Hopf bifurcations analysis of a three-dimensional nonlinear system

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1 BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Number 358), 28, Pages ISSN Hopf bifurcations analysis of a three-dimensional nonlinear system Mircea Craioveanu, Gheorghe Tigan Abstract. Bifurcations analysis of a 3D nonlinear chaotic system, called the T system, is treated in this paper, extending the work presented in [5] and [6]. The system T belongs to a class of cvasi-metriplectic systems having the same Poisson tensor and the same Casimir. Mathematics subject classification: 34D2, 37L1, 37G1. Keywords and phrases: Stability, Hopf bifurcations, Lorenz systems. 1 Introduction The nonlinear differential systems are studied both from theoretical point of view and from the point of view of their potential applications in various areas. The nonlinear systems present many times the property of sensitivity with respect to the initial conditions some authors consider this property sufficient for a system to be chaotic). Applications of such systems have been found lately in secure communications [1, 4]. Of the pioneering papers which proposed to use the chaotic systems in communications are the papers of Pecora and Carrol [8, 9]. Consequently, an appropriate chaotic system can be chosen from a catalogue of chaotic systems to optimize some desirable factors, idea suggested in [4]. These facts led us to study a new 3D polynomial differential system given by: ẋ = ay x), ẏ = c a)x axz, ż = bz + xy, 1) with a,b,c real parameters and a. Call it the T system. Some results regarding the T system are already presented in [5] and [6], where we pointed out two particular cases. The system T is related to the Lorenz, Chen and Lü system [3] being a small generalization of the latter one. The paper is organized as follows. In the first Section we recall some results regarding the stability of equilibria. In Section 2 we present the pitchfork and Hopf bifurcations occurring at the equilibrium points in the general case. Also, we show that the T system belongs to a class of cvasi-metriplectic systems. 2 Equilibrium points of the system Because the dynamics of the system is characterized by the existence and the number of the equilibrium points as well as of their type of stability, we recall in the c Mircea Craioveanu, Gheorghe Tigan, 28 57

2 58 MIRCEA CRAIOVEANU, GHEORGHE TIGAN y t z t x t 1 y t z t x t 2 Figure 1. a) The orbit of the Lorenz system for a = 1, b = 8/3, c = 28 and the initial condition.4,.3,.52) left); b) The orbit of the Chen system for a = 35, b = 3, c = 28 and the initial condition.1,.1,.4) right) y t z t x t y t -1 2 z t 1-5 x t 5 Figure 2. a) The orbit of the Lü system for a = 36,b = 3,c = 19 and the initial condition 1,.1,4) left) ; b) The orbit of the T system for a,b,c) = 2.1,.6,3) and the initial condition.1,.3,.2) right)

3 HOPF BIFURCATIONS ANALYSIS following the equilibrium points of the system T and their stability. Proposition 1. If isolated points: b c a) >, then the system T possesses three equilibrium a O,,),E 1 x,x,c/a 1),E 2 x, x,c/a 1) b where x = a c a), and for b, b c a), the system T has only one a isolated equilibrium point, namely O,,), Theorem 1. For b the following statements are true: a) If a >,b >,c a), then O,,) is asymptotically stable; b) If b < ) or a < ) or a >,c > a), then O,,) is unstable. For the other two equilibria, E 1,2 ±x, ±x,c/a 1), for b/ac a) >, using the Routh-Hurwitz conditions we have: Theorem 2. [5]) If the conditions a + b >, abc a) >, b2a 2 + bc ac) > hold, the equilibrium points E 1,2 ±x, ±x,c/a 1), are asymptotically stable. 3 Pitchfork and Hopf bifurcations of the system T Consider the parameter a as bifurcation parameter. a) Bifurcations at O,,) Proposition 2. [7]) If β = a c = the equilibrium O,,) of the system T undergoes a pitchfork bifurcation that generates the asymptotic stable equilibrium point O,,) if a > c, and for a < c three equilibria: O,,) unstable), E 1,2 ±x, ±x,c/a 1) locally stable). Notice that the equilibrium O,,) can not undergo a Hopf bifurcation because the roots of the characteristic polynomial of the Jacobian matrix of the system T at O,,) are λ 1 = b, λ 2 = 1 a ) 4ac 3a 2 2, λ 3 = 1 a + ) 4ac 3a 2 2 and the last two roots can not be purely imaginary because a. b) Bifurcations of the equilibria E 1 and E 2. We observe that the characteristic polynomial in this case is: f λ) := λ 3 + λ 2 a + b) + bcλ + 2abc a) 2) Because abc a) >, the system T does not undergo pitchfork bifurcations at E 1,2, so we study the Hopf bifurcations at these points. The following proposition characterizes the imaginary roots of 2).

4 6 MIRCEA CRAIOVEANU, GHEORGHE TIGAN Proposition 3. Consider b c a) >. The polynomial 2) has one real negative a root and two purely imaginary roots if and only if a,b,c) Ω, where Ω = {a,b,c) R 3 a > b >,2a 2 + bc = ac}. In this case the roots are: λ 1 = a b, λ 2,3 = ±iω,ω := bc. In the following we show that the system T undergoes a Hopf bifurcation at E 1 for E 2 is similar). Remember that a is the bifurcation parameter. From 2a 2 +bc = ac one gets a = a s := c ± c 2 8bc. Denote λ := α a) ±iω a) 4 the complex roots depending on the bifurcation parameter a of 2). If a = a s and a,b,c) Ω, from the above Proposition 3, we have λ 1 = a b and λ 2,3 = ±iω with ω = bc. From 2) it follows that: dλa) Re da a=as,λ=i bc ) ) bc 4ab = Re 2bc 2i = bc a + b) bc 4ab 2bc + 2a + b) 2 for c 8b because c 4a = ± c 2 8bc, b,c > ). In the following we compute the index number K from the Hopf bifurcation theorem [2], employing the central manifold theory. Using the transformation x,y,z) X 1,Y 1,Z 1 ), the system T leads to: x = X 1 + x, y = Y 1 + x, z = Z 1 + c/a 1, 3) X 1 = ay 1 X 1 ), Y 1 = ax Z 1 ax 1 Z 1, Z 1 = x X 1 + Y 1 ) bz 1 + X 1 Y 1 4) or, in the normal form: X 1 Y 1 Z 1 = a a ax x x b X 1 Y 1 Z 1 + ax 1 Z 1 X 1 Y 1. 5) The eigenvalues of the Jacobian matrix J, a a J = ax x x b are, respectively λ 1 = a b <, λ 2,3 = ±iω,ω = bc, b >, c >, with corresponding eigenvectors: v 1 1, c + 2a, 2 ) c 2a b, v 2 a c c ω qx + iqx, a ) ω x,i,

5 HOPF BIFURCATIONS ANALYSIS v 3 a ω qx iqx, a ) ω x, i a 2 with q = ω 2 + a 2. Then the vectors v 2 = v 2 + v 3 2 v 3 = v 2 v 3 c + 2a = qx,,1) and v 1 1,, 2 2i c c transformation: where or, equivalently, where X 1 Y 1 Z 1 = P X Y Z 1 a ω qx qx c + 2a P = a c ω x 2 c 2a b 1 c X Y Z = P 1 X 1 Y 1 Z 1 a ω qx, a ) ω x,, = ) c 2a b lead us to the c qc qx c P 1 = 1 c + 2a d ω c + 2 c 2a bqx ω q c 2a ω ax ax a 2 c 2a b 2q c 2a b c + qc 2qa with d = c + qc 2qa + 2 c 2a bqx. Then the system 5) reads: or, equivalently where Ẋ Ẏ Ż Ẋ Ẏ Ż = = P 1 JP X Y Z + P 1 2a c + a c ω ω ax 1 Z 1, 6) X 1 Y 1 X Y Z + g 1 g 2 g 3, 7) g 1 g 2 g 3 = P 1 a X a ) ω qx Y + qx Z 2 ) c 2a bx + Z c X a ) ω qx c + 2a Y + qx Z X a ) c ω x, 8) Y

6 62 MIRCEA CRAIOVEANU, GHEORGHE TIGAN or g 1 = 1 X d qca a ) ω qx Y + qx Z 2 ) c 2a bx + Z c 1 d qx c ax a ω qx Y + qx Z g 2 = 1 c + 2 c 2a bqx d 1 d x 2a qc ω a g 3 = 2 d q c 2a ba ) c + 2a c ω X a ω qx Y + qx Z ax a ω qx Y + qx Z ) c + 2a X a ) ω x Y, ) 2 c c 2a bx + Z ) + c X a ) ω x Y, X a ) ω qx Y + qx Z 2 ) c 2a bx + Z + c + 1 ax d c + qc 2qa) a ) ω qx c + 2a Y + qx Z X a ) c ω x Y. Consider the 2-dimensional central manifold at the origin given by: X = hy,z) = AY 2 + BY Z + CZ ) Then, replacing 9) in 7) and taking into account that BY Ż + 2CZŻ, obtained from 9), one gets: Ẋ = 2AY Ẏ + BẎ Z + where Ẋ = BZ 2 ω BY 2 ω 2CωY Z + 2AωZY ) On the other hand, from 7) we have Ẋ = α 1 Y 2 + α 2 Y Z + α 3 Z ) α 1 = 2aA + 2 a2 c A 1 d q2 x 3 c a2 ω 2, α 2 = 1 d q2 x 3 c a ω 1 d q2 ca 2 1 ω x 2aB + 2 a2 c B, α 3 = 2 a2 c C + 1 d q2 cax 2aC. Then, equalling the coefficients of the terms Y 2,Z 2,Y Z in the above relations 1) and 11), one gets Y 2 : ωb = 2aA + 2 a2 c A 1 d q2 x 3 c a2 ω 2, Y Z : 2ωA 2Cω = 1 d q2 x 3 c a ω 1 d q2 ca 2 1 ω x 2aB + 2 a2 c B, Z 2 : Bω = 2 a2 c C + 1 d q2 cax 2aC.

7 HOPF BIFURCATIONS ANALYSIS Finally we get A = 1 4 q2 c 2 ω 2 ca 3 + ω 2 c 2 a 2 ω 4 c 2 + ω 2 cx 2 x a2 4x 2 a4 c + 2x 2 a3 c 2 + 2x 2 a5 ω 2 c 2 2ca 3 + c 2 a 2 + a 4 ) dω 2, a + c) B = 1 2 q2 x c 22x2 ca + a3 ca 2 + ω 2 c x 2 a2 ωdω 2 c 2 2ca 3 + c 2 a 2 + a 4 ) a, C = 1 4 q2 c 2 2x 2 x c2 a + 5ca 3 3c 2 a 2 ω 2 c 2 cx 2 a2 2a 4 ω 2 c 2 2ca 3 + c 2 a 2 + a 4. )d a + c) Hence, the system 8) restricted to the central manifold is given by: ) ) ) Ẏ ω Y g = + 2 ) Y,Z) Ż ω Z g 3, Y,Z) where g 2 Y,Z) = g 2 hy,z),y,z), g 3 Y,Z) = g 3 hy,z),y,z). Now K can be computed as follows [2]: KY,Z) = 1 [ 3 g 2 16 Y g 2 Y Z g 3 Y 2 Z + 3 g 3 Z [ 2 g 2 2 g 2 16ω Y Z Y g 2 ) Z 2 2 g 3 2 g 3 Y Z 2 g 2 2 g 3 Y 2 Y g 2 2 g 3 Z 2 Z 2 ]. ] + Y g 3 Z 2 ) Then K = K,) = 1 8d x caqc + 1 4d x a 2 qc 1 4d x cq 2 C + 1 d a c 2a bcq+ + 1 x d c acq2 b 1 8d caqx 1 4dω a2 Bqx + 1 8dω cabq2 x + 1 4dω a3 Bqx 1 2dωc q2 a 2 Bbx 1 2dωc q2 a 3 Bx 1 8dω ca2 Bx C d cqx + 8dω ca2 Bqx 3 8 C d aqx 1 8d 2 ω 4c2 q 4 x 4 a3 1 2d 2 ω 4q4 a 5 x 4 1 2d 2 ω 4q4 c 2a ba 5 x d 2 ω 4cq4 c 2a ba 4 x d 2 ω 4cq3 c 2a ba 4 x d 2 ω 2cq4 a 3 x 2 1 d 2 ω 2q4 a 4 x 2 1 2d 2 ω 2q4 a 3 x 2 b + 1 2d 2x2 caq 4 1 d 2x2 a 2 q 4 1 2d 2x2 aq4 b 1 8d 2x2 c2 a 2 q3 ω q3 4d 2x2 ca3 ω 2 1 8d 2c2 aq 2 1 8d 2x4 c2 a q4 ω d 2x4 ca 2 q4 ω 2 1 4d 2x3 cq 4 c 2a b 1 8d 2x2 c 2 q 3 1 2d 2 x4 a 3 q4 ω 2+

8 64 MIRCEA CRAIOVEANU, GHEORGHE TIGAN + 1 2d 2x3 aq 4 c 2a b + 1 4d 2x2 caq d 2 ω 4cq4 x 4 a d 2 ω 2cq3 x 3 a2 c 2a b + 1 8d 2 ω 4c2 q 2 x 4 a3 + A d + 1 x d c a2 q 2 C 1 8d x c a q2 ωb A d + 1 A 4 d x cq d x x c aq2 ωb + 1 2d Concluding we have the theorem: c q2 a 2 A d c 2a baq + 1 8dx cωb+ x c q2 ab 3 A 8 d x cqa + 3 A 4 d x qa 2 + x c ωbq2 b + 1 4d Aaqx + 1 8dω cabqx. Theorem 3. If a = a s := c ± c 2 8bc, c 8b, the equation 2) has a negative solution λ 1 = a b < together with two purely imaginary roots λ 2,3 = ±iω,ω = 4 bc dλa) ) such that R := Re. Consequently, if K, the equilibrium da a=as,λ=iω E 1 of the system T undergoes a Hopf bifurcation. In addition, the periodic orbits that bifurcate from the equilibrium E 1 for a in the neighborhood of a s, are stable if K <, and unstable if K >. The direction of bifurcation are above bellow) a s if RK < RK > ). Remark 1. In the particular case a = 2.1, b = 1.86, c = 3 we have K = , R =.28. So the periodic orbits that bifurcate from the equilibria E 1,2 are stable and the bifurcation is above a s. In the following we show that the system T belongs to a class of cvasi-metriplectic systems. Definition 1. Consider X a vector field on R 3. The vector field X is called cvasimetriplectic, if there exists a Poisson tensor field P, a cvasi-metric tensor field g, a Hamilton function H and a Casimir function S associated to P, such that: Consider the vector field X = x 1,x 2,x 3 ) given by: X = P H + g S. 12) x 1 = a 1 x + a 1 y, x 2 = a 2 x + a 3 y a 4 xz + a 6, x 3 = a 5 z + xy, 13) with a i,i = 1,6, real numbers. Remark If a 1 = a,a 2 = c,a 3 = 1,a 4 = 1,a 5 = b,a 6 =, the dynamical system associated to X is the Lorenz system. 2. If a 1 = a,a 2 = c a,a 3 = c,a 4 = 1,a 5 = b,a 6 =, the dynamical system associated to X is the Chen or Chen 1) system. 3. If a 1 = a,a 2 =,a 3 = c,a 4 = 1,a 5 = b,a 6 =, the dynamical system associated to X is the Lü system.

9 HOPF BIFURCATIONS ANALYSIS If a 1 = a,a 2 = c a,a 3 =,a 4 = a,a 5 = b,a 6 =, the dynamical system associated to X is the T system. 5. If a 1 = a,a 2 = c a,a 3 = c,a 4 = 1,a 5 = b,a 6 = m, the dynamical system associated to X is the Chen 2 system. Proposition 4. The vector field X 13) has the cvasi-metriplectic representation given by: X = P H + g S 14) where P = a 1 a 1 x x,g = a 1 ε a 3y + a 6 a 3y + a 6 a 1 a 1 a 5 z a 1 15) Hx,y,z) = 1 2 y2 + a 4 z 2 ) a 2 z, Sx,y,z) = 1 2 x2 a 1 z with ε R. The proof is immediately. Observe that the Poisson tensor field P and the Casimir S are the same for the all above five systems, consequently, the systems belong to the same class of cvasi-metriplectic systems. In addition, for the system T, the tensor g is in diagonal form. 4 Conclusions In this paper we further investigated a nonlinear differential system with three equilibrium points, origin and another two. In the origin, the system displays a pitchfork bifurcation and in the other two equilibrium points a Hopf bifurcation. The system belongs to a class of cvasi-metriplectic systems. 5 Acknowledgements The author was supported by the Center of Advanced Studies in Mathematics at the Mathematics Department of Ben Gurion University and partially by grants ISF 926/4 and ISF 273/7. References [1] Alvarez G., Li S., Montoya F., Pastor G., Romera M. Breaking projective chaos synchronization secure communication using filtering and generalized synchronization. Chaos, Solitons and Fractals, 25, 24, [2] Chang Y., Chen G. Complex dynamics in Chen s system. Chaos, Solitons and Fractals, 26, 27,

10 66 MIRCEA CRAIOVEANU, GHEORGHE TIGAN [3] Lü J., Chen G. A new chaotic attractor coined. Int. J. of Bif. and Chaos, 22, 123), [4] Sprott J.C. Some simple chaotic flows. Physical Review E, 1994, 52), R647 R65. [5] Tigan G. Analysis of a dynamical system derived from the Lorenz system. Scientific Bulletin of the Politehnica University of Timisoara, Tomul 564), Fascicola 1, 25, [6] Tigan G. Bifurcation and the stability in a system derived from the Lorenz system. Proceedings of The 3-rd International Colloquium, Mathematics in Engineering and Numerical Physics, 24, [7] Tigan G. On a three-dimensional differential system. Matematicki Bilten, Tome 3 LVI), 26, [8] Pecora L.M., Carroll T.L. Synchronization in chaotic systems. Phys Rev Lett, 199, 648), [9] Pecora L.M., Carroll T.L. Driving systems with chaotic signals. Phys Rev A, 1991, 444), Mircea Craioveanu West University of Timisoara Department of Mathematics B-dul V.Parvan, 4, Timisoara, Timis Romania craiov@math.uvt.ro Received June 23, 27 Gheorghe Tigan Politehnica University of Timisoara Department of Mathematics Pta Victoriei, 2, Timisoara, Timis Romania. gheorghe.tigan@mat.upt.ro