Hopf bifurcations in an extended Lorenz system

Transcription

1 Zhou et al. Advances in Difference Equations :28 DOI /s R E S E A R C H Open Access Hopf ifurcations in an extended Lorenz system Zhiming Zhou 1,2*, Gheorghe Tigan 3 and Zhiheng Yu 4 * Correspondence: zzmcm@sina.com 1 Department of Mathematics, Shanghai Normal University, Shanghai, China 2 Department of Mathematics, Huei University of Science and Technology, Huei, China Full list of author information is availale at the end of the article Astract In this work, we study Hopf ifurcations in the extended Lorenz system, ẋ = y, ẏ = mx ny mxz px 3, ż = az + x 2, with five parameters m, n, p, a, R. For some values of the parameters, this system can e transformed to the classical Lorenz system. In this paper, we give conditions for occurrence of Hopf ifurcation at the equilirium points. We find totally three limit cycles, each of them located around one of the three equiliria of the system. Numerical simulations illustrate the validity of these conditions and the existence of limit cycles. MSC: 37G15; 70K50 Keywords: extended Lorenz system; Hopf ifurcation; center manifold; Lyapunov constant; degeneration 1 Introduction At 1963, E.N. Lorenz presented the first chaotic system, now called the Lorenz chaotic attractor, which can e formulated as follows [1]: Ẋ = σ Y X, Ẏ = ρx Y XZ, Ż = rz + XY. 1.1 Since then, chaotic attractors attract more and more interest, and many other chaotic attractors have een given, such as Rössler system [2],Chua scircuit [3], Chen system [4], Lü system [5], T system [6], and so on. Various dynamical ehaviors of chaotic systems are revealed. Recently, there has een increasing attention aout hidden attractors ecause of their rich ut distinctive dynamical ehaviors [7 15]. Chaos ecomes more and more popular notion in the nonlinear science. The main purpose of the article is to study the Hopf ifurcation of the following system given y [16]: ẋ = y, ẏ = mx ny mxz px 3, ż = az + x 2, 1.2 The Authors This article is distriuted under the terms of the Creative Commons Attriution 4.0 International License which permits unrestricted use, distriution, and reproduction in any medium, provided you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Zhou et al. Advances in Difference Equations :28 Page 2 of 10 where m, n, p, a, R are parameters of system 1.2. The Lorenz system 1.1 cane transformed into system 1.2 y the change of variales x = 2X, y = 2X + Y σ, z = X 2 σ 2σ r +ρ 1Z with a = r, =, m = σ ρ 1,p =1,n = σ +1.System1.2 wasintroduced recently in [16] astheextended Lorenz system. Few results are reported on σ ρ 1 this system so far. In [16], the authors studied the irth of the Lorenz attractor in the extended Lorenz system and revealed relations etween the extended Lorenz system and the Lorenz system. The extended Lorenz system 1.2 is invariant under the transformation x, y, z x, y, z. So the orits of system 1.2 are symmetric with respect to the z-axis. The origin O0,0,0 is an equilirium of system 1.2 regardless the values of the parameters. If ap + m 0and am ap+m 0, then the system has two other equiliria Ax 0,0,z 0 and A x 0,0,z 0 withx 0 = am ap+m and z 0 = m ap+m.thejacoianmatrixato is m n a with the eigenvalues 2 1 n2 +4m n, 1 n2 +4m 1 n,and a. Thisimpliesthat,for 2 m < n2, the Jacoian matrix has a pair of complex eigenvalues. If additionally we assume 4 that a > 0, then the system may undergo a Hopf ifurcation around O at the critical value n =0. This paper is organized as follows. In Section 2, we analyze the Hopf ifurcation of the extended Lorenz system at the origin. In Section 3, westudythehopfifurcationatthe nontrivial equiliria A and A. Numerical simulations illustrate the validity of the theoretical analysis in the last sections. 2 Hopf ifurcation of the origin In this section, we analyze the Hopf ifurcation at the origin of coordinates. Theorem 1 If n =0, m <0, a >0, then the extended Lorenz system 1.2 undergoes Hopf ifurcation at the origin. When <0, the Hopf ifurcation is supercritical, whereas if >0,then the Hopf ifurcation is sucritical. When =0,the extended Lorenz system 1.2 has no limit cycles near the origin of coordinates. Proof Whenever m < n2,system1.2 has a pair of conjugate complex eigenvalues 4 λ 1 := 1 2 n i n 2 +4m, λ 2 := 1 2 n 1 2 i n 2 +4m, and a real eigenvalue λ 3 := a. Hence,system1.2 hasasimplepairofpurelyimaginary eigenvalues as n = 0 and no other eigenvalues with zero real parts since a 0. Therefore, condition H1 of Theorem in [17], pp , is satisfied. Next, it is ovious that d Reλ 1 dn = 1 n=0 2 <0, so that condition H2 of Theorem in [17], pp , is also satisfied.

3 Zhou et al. Advances in Difference Equations :28 Page 3 of 10 We further compute the first Lyapunov coefficient. By applying the linear change x u y = 1 2 n 2 1 n2 4m 0 v z w we transform system 1.2into u 1 2 n 2 1 n2 4m 0 u 1 v = 2 n2 4m 1 2 n 0 v + ẇ 0 0 a w 0 2pu 3 +2muw n 2 4m u 2. Moreover, let η = u + iv,wherei 2 = 1, from which it follows that η = u iv, u = η η v =. After a simple calculation, we have 2 i η = 1 2 n + 2 i i p n2 4m η + n2 4m 4 η + η3 + mwη + η, ẇ = aw + 4 η 2 +2η η + η 2. η+ η 2,and 2.1 When n =0,m = ω 2 <0ω > 0 is an aritrary real constant, and a 0,system2.1 satisfies the center manifold theorem [18]. Thus, the center manifold W c has the representation w = Wη, η= 1 2 w 20η 2 + w 11 η η w 02 η 2 + O η with unknown w ij C.SinceW must e real, w 11 is real, and w 20 = w 02. Let us rewrite system 2.1atn =0: η = i mηi + 2 m p 4 η + η3 + mwη + η, ẇ = aw + 4 η2 +2η η + η Sustituting 2.2into2.3 2 and using 2.3 1,weget,atthequadraticlevel, a +2i mw 20 = 2, aw 11 = 2, a 2i mw 02 = 2. Thus, w 20 = 2a +2i m, w 11 = 2a, w 02 = 2a 2i m, and the center manifold W c is w = Wη, η= 4a +2i m η2 + 2a η η + 4a 2i m η2 + O η

4 Zhou et al. Advances in Difference Equations :28 Page 4 of 10 Now the restriction of 2.3toitscentermanifold,uptocuicterms,canewrittenas follows: η = i mη + 3p a + i m [ p 4 + 4a 2i m 4a +2i m η η 2 + 3p η a + p 4 + 4a 2i m η 2 η 4a +2i m ] η 3 +, 2.5 where we omit higher-order terms when considering the Hopf ifurcation. Now we can compute the first Lyapunov coefficient. There are three methods to compute the Lyapunov constant: the Poincaré map, the Poincaré formal series, and the norm form method. According to Han s ook [19], pp , these three methods are equivalent to each other, and the Lyapunov constants otained y the methods are the same up to a positive constant. Here we adopt the formal series method ecause it is easier to compute y algera system. η+ η η η Using the relationship u = and v = 2 2 i,wehave u = mv, v = 2.6 mu + 2a3 p+a 2 4amp 2m ma u am a 2 4m u2 v + 4 m a 3 4am uv2 + Or 4, where r = u 2 + v 2. By using the Maple program edited according to the formal series algorithm we otain the first Lyapunov constant L 1 = 2 ma 2 4m 2.7 and the formal series F = u 2 + v 2 + a3 p + a 2 4amp 2mu 4 maa 2 4m + u 3 v ma2 4m uv 3 ma2 4m 2 v4 a 2 4m +. Since m < 0, the denominator of 2.7 is positive, so L 1 and the parameter have the same sign. Thus, we have the result of the second part of Theorem 1. When = 0, the center manifold of system 1.2 isw = 0, and the restriction of system 1.2 in the center manifold ecomes ẋ = y, ẏ = mx ny px If n =0,thensystem2.8 is a Hamilton system with the Hamiltonian function Hx, y= 1 2 y2 m 2 x2 + p 4 x4. There exists a family of periodic orits given y L h : Hx, y =h, h 0, +. Because the limit cycle is an isolated periodic orit, system 2.8and,equivalently, system 1.2 have no limit cycles in this case. If n 0, then limit cycles also do not exist ecause Ḣ = ny 2 is a constant function. The last part of Theorem 1 is proved. To illustrate the conclusion, we show in Figure 1 the limit cycle in the phase space for a =1, = 1,m = 1n =0,p =2.

5 Zhou et al. Advances in Difference Equations :28 Page 5 of 10 Figure 1 The limit cycle of the extended Lorenz system; a =1, = 1,m = 1. 3 Hopf ifurcation of the nontrivial equiliria Now we analyze the Hopf ifurcation at the nontrivial equiliria A x 0,0,z 0 and Ax 0,0,z 0, where x 0 = am ap+m and z 0 = m. Due to the symmetry, we only analyze ap+m the equilirium Ax 0,0,z 0. First, we move the equilirium to the origin of the coordinates under the following change of variales: X = x x 0, Y = y, Z = z z 0. The extended Lorenz system 1.2thusecomes: Ẋ = Y, Ẏ = 2amp ap+m X ny mx 0Z 3px 0 X 2 mxz px 3, Ż =2x 0 X az + X The Jacoian matrix of the linearization of system 3.1 at the origin is amp ap+m n mx 0 2x 0 0 a with the characteristic equation Pλ λ 3 + a 1 λ 2 + a 2 λ + a 3 =0, 3.2 where a 1 = a + n, a 2 = an + 2amp ap + m, a 3 =2am. When we consider the Hopf ifurcation of a system at an equilirium, we hope that the Jacoian matrix of the linear part of the system at the equilirium has a pair of purely imaginary eigenvalues and no other eigenvalues with zero real part. As to three-dimensional

6 Zhou et al. Advances in Difference Equations :28 Page 6 of 10 system, this means that the characteristic polynomial of the Jacoian matrix should e formulated as Pλ =λ + p 1 λ 2 + p 2 withp 1 0andp 2 > 0. Expanding this form, we know that p 1 and p 2 are the coefficients at λ 2 and λ, respectively, and the constant term of the polynomial is p 1 p 2. At this moment, the three eigenvalues of the Jacoian matrix are λ 1 = p 1 and λ 2,3 = ±i p 2 [20]. By this fact we have the following proposition. Proposition 1 The polynomial 3.2 has two purely imaginary roots if and only if a 1 a 2 = a 3, a 1 0,and a 2 >0,that is, a + n 0, an + 2apm ap + m >0, p = 2am 2 2amn + a 3 n + a 2 n In this case, the roots are λ 1 = a nandλ 2,3 = ±iω, where ω = an + 2amp ap+m. We choose the parameter p as the ifurcation parameter. Setting λ = λp and applying the implicit function theorem to 3.2, we have dλ dp and dλ Re dp λ=iω,p= 2am 2 2amn+a 3 n+a 2 n 2 = dp dp = dp dλ an 2 2m + a 2 + an 2 an +2manL +2mn 3 +2a 3 m +4am 2 0 whenever an 0and2m + a 2 + an 0,whereL =8am + a 3 +4a 2 n +8mn +4an 2 + n 3. Next, we computer the first Lyapunov constant of system 3.1. We consider system 3.1 at the critical parameter values 3.3 and perform the linear change X U Y = 0 ω a n V, Z 2aqx 0 2ωqx 0 2 n x 0 W where q =.Thensystem3.1ecomes a 2 +ω 2 U 0 ω 0 U 0 V = ω 0 0 V + GU, V, W, 3.4 Ẇ 0 0 a n W HU, V, W where GU, V, W= 3px 0 +2amqx 0 U 2 +2mqωx 0 UV 6px 0 +2amqx 0 2mx 0 UW n +2mqωx 0 VW 3px 0 2mx 0 n pu 3 3pU 2 W 3pUW 2 pw 3, HU, V, W= U 2 +2UW + W 2. W 2

7 Zhou et al. Advances in Difference Equations :28 Page 7 of 10 Introduce ξ = U + iv, ξ = U iv,wherei 2 = 1.ThenU = ξ+ ξ 2 and V = ξ ξ 2i.Wetransform system 3.4intothecomplexform { ξ = ωξi + g 1 ξ, ξ, W+ig 2 ξ, ξ, W, Ẇ = a + nw + hξ, ξ, W, 3.5 where g 1 ξ, ξ, W= 1 2 ωmqx 0ξ ω mqx 0ξ 2 ωmqx 0 ξ W + ω mqx 0 ξw, g 2 ξ, ξ, W= 3 4 px amω qx 0 ξ px amω qx 0 ξ mx 0 n 3px 0 W 2 + amω qx px 0 ξ ξ mx0 + 3px 0 amqx 0 ξ W + n 1 8 pξ p ξ 3 pw pξ 2 ξ 3 8 pξ ξ 2 mx0 3px 0 amqx 0 ξw n 3 4 pwξ2 3 4 pw ξ pw 2 ξ 3 2 pw 2 ξ 3 2 pwξ ξ, hξ, ξ, W= 1 4 ξ 2 +2ξξ + ξ 2 +4ξW +4ξW +4W 2. The center manifold W c now has the representation W = Wξ, ξ, W= 1 2 W 20ξ 2 + W 11 ξ ξ W 02 ξ 2 + O ξ Thus, we have Ẇ = W 20 ξ ξ + W 11 ξ ξ + W 11 ξ ξ + W 02 ξ ξ + O ξ 3. Sustituting equation into the last formula, we get Ẇ = W 20 ωiξ 2 W 02 ωi ξ 2 + O ξ 3. According to equation 3.5 2, the differential of W reads 1 Ẇ = aw nw aw 11 nw 11 1 ξ aw nw 02 ξ 2 ξ ξ + O ξ 3. The two formulas are identical, and we equate the coefficients to get W 20 = 2a + n+4ωi, W 11 = 2a + n, W 02 = 2a + n 4ωi.

8 Zhou et al. Advances in Difference Equations :28 Page 8 of 10 Thus, we otain the representation of center manifold W c W = Wξ, ξ, W= ξ 2 4a + n+8ωi + ξ ξ 2a + n + ξ 2 4a + n 8ωi + O ξ and the restriction of system 3.5tothecentermanifold ξ = ωξi + g 20 ξ 2 + g 11 ξ ξ + g 02 ξ 2 + g 30 ξ 3 + g 21 ξ 2 ξ + g 12 ξ ξ 2 + g 03 ξ 3 + O ξ, ξ 4, where g 20 = 1 2 ωmqx 0 i 1 2 amqx 0 i 3 4 px 0, g 11 = iamqx 0 i 3 2 px 0, g 02 = 1 2 ωmqx 0 i 1 2 amqx 0 i 3 4 px 0, g 30 = g 21 = ω mqx 0 4a + n +2iω iamqx 0 4a + n +2iω + i 2 mx 0 4na + n +2iω 3 ipx 0 4a + n +2iω 1 8 ip, ω mqx 0 4a + n +2iω iamqx 0 4a + n +2iω + i 2 mx 0 4na + n +2iω 3 ipx 0 4a + n +2iω 3 8 ip + i2 mx 0 2na + n + ω mqx 0 2a + n 3 ipx 0 2a + n iamqx 0 2a + n, ω mqx 0 g 12 = 4 a n +2iω + iamqx 0 4 a n +2iω i 2 mx 0 4n a n +2iω + 3 ipx 0 4 a n +2iω 3 8 ip + i2 mx 0 2na + n ω mqx 0 2a + n 3 ipx 0 2a + n iamqx 0 2a + n, g 03 = ω mqx 0 4 a n +2iω + iamqx 0 4 a n +2iω i 2 mx 0 4n a n +2iω + 3 ipx 0 4 a + n+2iω 1 8 ip. Returning to the real form, we have { U = ωv, V = ωu + k 20 U 2 + k 11 UV + k 30 U 3 + k 21 U 2 V + k 12 UV 2 + k 03 V 3 + Or 4, 3.8 where the coefficients k ij are given as follows: k 20 = 2amq +3px 0, k 11 =2qmωx 0, 1 k 30 = 2 a 3 mnqx na + na 2 +2an + n 2 +4ω a 3 pn +4a 2 mqx 0 n 2 +6a 2 px 0 n 2 2 mx 0 a 2 +3a 2 pn 2 +4amnω 2 qx 0 +4apnω 2 4a 2 mnx 0 +2amqx 0 n 3 +3apn 3 +12apx 0 n 2 +4pn 2 ω mx 0 ω 2 +12nω 2 px mn 2 x 0 + pn 4 +6px 0 n 3,

9 Zhou et al. Advances in Difference Equations :28 Page 9 of 10 k 21 = 2ω x 0a 2 mqn +6apn 2am +6pn 2 2mqnω 2 2mn mqn 3, na + na 2 +2an + n 2 +4ω 2 k 12 = 4ω2 x 0 m 3pn + mqn 2 na + na 2 +2an + n 2 +4ω 2, k 4ω 3 mqx 0 03 = a + na 2 +2an + n 2 +4ω 2. By using the formal series method we find the first Lyapunov constant where L 1 = x 0 K 1 + K 2 x 0 4nωa + na + n 2 +4ω 2, K 1 = mn 3 qω 6pn 2 ω +8qmnω 3 6anpω +2nm 2 ω qmna 2 ω +2a 2 mω, K 2 =2am 2 n 4 q 2 +3mn 4 pq +9amn 3 pq +6a 2 m 2 n 3 q 2 +9a 2 mn 2 pq +6a 3 m 2 n 2 q 2 +12mn 2 pqω 2 +8am 2 n 2 q 2 ω 2 +3mnpqa 3 +12amnpqω 2 +2nm 2 a 4 q 2 +8nm 2 a 2 q 2 ω 2. Therefore, we have the following: Theorem 2 If L 1 0,then the extended Lorenz system 1.2 undergoes Hopf ifurcation at the equilirium A. When L 1 <0,the Hopf ifurcation is supercritical, and system has a stale limit cycle. When L 1 >0,the Hopf ifurcation is sucritical, and system has an unstale limit cycle. Due to the symmetry, the same result can e otained at the other nontrivial equilirium A. So we find two limit cycles around the two equiliria A and A,respectively. To illustrate this conclusion, we show in Figure 2 the limit cycle around the equilirium A inthephasespacefora =1, =2,m =3,n =2,p = 2. The initial value is , , In this case, the first Lyapunov constant is L 1 = Figure 2 The limit cycle around Ax 0,0,z 0 of the extended Lorenz system, a =1, =2,m =3,n =2, p =2.

10 Zhou et al. Advances in Difference Equations :28 Page 10 of 10 4 Conclusion In this paper, we analyze the existence of Hopf ifurcation in an extended Lorenz system. We show that a nondegenerate Hopf ifurcation arises at each of the three equilirium points, the origin and two other nontrivial symmetric equiliria. We find totally three limit cycles, one for each of the equiliria. In each case, we reduce the system to a two-dimensional center manifold and compute the first Lyapunov constant L 1. Due to the complex form of the analytic expression of the Lyapunov constant and the multitude of parameters, we cannot fully analyze the sign of L 1 at the nontrivial equiliria. We illustrate the results y numerical simulation using Matla R2012a for some particular parameters. Competing interests The authors declare that they have no competing interests. Authors contriutions The idea of this research was introduced y the second author. The first author made the main contriutions to the specific computation and proof. The third author provided the numerical simulations. Author details 1 Department of Mathematics, Shanghai Normal University, Shanghai, China. 2 Department of Mathematics, Huei University of Science and Technology, Huei, China. 3 Department of Mathematics, Politehnica University of Timisoara, Timisoara, Romania. 4 School of Mathematics, Southwest Jiaotong University, Sichuan, China. Acknowledgements The paper was supported y FP7-PEOPLE-2012-IRSES Received: 25 July 2016 Accepted: 14 January 2017 References 1. Lorenz, EN: Deterministic nonperiodic flow. J. Atmos. Sci. 20, Rössler, O: An equation for hyperchaos. Phys. Lett. A 2-3, Chua, L, Komura, M, Matsumoto, T: The doule scroll family. IEEE Trans. Circuit Syst. I 33, Chen, G, Ueta, T: Yet another chaotic attractor. Int. J. Bifurc. Chaos 9, Lü, J, Chen, G: A new chaotic attractor coined. Int. J. Bifurc. Chaos 12, Tigan, G: On a three-dimensional differential system. Mat. Bilt. 30, Kuznetsov, AP, Kuznetsov, SP, Stankevich, NV: A simple autonomous quasiperiodic self-oscillator. Commun. Nonlinear Sci. Numer. Simul Kuznetsov, NV, Leonov, GA, Vagaitsev, VI: Analytical-numerical method for attractor localization of generalized Chua s system. In: 4th Int. Workshop on Periodic Control Systems, vol. 4, pp Kuznetsov, NV, Kuznetsova, OA, Leonov, GA, Vagaytsev, VI: Hidden attractor in Chua s circuits. In: ICINCO Proc. 8th Int. Conf. Informatics in Control, Automation and Rootics, pp Kuznetsov, NV, Leonov, GA, Seledzhi, SM: Hidden oscillations in nonlinear control systems. In: Proc. 18th IFAC World Congress, vol. 18, pp Wei, Z, Wang, R, Liu, A: A new finding of the existence of hidden hyperchaotic attractors with no equiliria. Math. Comput. Simul. 100, Wei, Z, Zhang, W: Hidden Hyperchaotic Attractors in a Modified Lorenz-Stenflo System with Only One Stale Equilirium. Int. J. Bifurc. Chaos 2410, Wei, Z, Zhang, W, Wang, Z, Yao, M: Hidden attractors and dynamical ehaviors in a extended Rikitake system. Int. J. Bifurc. Chaos 252, Wei, Z, Yu, P, Zhang, W, Yao, M: Study of hidden attractors, multiple limit cycles from Hopf ifurcation and oundedness of motion in the generalized hyperchaotic Rainovich system. Nonlinear Dyn. 82, Wei, Z, Zhang, W, Yao, M: On the periodic orit ifurcating from one single non-hyperolic equilirium in a chaotic jerk system. Nonlinear Dyn. 82, Ovsyannikov, I, Turaev, D: Lorenz attractors and Shilnikov criterion. ArXiv preprint arxiv: Guckenheimer, J, Holmes, P: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York Kuznetsov, Y: Elements of Applied Bifurcation Theory. Springer, New York Han, M: Bifurcation Theory of Limit Cycles. Science Press, Beijing Liu, W: Criterion of Hopf ifurcations without using eigenvalues. J. Math. Anal. Appl. 182,