Abstract: This paper applies the center manifold theorem to reduce the dimensions of the Qi three-dimensional system. Local bifurcation phenomena are analyzed, including the pitchfork and Hopf bifurcations of the chaotic system. The Poincaré map is also investigated. The analyses demonstrate the rich dynamics of the Qi chaotic system. Finally, the frequency spectral analysis shows that the system has a broad frequency bandwidth, which is desirable for engineering applications such as secure communications

Sun, Yanxia

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Sun, Yanxia.

Name not available

Analysis of the Qi three dimensional chaotic system.

University of Johannesburg Institutional Repository

ISSN 0972-1118 VOLUME 11 NUMBER 1 February 2009 FAR  EAST  JOURNAL  OF  DYNAMICAL SYSTEMS Alejandro Mesón and Fernando Vericat: Estimates of large deviations in dynamical systems by a non-additive thermodynamic formalism 1 V. Sundarapandian: Necessary and sufficient conditions for local and global asymptotic stability of nonlinear cascade systems 17 Shiro Ishikawa: Ergodic problem in quantitative language 33 Lasha Ephremidze and Nobuhiko Fujii: The John-Nirenberg inequality for ergodic systems 49 S. Khademloo and G. A. Afrouzi: Existence of positive solutions of some quasilinear elliptic equation 57 Yan-Long Li, Wu Min and Wei Zhang: Influence of network topology on critical coupling parameter in synchronization of small world neural networks 65 Yanxia Sun, Guoyuan Qi, Barend J. Van Wyk and Zenghui Wang: Analysis of the Qi three dimensional chaotic system 77 Ying Huang and Zhengde Dai: Exact solitary wave solutions for a cubic 2D Ginzburg-Landau equation 95 Huimei Wang and Zhiwei Xing: Existence of positive periodic solution of an impulsive delay logistic model 107        PUSHPA PUBLISHING HOUSE  Vijaya Niwas, 198, Mumfordganj  Allahabad 211002, INDIA e-mail: arun@pphmj.com http://www.pphmj.com     amical Systems Vol. 5, No. 1, pp 1-99 June 2003   Far East Journal of Dynamical Systems Volume 11, Issue 1, 2009, Pages 77-94 Published online: February 23, 2009 This paper is available online at http://www.pphmj.com   2009 Pushpa Publishing House  :tionClassifica jectSub sMathematic 2000 37D45. Keywords and phrases: bifurcation, chaos, Hopf bifurcation, center manifold theorem, frequency spectral analysis. This work was supported by grants from Innovative Research Foundation of Tshwane University of Technology, the National Scientific Foundation of China (10772135, 60774088) and the Scientific Foundation of Tianjin City, P. R. China (07JCYBJC05800), the Scientific and Technological Research Foundation of Ministry of Education, P. R. China (207005). Received November 4, 2008 ANALYSIS OF THE QI THREE DIMENSIONAL CHAOTIC SYSTEM YANXIA SUN1, GUOYUAN QI1, BAREND J. VAN WYK1 and ZENGHUI WANG1,2 1F’SATLE Tshwane University of Technology Pretoria 0001, South Africa 2Department of Automation Shandong University of Science and Technology Qingdao 266510, P. R. China Abstract This paper applies the center manifold theorem to reduce the dimensions of the Qi three-dimensional system. Local bifurcation phenomena are analyzed, including the pitchfork and Hopf bifurcations of the chaotic system. The Poincaré map is also investigated. The analyses demonstrate the rich dynamics of the Qi chaotic system. Finally, the frequency spectral analysis shows that the system has a broad frequency bandwidth, which is desirable for engineering applications such as secure communications. YANXIA SUN et al. 781. Introduction With the advancement of chaos research, more and more chaos phenomena have been discovered in the mathematics, engineering and other fields. Many researchers have focused on the demonstration of chaos phenomena using simulation. The theoretical analyses of local and global characteristics and phenomena significantly have contributed to our understanding of the essence of chaos. Qi et al. [7] proposed a three-dimensional nonlinear chaotic system which exhibited a rich dynamics. This system is quite different from previous systems such as the Lorenz [6], Alligood et al. [1], Rossler [10], Chen [3] and series Lorenz systems [5]. Every equation of the Qi system has a nonlinear term, making it more complex in dynamics than the former systems. In this paper, two types of bifurcations are analyzed for the original Qi system, i.e., the pitchfork and Hopf bifurcations. The conditions of their existences are developed in detail by using the center manifold theorem and bifurcation theory. The Poincaré map and power spectrum exhibit the rich dynamic character of the Qi system [7]. 2. Qi Three-dimensional Chaotic System and its Equilibrium Properties The Qi system [7] is a three-dimensional continuous system described by ( ) ,yzxyax +−=  ,xzycxy −−=  ,bzxyz −=  (1) where ba,  and c are all real constant parameters. This system has five equilibria. Note that system (1) is invariant under the coordinate transformation ( ) ( ),,,,, zyxzyx −−→  i.e., the system is symmetrical about z-axis. Figure 1 shows the spectrum of Lyapunov exponents of system (1) … QI THREE DIMENSIONAL CHAOTIC SYSTEM 79with respect to parameter c. The largest Lyapunov exponent can reach a value of 10, which is big different from those of the Lorenz system and the Chen system with 0.9 and 2, respectively. As can be seen from Figure 1, the nonlinear system displays chaos, and the positive Lyapunov exponents are high in a large range of parameters.  Figure 1. Spectrum of Lyapunov exponents of system (1) with ,35=a  .38=b  Let ( ) ,,24,2 222222 bcabcfcacaacbebcabcabd +=+++−=++−=  (2) ,22,22 21aedxaedx−=+=  ,2,2 21efaaedbcyefaaedbcy −−=++=  ( ) ( ),, 21efcedzefcedz −−=++=  (3) the following five equilibria are found: [ ] ( ) ( ),,,,,,,0,0,0 111311121 zyxSzyxSS −−===  ( ) ( ).,,,,, 22252224 zyxSzyxS −−==  (4) YANXIA SUN et al. 80It can easily be verified that when 22,,0 yxed <−  are complex. So only 21, SS  and 3S  are the equilibria of the system. By linearizing system (1) at ,1S  one obtains the Jacobian  .00010−−−bcaa (5) And its characteristic equation,  ( ) ( ) ( ( ) ) .012 =−+λ++λ+λ=λ acaabf  (6) It is obvious that b−  is one of the roots of (6), and the other two roots do not always have negative real parts according to the Routh-Hurwitz condition. 3. Bifurcations of Origin 3.1. Pitchfork bifurcation of 1S  From (6) with ,1=c  we have ( ) ,0,01,0 321 <−=λ<+−=λ=λ ba  and the corresponding eigenvectors are  [ ] [ ] [ ] .1,0,0,0,1,,0,1,1 321 TTT vavv =−==  (7) According to the center manifold theorem [12], the topological structure of system (1) will change and generate pitchfork bifurcation because of ,01 =λ  where c is the local bifurcation parameter at 1. Within the neighborhood of ,1=c  let ξ+= 1c  with ξ sufficiently small. Then we have ( ) ( )1412121 21 −++++−=λ caaa  ( ) +ξ++−+=2141121aaa ( ) ( )  ξ++ξ++−+= 22121121oaaa ( ) ( ),12ξ++ξ= oaa (8) … QI THREE DIMENSIONAL CHAOTIC SYSTEM 81where Taylor expansion is used. By utilizing the eigenvectors to construct a new vector space, we obtain the following transformation:  .10001101−=wvuazyx (9) Thus the system is transformed into the form  ( ) ,00010000321+−+−=gggwvubawvu (10) where (( ) ( ) ),111 21 ++−−=auvauvag  ( )( ),1122 +−+=avwauwg  ( ) .1223 uvaavug −+−=  Since ,0,,0 321 <λλ=λ  there exists a center manifold which is tangent to the u axis. The center manifold is expressed as ( ) {( ) ( ) ( ) ,,, 2131 δ<==|∈= uuhwuhvRwvuSW c  ( ) ( ) }.2,1,00,00 === iDhh ii  (11) Here δ is sufficiently small. To seek the center manifold ( ),1SW c  we assume that ( )uh1  and ( )uh2  have the forms: ( ) ,4131211 "+++== ucubuauhv  ( ) .4232222 "+++== ucubuauhw  (12) Substituting (12) into the second and third equations of (10), respectively, and comparing the coefficients of ,,, 432 uuu  we have two sets of YANXIA SUN et al. 82equations. The first set of equations is  ( )( )( )( ) ( ) ( )+−++++−=−++−+=+−=.111211211,1120,10212121121aaaaabcaaaaabaaaaa (13) The second set of equations is  ( )( ) ( )−−−=+−=−−=−.1112,01,01212122212aabcbaaaabbaaba (14) Solving equations (13) and (14) simultaneously, we have ( ) ,0,12,0 1211 =+== cbaba   ( )( ) .112,0,1322222bababacbba +−−+===  (15) Thus the following center manifold equations are obtained: ( ) ( ) ( ),012 5321uubauhv ++==  ( ) ( )( ) ( ).01121 5432222 uubababaubuhw ++−−++==  (16) Finally, substituting (16) into (10), a one-dimensional (1-D) reduced vector field, ( ) ( ) ( ) ( ) ( ),11111 53531 uoubaauaauoubaauu ++−++ξ=++−+λ=  (17) 0=ξ   is obtained based on the center manifold theory, which can be used to investigate the bifurcation. We now can compute the center manifold with … QI THREE DIMENSIONAL CHAOTIC SYSTEM 83an accuracy of order 3, which is sufficient to answer questions of stability. Let ( ) ,0, =ξ= ufu  and ignoring the term ( )5uo  in (17), we have ( )( ) ( ),0,,00,00,00,0=ξ∂∂∂∂= fuff  ( ) ( ) ( )( )( ) .0116,01,00,0330,020,022≠+−=∂∂≠+=ξ∂∂∂=∂∂baaufaaufuf According to the theorem [12], the equilibrium point ( ) ( )0,0, =ξu  undergoes a pitchfork bifurcation at .0=u  Furthermore, let  ( ) ( ) ,01113 =+−++ξubaauaa (18) we have ( ) ( ).1,1,0 3,21 >−ξ±== aaabuu  Hence, about the equilibria of (18), under the condition 1>a  we have (1) when ,01 <ξ<−  i.e., ,10 << c  the reduced 1-D system (18)    has only one equilibrium at origin 01 =u  because 3,2u  are a pair of imaginary numbers; (2) when ,0>ξ  i.e., 1>c  it has three equilibria at 3,21 ,0 uu =  ( ) ( ),1,1 >−ξ±= aaab respectively. Furthermore, we have  ( )<ξ<>ξ>+ξ=∂ξ∂= ,0,0,0,01,1aauufuu (19)  ( ) .0,012,3,2>ξ<+ξ−=∂ξ∂= aauufuu (20) Therefore, when ,01 <ξ<−  the only one equilibrium 1u  of system (18) is a sink. At ,0=ξ  i.e., ,1=c  system (18) undergoes pitchfork bifurcation from one equilibrium into three equilibria. When ,0>ξ  the equilibrium YANXIA SUN et al. 841u  becomes a source, and another two equilibria 3,2u  come out and are sinks. From (18), with the parameter changing of the system, the range of ξ accordingly changes. The diagram of the pitchfork bifurcation of equation (18) is shown in Figure 2. The blue lines denote the trajectories of stable equilibrium with parameter ξ increasing, and the red line denotes the trajectory of the unstable equilibrium. It is notable that in the neighborhood of 0,1 =ξS  corresponds to the bifurcation parameter .1=c  Therefore, system (1) undergoes a pitchfork bifurcation at .1=c  Correspondingly, the system has only one globally stable equilibrium when .01 <ξ<−  It has three equilibria in which the origin is stable, while another two non-zero equilibria are unstable when .0>ξ  The diagram of pitchfork bifurcation of system (1) with parameter changing is shown in Figure 3 where the curves with arrows show that although the two orbits of system (1) start from the neighborhood of the origin (a saddle) are attracted into the non-zero equilibria, respectively.  Figure 2. The diagram of the pitchfork bifurcation based on system (18). … QI THREE DIMENSIONAL CHAOTIC SYSTEM 85 Figure 3. The diagram of the pitchfork bifurcation based on system (1). We therefore obtain the following theorem: Theorem 1. Under condition 1>a  and ξ sufficiently small, on the line ,1=c  system (1) undergoes a pitchfork bifurcation at origin [ ].0001 =S  For ξ+= 1c  and ,01 <ξ<−  there is one equilibrium [ ],0001 =S  and 1S  is a sink. For ξ+= 1c  and ,0>ξ  two new equilibria ( )1112 ,, zyxS =  and ( )1113 ,, zyxS −−=  emerge and are sinks while 1S  becomes a source. 3.2. Hopf bifurcation of 1S  In this section, we deal with another kind of bifurcation at origin [ ]0001 =S  of system (1) using an analytical method. From (6), we have these three eigenvalues at origin.  ( ) ( ) iwaacaaa ±σ=++−±−−=λ 21421 22,1  and .3 b−=λ  (21) The existence of Hopf bifurcation of equilibrium has three following conditions [12]: (1) The eigenvalues cross the imaginary axis transversely. (2) The coefficient 1Λ  in the Poincaré-Andronov-Hopf normal form is nonzero. YANXIA SUN et al. 86It is easy to get  ( ),211 −=σ=ρ daad (22) so the first condition of the Hopf bifurcation is satisfied. Now we investigate the second condition. Suppose the characteristic equation (6) has pure imaginary roots ( ).02,1 >±=λ wiw  When 10 −== aa  the Jacobian matrix of system (1) has a pair of imaginary eigenvalues and one negative real eigenvalue, i.e.,  ( ) bccwawiw −=λ>−==±=λ 30002,1 ,1,1,  (23) and the corresponding eigenvectors  [ ] .100,01,01 321 TTTvcwivcwiv =−==  (24) By utilizing the real generalized eigenvectors as the basis of new coordinates, we obtain the transformation  .1000001112ωω−ω=zyxzyx (25) System (1) becomes  .000000321111111+−ωω−=fffzyxbzyx (26) Here ( ) ,1 1111 zxycf −ω=  ( ) ,11 111112 zxyczxf −ωω+ω−=  .1132123 yxxf ω−ω=  … QI THREE DIMENSIONAL CHAOTIC SYSTEM 87Thus according to the center manifold theorem, there exists a center manifold for (26), which could be represented locally by ( ) {( ) 131111 zRzyxSW c |∈=  ( ) ( ) ( ) },00,0,00,0,,,, 1111 ==δ<= Dhhyxyxh  (27) where δ is sufficiently small. We assume that ( )111 , yxhz =  .31721161215314213112211 "ydyxdyxdxdydyxdxd ++++++=  (28) The center manifold can be approximately computed by substituting (28) into (26) and comparing the coefficients, to obtain ( ) ,22,22,2 222332223222321 ω+ω−ω=ω+ω−ω=ω+ω+ω=bbbdbbdbbd  .0,0,0,0 7654 ==== dddd  (29) The vector field reduced to the center manifold is therefore given by  ( )( ) ,,,00112111110011 +ωω−=yxfyxfyxyx (30) where ( ) ( )( ),,,,, 11111111 yxhyxfyxf =  ( ) ( )( ).,,,, 11112112 yxhyxfyxf =  (31) The index number 1Λ  can be computed as [ ]22111 111111111111161yyyyxxyyxxxx ffff +++=Λ  [ ( ) ( )2221110 111111111111161yyxxyxyyxxyx ffffff +−+ω+  ]212111111111 yyyyxxxxffff +−  .22222161 321  −ω−ω−ω+=cddcccd (32) YANXIA SUN et al. 88To meet the condition of index number ,01 ≠Λ  we have  .022222 321 ≠−ω−ω−ω+cddcccd (33) Therefore, the second condition of Hopf bifurcation is met. From (22) and (33), we have the conditions of Hopf bifurcation as follows:  .022222,1 321 ≠−ω−ω−ω+>cddcccdc  (34) Theorem 2. If the parameters of system (1) meet condition (34), system (1) undergoes a Poincaré-Anddronov-Hopf bifurcation (Hopf bifurcation) at origin [ ].0001 =S  A transition from sink to periodic motion occurs. Moreover, since ,0211 <−=ρ  the periodic solution emerging after ,1−>a  is stable if ,01 <Λ  and is unstable if .01 >Λ  For example, when ,10,1 === cba  we have ,0257.01 −=Λ  =ρ1  ( ).0211<−=α=adaad According to Theorem 2, the stable periodic orbits emerge from zero equilibrium when .1−<a  Select [ ]05.0,05.0,05.0  as the initial states in the neighborhood of the origin. When 1−>a  the zero equilibrium is stable, and thus the system orbit is attracted to zero as shown in Figure 4(a), when ,1−<a  the system orbit is attracted to a stable periodic orbit, at the same time, the zero equilibrium becomes unstable as shown in Figure 4(b). When 1−<a  the limit circle is shown clearly in Figure 5.  (a) … QI THREE DIMENSIONAL CHAOTIC SYSTEM 89 (b) Figure 4. Hopf bifurcation in system (1) according to Theorem 2. (a) ,195.0 −>−=a  (b) .105.1 −<−=a   Figure 5. The limit circle omitting the transient response when =a  .105.1 −<−  4. Poincaré Map of the Chaotic Attractor As an important analysis technique, the Poincaré map can reveal bifurcation and folding properties of chaos. When ,35=a  ,38=b  YANXIA SUN et al. 90,206=c  one may take the equilibrium value of ,2S  i.e., ,, 11 yyxx ==  and 1zz =  as crossing planes, respectively, where ,4622.611 =x  ,9314.81 =y  and .8547.2051 =z  Figures 6(a-c) show the Poincaré maps on different crossing planes. They indicate that the system has extremely rich dynamics.  (a) Poincaré map on the crossing section .1xx =   (b) Poincaré map on the crossing section .1yy =  … QI THREE DIMENSIONAL CHAOTIC SYSTEM 91 (c) Poincaré map on the crossing section .1zz =  Figure 6. Poincaré maps on different crossing sections when ,35=a  .206,38 == cb  5. Frequency Spectral Analysis Many proposed chaos-based encryption schemes have been totally or partially broken by different attacks [8, 9, 11]. One of the reasons is that the degree of randomness and disorder of simple chaotic signals are not high enough as reflected by their narrow bandwidths. Recall that the bandwidths of the Lorenz and Chen systems are approximately between 0 and 5.8Hz and therefore cannot be used to completely mask the messages in real communication applications. Once intercepted, there is a high possibility that the messages can be extracted. Figures 7(a) and (b) show the frequency spectra of the Lorenz and Chen systems, respectively. However, the frequency spectrum of the Qi system has very wide bandwidth as shown in Figure 8. The bandwidth of signal y is about 35Hz, which is nearly 10 times wider than that of the Lorenz system, which could be advantageous to secure communication applications. YANXIA SUN et al. 92 (a) The Lorenz system with .28,38,10 === cba   (b) The Chen system with .28,38,35 === cba  Figure 7. The frequency spectra generated numerically from several typical chaotic systems. … QI THREE DIMENSIONAL CHAOTIC SYSTEM 93 (a) spectrum of y.  (b) spectrum of x. Figure 8. The frequency spectra generated numerically from Qi chaotic systems with .206,38,35 === cba   YANXIA SUN et al. 946. Conclusion In this paper, the pitchfork and Hopf bifurcations, the poincaré map and frequency spectrums of the Qi three dimensional chaos system have been investigated. Based on a rigorous theoretical analysis, the stability of the fixed points when the bifurcation parameter passes the pitchfork bifurcation point has been analyzed. The bifurcation of the period cycle emerging from the zero equilibrium as can be seen from the Hopf bifurcation analysis was also investigated. The frequency spectral analysis shows that the system has a very wide frequency bandwidth, which is very desirable for engineering applications such as secure communications. References  [1] K. Alligood, T. Sauer and J. Yorke, Chaos: An Introduction to Dynamical Systems, 3rd ed., Springer-Verlag, New York, 2000.  [2] G. Chen and J. Lü, Dynamic Analysis, Control and Synchronization for Lorenz System Family, Science Press, Beijing, 2003.  [3] G. Chen and T. Ueta, Yet another chaotic attractor, Int. J. Bifur. Chaos 9 (1999), 1465-1466.  [4] M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, New York, Heidelberg, Berlin, 1974.  [5] T. Li, G. Chen and Y. Tang, On stability and bifurcation of Chen’s system, Chaos Solutions Fractals 19 (2004), 1269-1282.  [6] J. Lu, D. Cheng and S. Celikovsky, Bridge the gap between the Lorenz system and the Chen system, Int. J. Bifur. Chaos 12 (2002), 2917-2926.  [7] G. Qi, G. Chen and S. Du, Analysis of a new chaotic system, Physic A 352 (2005), 295-308.   [8] G. Qi, M. A. van Wyk and B. J. van Wyk, On a new hyperchaotic system, Phys. Lett. A 372 (2007), 124-136.  [9] G. Qi, G. Chen, M. A. van Wyk, B. J. van Wyk and Y. Zhang, Four-wing chaotic attractor generated from a new 3-D quadratic chaotic system, Chaos Solitons Fractals, 2007 (in press and available online).  [10] O. Rossler, An equation for hyperchaos, Phys. Lett. A 71 (1979), 155-157.  [11] T. Ueta and G. Chen, Bifurcation analysis of Chen’s attractor, Int. J. Bifur. Chaos 10 (2000), 917-931.  [12] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 2003. 

Analysis of the Qi three dimensional chaotic system

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Analysis of the Qi three dimensional chaotic system

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