 =1,d 2 =3andτ =

11 Multi-Scroll Chaotic and Hperchaotic Attractors Generated from Chen Sstem Fig. 12. (Continued) Fig. 13. The phase portraits of sstem (4) withd =1,d 1 =1,d 2 =3andτ =

12 X. Liu et al Fig. 14. The phase portraits of sstem (4) withd =.2,d 1 =1,d 2 =andτ = Fig.. The phase portraits of sstem (4) withd =.2,d 1 =1,d 2 =3andτ =

13 Multi-Scroll Chaotic and Hperchaotic Attractors Generated from Chen Sstem Fig.. (Continued) Fig. 16. The phase portraits of sstem (4) withd =1,d 1 =.2,d 2 =2andτ =

14 X. Liu et al Fig. 17. The phase portraits of sstem (4) withd =1,d 1 =.4,d 2 =andτ =1. (ii) When d =1,d 1 =1,d 2 =andτ =.8, sstem (4) generates comple chaotic dnamic behaviors(showninfig.11). (iii) When d =1,d 1 =1,d 2 =3andτ =., sstem (4) generates comple chaotic dnamic behaviors(showninfig.12). (iv) When d =1,d 1 =1,d 2 =3andτ =.8, sstem (4) generates comple chaotic dnamic behaviors(showninfig.13). (ii) When d =.2, d 1 =1,d 2 =3andτ =., sstem (4) generates comple chaotic dnamic behaviors(showninfig.). (iii) When d =1,d 1 =.2,d 2 =2andτ =, sstem (4) generates comple chaotic dnamic behaviors(showninfig.16). (iv) When d =1,d 1 =.4,d 2 =andτ =1, sstem (4) generates comple chaotic dnamic behaviors(showninfig.17)..2. Special case II : d d 1 (i) When d =.2,d 1 =1,d 2 =andτ =.1, sstem (4) generates comple chaotic dnamic behaviors(showninfig.14). 6. Conclusion We have presented a feedback control function to generate multi-scroll attractors from the Chen sstem. Then we have analed the dnamical

15 Multi-Scroll Chaotic and Hperchaotic Attractors Generated from Chen Sstem behavior of this new attractor. B varing the control parameters, we have achieved a set of new attractors with different scrolls. After that, we have generalied the multi-scroll attractors to generate more comple chaos and hperchaos b introducing time dela to the control function. This kind of control method ma also be used to other chaotic sstems. References Ahmad, W. M. & Sprott, J. C. [23] Chaos in fractional-order autonomous nonlinear sstems, Chaos Solit. Fract. 16, Ballinger, G. & Liu, X. [1997] On boundedness of solutions of impulsive sstems, Nonlin. Stud. 4, Chen, G. & Ueta, T. [1999] Yet another chaotic attractor, Int. J. Bifurcation and Chaos 9, Chua, L. O., Wu, C. W., Huang, A. & Zhong, G. [1993] A universal circuit for studing and generating chaos. I. Routes to chaos, IEEE Trans. Circuits Sst.-I 4, Cuomo, K. M. & Oppenheim, A. V. [1993] Circuit implementation of snchronied chaos with applications to communications, Phs. Rev. Lett. 71, Elwakil, A. S. & Kenned, M. P. [21] Construction of classes of circuit-independent chaotic oscillatorsusing passive-onl nonlinear devices, IEEE Trans. Circuits Sst.-I 48, Elwakil, A. S. & Oogu, S. [26] Multi-scroll chaotic oscillators: The nonautonomous approach, IEEE Trans. Circuits Sst.-II 3, Farmer, J. [1982] Chaotic attractors of an infinitedimensional dnamical sstem, Phsica D 4, Goedgebuer, J., Larger, L. & Porte, H. [1998] Optical crptosstem based on snchroniation of hperchaos generated b a delaed feedback tunable laser diode, Phs. Rev. Lett. 8, Grassi, G. & Mascolo, S. [1999] Snchroniing hperchaotic sstems b observer design, IEEE Trans. Circuits Sst.-II 46, Hartle, T., Loreno, C. & Qammer, H. K. [199] Chaos in a fractional order Chua s sstem, IEEE Trans. Circuits Sst.-I 42, Ikeda, K. & Matsumoto, K. [1987] High-dimensional chaotic behavior in sstems with time-delaed feedback, Phsica D 29, Khadra, A., Liu, X. & Shen, X. [23] Application of impulsive snchroniation to communication securit, IEEE Trans. Circuits Sst.-I, Kocarev, L., Halle, K. S., Eckert, K., Chua, L. O. & Parlit, U. [1992] Eperimental demonstration of secure communications via chaotic snchroniation, Int. J. Bifurcation and Chaos 2, Li, C. & Chen, G. [24] Chaos and hperchaos in the fractional-order Rossler equations, Phsica A 341, 61. Li, Y., Liu, X. & Zhang, H. [] Dnamical analsis and impulsive control of a new hperchaotic sstem, Math. Comput. Model. 42, Liu, X. [26] Stabilit of linear dela sstems via impulsive control, Int. J. Dn. Contin. Discr. Impul. Sst. 13, Liu, X., Teo, K. L., Zhang, H. & Chen, G. [26] Switching control of linear sstems for generating chaos, Chaos Solit. Fract. 3, Lu, H. & He, Z. [1996] Chaotic behavior in first-order autonomous continuous-time sstems with dela, IEEE Trans. Circuits Sst.-I 43, Lu, J., Han, F., Yu, X. & Chen, G. [24] Generating 3-D multi-scroll chaotic attractors: A hsteresis series switching method, Automatica 4, Namajunas, A., Pragas, K. & Tamasevicius, A. [199] An electronic analog of the Macke Glass sstem, Phs. Lett. A 21, Pecora, L. M. & Carroll, T. L. [199] Snchroniation in chaotic sstems, Phs. Rev. Lett. 64, Peng, J. H., Ding, E. J., Ding, M. & Yang, W. [1996] Snchroniing hperchaos with a scalar transmitted signal, Phs. Rev. Lett. 76, Rossler, O. E. [1979] An equation for hperchaos, Phs. Lett. A 71, 7. Tamasevicius, A., Mkolaitis, G. & Bumeliene, S. [26] Delaed feedback chaotic oscillator with improved spectral characteristics, Electron. Lett. 42, Wang, L. & Yang, X. [26] Generation of multi-scroll delaed chaotic oscillator, Electron. Lett. 42, Yalcin, M., Sukens, J., Vandewalle, J. & Oogu, S. [22] Families of scroll grid attractors, Int. J. Bifurcation and Chaos 12, Yalcin, M. [27] Multi-scroll and hpercube attractors from a general jerk circuit using Josephson junctions, Chaos Solit. Fract. 34, Yalcin, M. & Oogu, S. [27] N-scroll chaotic attractors from a first-order time-dela differential equation, Chaos 17, Yan, Z. Y. & Yu, P. [28] Hperchaos snchroniation and control on a new hperchaotic attractor, Chaos Solit. Fract. 3, Yang, T. & Chua, L. O. [1996] Ecure communication via chaotic parameter modulation, IEEE Trans. Circuits Sst.-I 43,