Advances in Systems Science and Applications (2012) Vol.12 No.4 347-352 A Universal Nonlinear Control Law for the Synchronization of Arbitrary 3-D Continuous-time Quadratic Systems Zeraoulia Elhadj1 and J.C.Sprott2 1Department of Mathematics, University of Tébessa, 12002, Algeria. 2Department of Physics, University of Wisconsin, Madison, WI 53706, USA Abstract In this letter we present a universal nonlinear control law for the synchronization of arbitrary 3-D continuous-time quadratic systems. This control law does not require any type of conditions on the considered systems. Keywords Synchronization, universal nonlinear control law, chaos. PACS num- bers: 05.45.-a, 05.45.Gg 1 Introduction Several methods have been successfully applied to chaos synchronization. For ex- ample, in [1] a method is introduced to synchronize two identical chaotic systems with different initial conditions. An adaptive control approach is presented in [2], a backstepping design was presented in [3], an active control method is pre- sented in [4-6], and a nonlinear control scheme was given in [7-9]. Consequently, there are many applications of chaos synchronization in physical, chemical, and ecological systems, and in secure communications as shown in [1-2,10-13]. In this letter, we apply nonlinear control theory to synchronize two arbitrary 3-D continuous-time quadratic systems. The proposed control law does not need any conditions on the considered systems, and hence it is a universal synchro- nization approach for general 3-D continuous-time quadratic systems. In other words, the present letter is concerned with synchronization of nonlinear system- s in the framework of nonlinear observers. The investigation is restricted to a pair of quadratic three dimensional systems, for which a control feedback can be chosen in such a way that global asymptotic stability of the error system can be established in the framework of classical Lyapunov theory. This restriction is justified by the importance of this type of systems in real applications [14] which is certainly a useful result. 2 Synchronization using a universal nonlinear control law In this section, we consider two arbitrary 3-D continuous-time quadratic systems. The one with variables x1, y1, and z1 will be controlled to be the new system given by 348 Zeraoulia Elhadj:A Universal Nonlinear Control Law for the Synchronization of Arbitrary...  x′1 = a0 + a1x1 + a2y1 + a3z1 + f1(x1, y1, z1) y′1 = b0 + b1x1 + b2y1 + b3z1 + f2(x1, y1, z1) z′1 = c0 + c1x1 + c2y1 + c3z1 + f3(x1, y1, z1) (1) where  f1(x1, y1, z1) = a4x 2 1 + a5y 2 1 + a6z 2 1 + a7x1y1 + a8x1z1 + a9y1z1 f2(x1, y1, z1) = b4x 2 1 + b5y 2 1 + b6z 2 1 + b7x1y1 + b8x1z1 + b9y1z1 f3(x1, y1, z1) = c4x 2 1 + c5y 2 1 + c6z 2 1 + c7x1y1 + c8x1z1 + c9y1z1 (2) and the one with variables x2, y2, and z2 as the response system  x′2 = d0 + d1x2 + d2y2 + d3z2 + g1(x2, y2, z2) + u1(t) y′2 = r0 + r1x2 + r2y2 + r3z2 + g2(x2, y2, z2) + u2(t) z′2 = s0 + s1x2 + s2y2 + s3z2 + g3(x2, y2, z2) + u3(t) (3) where  g1(x2, y2, z2) = d4x 2 2 + d5y 2 2 + d6z 2 2 + d7x2y2 + d8x2z2 + d9y2z2 g2(x2, y2, z2) = r4x 2 2 + r5y 2 2 + r6z 2 2 + r7x2y2 + r8x2z2 + r9y2z2 g3(x2, y2, z2) = s4x 2 2 + s5y 2 2 + s6z 2 2 + s7x2y2 + s8x2z2 + s9y2z2 (4) Here (ai, bi, ci)0≤i≤9 ⊂ R30 and (di, ri, si)0≤i≤9 ⊂ R30 are bifurcation parameters, and u1(t), u2(t), u3(t) are the unknown (to be determined) nonlinear controller such that two systems (1) and (3) can be synchronized. First, let us define the following quantities depending on the above two systems in which we can proceed with our proposed method:  ξ1 = a1 + d1 + a4(x1 + x2) + d4(x1 + x2) + a7y1 + a8z1 + d7y2 + d8z2 ξ2 = a2 + d2 + a5(y1 + y2) + d5(y1 + y2) + a9z1 + d9z2 ξ3 = a3 + d3 + a6(z1 + z2) + d6(z1 + z2) ξ4 = η1 + η2 + η3 ξ5 = b1 + r1 + b4(x1 + x2) + r4(x1 + x2) + b7y1 + b8z1 + r7y2 + r8z2 (5) and Advances in Systems Science and Applications (2012) Vol.12 No.4 349  ξ6 = b2 + r2 + b5(y1 + y2) + r5(y1 + y2) + b9z1 + r9z2 ξ7 = b3 + r3 + b6(z1 + z2) ξ8 = η4 + η5 + η6 ξ9 = c1 + s1 + c4(x1 + x2) + s4(x1 + x2) + c7y1 + c8z1 + s7y2 + s8z2 ξ10 = c2 + s2 + c5(y1 + y2) + s5(y1 + y2) + c9z1 + s9z2 ξ11 = c3 + s3 + c6(z1 + z2) + s6(z1 + z2) ξ12 = η7 + η8 + η9 (6) where  η1 = d4x 2 1 + d7x1y2 + d8x1z2 + d1x1 − a4x2 − a7x2y1 − a8x2z1 η2 = −a1x2 + d5y 2 1 + d9y1z2 + d2y1 − a5y 2 2 − a9y2z1 − a2y2 η3 = d6z 2 1 + d3z1 − a6z 2 2 − a3z2 − a0 + d0 η4 = r4x 2 1 + r7x1y2 + r8x1z2 + r1x1 − b4x 2 2 − b7x2y1 − b8x2z1 η5 = −b1x2 + r5y 2 2 + r9y1z2 + r2y1 − b5y 2 2 − b9y2z1 − b2y2 η6 = r6z 2 1 + r3z1 − b6z 2 2 − b3z2 − b0 + r0 η7 = s4x 2 1 + s7x1y2 + s8x1z2 + s1x1 − c4x 2 2 − c7x2y1 − c8x2z1 η8 = −c1x2 + s5y 2 1 + s9y1z2 + s2y1 − c5y 2 2 − c9y2z1 − c2y2 η9 = s6z 2 1 + s3z1 − c6z 2 2 − c3z2 − c0 + s0 (7) The above quantities comes from the formulation of the problem as the system in (8) below. Now let the error states be e1 = x2−x1, e2 = y2−y1, and e3 = z2−z1. Then the error system is given by e′1 = ξ1e1 + ξ2e2 + ξ3e3 + ξ4 + u1(t) e′2 = ξ5e1 + ξ6e2 + ξ7e3 + ξ8 + u2(t) e′3 = ξ9e1 + ξ10e2 + ξ11e3 + ξ12 + u3(t) (8) We propose the following universal control law for the system (3): u1 = −(ξ1 + 1)e1 − (ξ2 + ξ5)e2 − ξ4 u2 = −(ξ6 + 1)e2 − (ξ7 + ξ10)e3 − ξ8 u3 = −(ξ3 + ξ9)e1 − (ξ11 + 1)e3 − ξ12 (9) Then the two 3-D continuous-time quadratic systems (1) and (3) approach syn- chronization for any initial condition. Indeed, the error system (8) becomes 350 Zeraoulia Elhadj:A Universal Nonlinear Control Law for the Synchronization of Arbitrary...  e′1 = −e1 − ξ5e2 + ξ3e3 e′2 = ξ5e1 − e2 − ξ10e3 e′3 = −ξ3e1 + ξ10e2 − e3 (10) and if we consider the Lyapunov function V = e21+e22+e23 2 , then it is easy to verify the asymptotic stability of the error system (10) by Lyapunov stability theory since we have dV dt = −e21 − e22 − e23 < 0 for all (ai, bi, ci)0≤i≤9 ⊂ R30, (di, ri, si)0≤i≤9 ⊂ R30 and for all initial conditions. In particular, if the two sys- tems (1) and (3) are chaotic, then the control law (9) guarantees also their syn- chronization for any initial condition. A practical example of this situation can be found in [8]. On the other hand, any 3-D continuoustime quadratic chaotic system can be stabilized (controlled) to a stable 3-D continuous-time quadratic system that converges to an equilibrium point (to a 3-D continuous-time quadratic sys- tem that converges to a periodic solution). Furthermore, any 3-D continuous-time quadratic system can be chaotified to a chaotic 3-D continuous-time quadratic system. 3 Example The most known example of 3-D quadratic systems, is the original Lorenz system given by:  x′1 = a1x1 − a1y1 y′1 = b1x1 − y1 − x1z1 z′1 = −c3z1 + x1y1 (11) To apply the above method, we consider the one with variables x2, y2, and z2 as the response system  x′2 = d1x2 − d1y2 y′2 = r1x1 − y2 − x2z2 + u2(t) z′2 = −s3z2 + x1y2 + u3(t) (12) Thus, the universal control law for the system (12) is given by: u1(t) = −(ξ1 + 1)e1 − (ξ2 + ξ5)e2 − ξ4 u2(t) = e2 − ξ8 u3(t) = −ξ9e1 − (ξ11 + 1)e3 − ξ12 (13) Advances in Systems Science and Applications (2012) Vol.12 No.4 351 where  ξ1 = a1 + d1, ξ2 = −a1 − d1, ξ4 = d1x1 − a1x2 − d1y1 + a1y2 ξ5 = −z1 − z2, ξ6 = −2, ξ8 = −x1z2 + z1x2 − y1 + y2 ξ9 = y1 + y2, ξ11 = −c3 − s3, ξ12 = x1y2 − x2y1 − s3z1 + c3z2 (14) In particular, if the two systems (11) and (12) are chaotic, then the control law (13) guarantees their synchronization for any initial condition. 4 Conclusion We have presented a universal nonlinear control law (without any conditions) for the synchronization of arbitrary 3-D continuous-time quadratic systems. 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(2005), “Synchronization of two different chaotic systems: a new system and each of the dynamical systems Lorenz”, Chen and Lu, Chaos, Solitons & Fractals, Vol.25, pp.1049-1056. [13] J. Lu, X. Wu and J. L. (2002), “Synchronization of a unified chaotic system and the application in secure communication”, Phys. Lett. A, Vol.305, pp.365- 370. [14] G. Chen. (1999), Controlling Chaos and Bifurcations in Engineering Sys- tems, CRC Press, Boca Raton, FL. Corresponding author Zeraoulia Elhadj can be contacted at:zeraoulia@mail.univ-tebessa.dz, and zelhad- j12@yahoo.fr.