Chin. Phys. B Vol. 19 No. 12 2010) 120510 A new four-imensional chaotic system Chen Yong ) a)b) an Yang Yun-Qing ) a) a) Shanghai Key Laboratory of Trustworthy Computing East China Normal University Shanghai 200062 China b) Nonlinear Science Center an Department of Mathematics Ningbo University Ningbo 315211 China Receive 26 January 2010; revise manuscript receive 3 August 2010) A new four-imensional chaotic system with a linear term an a 3-term cross prouct is reporte. Some interesting figures of the system corresponing ifferent parameters show rich ynamical structures. Keywors: chaotic system Lyapunov exponent attractor PACC: 055 0200 1. Introuction In 1963 Lorenz foun the first chaotic attractor in a three-imensional 3D) autonomous system [1] later Rösslor constructe an even simpler three-imensional chaotic system. [2] Since then chaos as an important nonlinear phenomenon has been stuie in science mathematics engineering communities an so on. [3 12] As chaos is useful an has great potential applications in many technological isciplines the iscovery an the creation of chaos are important. In the past few years Chen [] constructe a 3D chaotic system via a simple state feeback to the secon equation in the Lorenz system followe by a closely relate Lü system constructe by Lü [5] an a unifie system [6] that combines Lorenz system Chen system an Lü system as its special cases. Some other 3D chaotic systems are also constructe. Recently Qi et al. [13 18] propose a new 3D chaotic system an D chaotic system with cubic terms. Here we report a new D chaotic system with a linear term an a cubic term which also takes on goo symmetries an similarities. 2. New D system an its properties The new D system is escribe by ẋ 1 = ax 1 b 1 x 1 x 2 x 3 ẋ 2 = bx 2 b 2 x 1 x 3 x ẋ 3 = cx 3 b 3 x 1 x 2 x ẋ = cx b x 1 x 2 x 3. 1) i) Symmetry The system is invariant for the following coorinate transformations: x 1 x 2 x 3 x ) x 1 x 2 x 3 x ) x 1 x 2 x 3 x ) x 1 x 2 x 3 x ) x 1 x 2 x 3 x ) x 1 x 2 x 3 x ) x 1 x 2 x 3 x ) x 1 x 2 x 3 x ). 2) So it is of symmetry. ii) Dissipation Since V = ẋ 1 x 1 + ẋ 2 x 2 + ẋ 3 x 3 + ẋ x = a + b + c + 3) when a + b + c + < 0 system 1) is issipative with an exponential contraction rate V t = a + b + c +. ) It means that a volume element V 0 is contracte into a V 0 e a+b+c+)t at time t. Therefore orbits near the chaotic attractor are ultimately restricte within a specific fractal-imensional subspace of zero volume. iii) Equilibria The equilibria of system 1) can be obtaine by solving the following equation: ax 1 b 1 x 1 x 2 x 3 = 0 bx 2 b 2 x 1 x 3 x = 0 Project supporte by the National Natural Science Founation of China Grant Nos. 10735030 11075055 an 9071801) the Shanghai Leaing Acaemic Discipline Project China Grant No. B12) the Program for Changjiang Scholars the Innovative Research Team in University of Ministry of Eucation of China Grant No. IRT 073) an the K. C. Wong Magna Fun in Ningbo University. Corresponing author. E-mail: ychen@sei.ecnu.eu.cn c 2010 Chinese Physical Society an IOP Publishing Lt http://www.iop.org/journals/cpb http://cpb.iphy.ac.cn 120510-1
Chin. Phys. B Vol. 19 No. 12 2010) 120510 cx 3 b 3 x 1 x 2 x = 0 cx b x 1 x 2 x 3 = 0. 5) By calculations one can fin 9 real equilibria incluing zero. Let cb1 bb 3 3 a p = ± b ab 2 b 3 b1 bab 2 p q = ± cb1 b 3 ap r = ± 6) then the equilibria except zero can be enote by S = p q r b ) pqr. 7) Let cb1 bb 3 3 a p 1 = b ab 2 b 3 cb1 bb 3 3 a p 2 = b ab 2 b 3 b1 bab 2 p 1 q 11 = q 12 = b1 bab 2 p 2 q 21 = q 22 = cb1 b 3 ap 1 r 11 = r 12 = r 21 = cb1 b 3 ap 2 r 22 = the equilibria can be enote as follows: b1 bab 2 p 1 b1 bab 2 p 2 cb1 b 3 ap 1 cb 1 cb1 b 3 ap 2 8) S 0 = 0 0 0 0) S 1 = p 1 q 11 r 11 b ) p 1 q 11 r 11 S 2 = p 1 q 11 r 12 b ) p 1 q 11 r 12 S 3 = p 1 q 12 r 11 b ) p 1 q 12 r 11 S = p 1 q 12 r 12 b ) p 1 q 12 r 12 S 5 = p 2 q 21 r 21 b ) p 2 q 21 r 21 S 6 = p 2 q 21 r 22 b ) p 2 q 21 r 22 S 7 = p 2 q 22 r 21 b ) p 2 q 22 r 21 S 8 = p 2 q 22 r 22 b ) p 2 q 22 r 22. 9) It can be seen that S 1 an S 2 are symmetric with respect to plane x 1 x 2 S 1 an S 3 are symmetric with respect to plane x 1 x 3 S 1 an S are symmetric with respect to plane x 1 x S 1 an S 5 are symmetric with respect to 0000) S 1 an S 6 are symmetric with respect to plane x 3 x S 1 an S 7 are symmetric with respect to plane x 2 x S 1 an S 8 are symmetric with respect to plane x 2 x 3. iv) Jacobian matrix By linearizing system 1) at S i = x 1 x 2 x 3 x ) one can obtain the Jacobian as follows: a b 1 x 3 x b 1 x 2 x b 1 x 2 x 3 b 2 x 3 x b b 2 x 1 x b 2 x 1 x 3 A i =. 10) b 3 x 2 x b 3 x 1 x c b 3 x 1 x 2 b x 2 x 3 b x 1 x 3 b x 1 x 2 If i = 0 x 1 = x 2 = x 3 = x = 0 then the eigenvalues of matrix A 0 are λ 01 = a λ 02 = b λ 03 = c λ 0 =. 11) Therefore when abc < 0 the equilibrium S 0 is a sale point. Given the values of x 1 x 2 x 3 x at S i i = 1... 8) one can calculate the eigenvalues of A i. By calculating it is seen that A i i = 1... 8) have the same eigenvalues. 3. Observation of new chaotic attractor By choosing the parameters from system 1) a great eal of ynamics can be observe which is liste together with some iscoveries as follows: I) a = 35 b = 10 c = 1 = 10 b 1 = 1 b 2 = 1 b 3 = 1 an b = 1. In this case a+b+c+ = 35+10 1 10 = 36 so the system is issipative an the eigenvalues of the Jacobian matrix at S 0 are λ 01 = 35 λ 02 = 10 λ 03 = 1 λ 0 = 10 12) an one can easily fin λ 02 = 10 > 0 implying that S 0 is a sale point. By calculating with Maple we can obtain the eigenvalues of the Jacobian matrix at S i i = 1... 8) as λ i1 = 3.903 + 12.1911i λ i2 = 3.903 12.1911i λ i3 = 3.9551 λ i =.025 13) the real part of λ i1 λ i2 is 3.9026006 > 0 so S i i = 1... 8) is also a sale point. By calculating with Matlab the Lyapunov exponents of this system with these parameters are obtaine to be l 1 =.361 l 2 = 0.0000 120510-2
Chin. Phys. B Vol. 19 No. 12 2010) 120510 l 3 = 3.72 l = 36.8879. 1) We can easily fin that the maximum Lyapunov exponent is positive so the system is chaotic. Figure 1 shows numerical results for projections on ifferent phase planes an phase spaces. Especially we can obtain two chaotic attractors when we choose ifferent initial values which can be seen in Figs. 1a) 1k). Fig. 1. Chaos system projections on ifferent phase planes an phase spaces with the parameters: a = 35 b = 10 c = 1 = 10 b 1 = 1 b 2 = 1 b 3 = 1 an b = 1. a) 3D view in the x 1 x 2 x space; b) 3D view in the x 1 x 2 x 3 space; c) 3D view in the x 2 x 3 x space; ) 3D view in the x 1 x 3 x space; e) Projection in the x 1 x 2 plane; f) Projection in the x 1 x 3 plane; g) Projection in the x 1 x plane; h) Projection in the x 2 x 3 plane; i) Projection in the x 2 x plane; j) Projection in the x 3 x plane; k) two coexisting chaotic attractor. When a varies from 35 to 20 the Lyapunov exponents are l 1 = 0.0022 l 2 = 2.2532 l 3 =.702 l = 1.08. 15) The maximum Lyapunov equals zero implying that the system has a perioic orbit. Figure 2 shows numerical results for projections on ifferent phase spaces. II) a = 15 b = 5 c = 1 = 9 b 1 = 1 b 2 = 1 b 3 = 1 an b = 1. Similar to the first case the system by choosing the above parameters is also chaotic. Here only the 3D view figure see Fig. 3) in the x 1 x 2 x space is given the other figures are omitte for the sake of concision. 120510-3
Chin. Phys. B Vol. 19 No. 12 2010) 120510 Fig. 2. Chaos system projections on ifferent phase planes an phase spaces with the parameters: a = 20 b = 10 c = 1 = 10 b 1 = 1 b 2 = 1 b 3 = 1 an b = 1. a) 3D view in the x 1 x 2 x 3 space; b) 3D view in the x 1 x 3 x space; c) Projection in the x 1 x 2 plane; ) Projection in the x 1 x 3 plane. Fig. 3. 3D view in the x 1 x 2 x space with parameters: a = 15 b = 5 c = 1 = 9 b 1 = 1 b 2 = 1 b 3 = 1 an b = 1. III) a = 10 b = 3 c = 1 = 2 b 1 = 1 b 2 = 1 b 3 = 1 an b = 1. Similar to the first case this system uner the above parameters is chaotic. The 3D view figures see Fig. ) in the x 1 x 2 x space with one initial value an two initial values are given the other figures are omitte for the sake of concision. Fig.. 3D view in the x 1 x 2 x space with parameters: a = 10 b = 3 c = 1 = 2 b 1 = 1 b 2 = 1 b 3 = 1 an b = 1: a) one initial value; b) two initial values. 120510-
Chin. Phys. B Vol. 19 No. 12 2010) 120510 In summary we first constructe a new D chaotic system an stuie its properties. Some interesting figures are given in which one can see that the new system possesses very rich ynamical structures. Hopf bifurcation Poincaré map synchronization an so on will be our further stuy. References [1] Lorenz E N 1963 J. Atmos. Sci. 20 130 [2] Rossler O E 1976 Phys. Lett. A 57 397 [3] Celikovsky S an Chen G 2002 Int. J. Bifur. Chaos 12 1789 [] Chen G an Ueta T 1999 Int. J. Bifur. Chaos 9 165 [5] Lü J an Chen G 2002 Int. J. Bifur. Chaos 12 659 [6] Lü J Chen G Cheng D an Celikovsky S 2002 Int. J. Bifur. Chaos 12 2917 [7] Zhang R an Yang S 2009 Chin. Phys. B 18 3295 [8] Chen Y an Yan Z 2003 Appl. Math. Mech. 2 256 [9] Yang Y an Chen Y 2009 Chaos Solitons an Fractals 39 2378 [10] Gu Q L an Gao T G 2009 Chin. Phys. B 18 8 [11] Tao C H Lu J A an Lü J 2002 Acta Phys. Sin. 51 197 in Chinese) [12] Zhou P Cao Y an Cheng X 2009 Chin. Phys. B 18 139 [13] Qi G Du S Chen G Chen Z an Yuan Z 2005 Chaos Solitons an Fractals 23 1671 [1] Qi G Chen G an Zhang Y 2006 Phys. Lett. A 352 386 [15] Qi G Chen G Du S Chen Z an Yuan Z 2005 Physica A: Stat. Mech. Appl. 352 295 [16] Zhou P Wei L an Cheng X 2009 Acta Phys. Sin. 58 5201 in Chinese) [17] Tang L Li J an Fan B 2009 Acta Phys. Sin. 58 16 in Chinese) [18] Hui Meng Zhang Y an Liu C 2008 Chin. Phys. B 17 3258 120510-5