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International Journal of Bifurcation and Chaos, Vol. 12, No. 3 (2002) 561–570 c World Scientific Publishing Company PARTIAL SYNCHRONIZATION OF NONIDENTICAL CHAOTIC SYSTEMS VIA ADAPTIVE CONTROL, WITH APPLICATIONS TO MODELING COUPLED NONLINEAR SYSTEMS DAVID J. WAGG Department of Mechanical Engineering, University of Bristol, Queens Building, University Walk, Bristol BS8 1TR, UK [email protected] Received February 23, 2001; Revised May 25, 2001 We consider the coupling of two nonidentical dynamical systems using an adaptive feedback linearization controller to achieve partial synchronization between the two systems. In addition we consider the case where an additional feedback signal exists between the two systems, which leads to bidirectional coupling. We demonstrate the stability of the adaptive controller, and use the example of coupling a Chua system with a Lorenz system, both exhibiting chaotic motion, as an example of the coupling technique. A feedback linearization controller is used to show the difference between unidirectional and bidirectional coupling. We observe that the adaptive controller converges to the feedback linearization controller in the steady state for the Chua– Lorenz example. Finally we comment on how this type of partial synchronization technique can be applied to modeling systems of coupled nonlinear subsystems. We show how such modeling can be achieved where the dynamics of one system is known only via experimental time series measurements. Keywords: Synchronization; adaptive; feedback linearization. 1. Introduction partial synchronization using an adaptive synchro- nization technique. The problem of synchronizing two identical dynam- A case of particular interest is when an addi- ical systems has been studied by many authors, for tional feedback signal exists between the two sys- example, Ashwin et al. [1994], Kozlov and Shalfeev tems such that the coupling is bidirectional and [1996], Ashwin [1998], and Yang and Duan [1998], the two systems interact dynamically, giving rise following the work of Pecora and Carroll [1990]. to a complex dynamical behavior. This has appli- When this is achieved using adaptive control type cations to dynamic substructuring, where systems methods, the process is referred to as adaptive syn- are modeled by coupling a set of interacting sub- chronization [John & Amritkar, 1994; Boccaletti structures together [Ohayon et al., 1997; Wagg & et al., 1997; Fradkov & Markov, 1997; Dedieu & Stoten, 2001]. We demonstrate this concept us- Ogorzalek, 1997]. More recently, the concept of ing both a feedback linearization controller and an partial synchronization between two or more sim- adaptive feedback linearization controller. ilar chaotic systems has been studied [Hasler, 1998; In addition, we demonstrate how the adaptive Yanchuk et al., 2001]. In this paper we consider controller can be designed when coupling single and coupling two nonidentical dynamical systems via multiple variables from each of the nonidentical 561 562 D. J. Wagg nonlinear systems. We show that this type of adap- tive controller is stable for such a coupled system. ff 11 12 This is demonstrated using the example of coupling a Lorenz system with a Chua system; for similar examples see [Di Benardo, 1996]. In this example, we observe that the steady state adaptive controller ff converges to the feedback linearization controller. 21 22 Finally we discuss applications to modeling dy- namical systems composed of a set of coupled non- linear dynamical systems. We discuss how partial Signals from the two synchronization can be used to achieve this type of ff systems coupled via 11 22 modeling. We also discuss how the concepts of syn- partial synchronization chronizing dynamical systems [Ashwin, 1998] can be used to monitor the performance of the controller Synchronization f f producing the coupling and hence the modeling pro- 12 21 cess itself, using the Lorenz Chua system as an Fig. 1. Schematic representation of a system formed by par- example. tial synchronization. 2. Partial Synchronization for To achieve partial synchronization, we need to Nonidentical Systems synchronize the dynamics of f12 and f21.Thuswe add a controller, to the coupled system, such that We consider two nonidentical dynamical systems, Eq.(2)canbewrittenas one with state variable x ∈Rp, and the second, q with state variable y ∈R, with governing equa- x˙n(t)=f11(xn,xs,t) tions of the general form x˙s(t)=f12(xn,xs,t)+g(u, t) (3) y˙ (t)=f (y ,y,t) x˙(t)=f1(x, t) , s 21 n s (1) y˙ (t)=f (y ,y,t) y˙(t)=f2(y, t). n 22 n s · In general, we consider that the dynamics of the two where u is the control signal, and g( ) represents systems are nonlinear and that there is no cross cou- the controller function. In this form, the dynam- pling between the two sets of state variables. We de- ics of f21 can be thought of as the reference model [Landau, 1979], which we want f +g(u, t) to repli- fine a coordinate subset of x, x ∈Rn, and similarly 12 s cate and f represents the plant. y ∈Rn, which represent the coordinates which re- 12 s So, in the formulation of Eq. (3), a part of sys- quire synchronization to achieve coupling between tem 1 will be forced to behave like part of system 2. the two systems. So, we will consider the class of However, for bidirectional coupling, system 1 will systems for which Eq. (1) can be expressed as also have an influence on the behavior of systems 2. In this case, an additional feedback signal between x˙ n(t)=f11(xn,xs,t) f1 and f2 can be used to represent the coupling. x˙s(t)=f12(xn,xs,t) (2) We represent it by adding a coupling function to y˙s(t)=f21(yn,ys,t) f21, such that y˙n(t)=f22(yn,ys,t) x˙ n(t)=f11(xn,xs,t) where xn = {xi ∈ x : xi ∈/ xs} and xi denotes the x˙s(t)=f12(xn,xs,t)+g(u, t) { ∈ ∈ (4) ith element of x, and likewise yn = yi y : yi / y˙s(t)=f21(yn,ys,t)+c(xn,xs,t) ys}.Thenifxs→ysas t →∞we say that the sys- y˙n(t)=f22(yn,ys,t) tem is partially synchronized. When such partial synchronization occurs a coupled system is formed In the case where f1 is a physical system and f2 which is shown schematically in Fig. 1. The case is an analytical model the dynamics of f11 can be where xs = x and ys = y is the standard synchro- assumed to be unknown, and c(xn,xs,t)would nization problem [Pecora & Carroll, 1990]. typically be a recorded time series from f11.The Partial Synchronization of Nonidentical Systems 563 functions f22 and f21 must be known explicitly, so therefore e is scalar in this case. Then we can write that they can be computed numerically, and the the error dynamics as structure of f12 must be known. Knowledge of spe- cific parameter values is not required, as the adap- e˙(t)=−λe + L − g(u, t) , (8) tive controller can be applied without this informa- tion. If c = 0 the coupling between the two systems where L =∆f+λe+c,andλ>0. This type of for- (via partial synchronization) is effectively unidirec- mulation is possible with a wide variety of both lin- tional, whereas if c =6 0 the coupling is bidirectional; ear and nonlinear systems [Di Benardo, 1996], and examples will be discussed in Sec. 3.1, 4.1 and 5.1. this requirement is therefore not overly restrictive. We note also that the analysis in this section is for It is clear from Eq. (8) that (L − g1(u, t)) → 0, and autonomous systems, however it is possible to ap- λ>0 will stabilize the required equilibrium, e =0. ply this analysis to some nonautonomous systems Therefore L is the feedback linearization controller [Wagg & Stoten, 2001] which we briefly discuss in for the system [Di Benardo, 1996]. Sec. 5.1. For the class of systems considered in this work, L can be expressed as L = k∗ξ, where k∗ represents 2.1. Controller design a set of (constant) parameters, and ξ the vector of coupling variables. For such systems we use an To design a controller for the system we first reduce adaptive controller which has essentially the same Eq. (4) to the form form as L, g(u, t)=u=k(t)ξ, where k(t)isthe adaptive gain. Using these definitions enables us to x˙ s(t)=f12(d1(t),xs,t)+g(u, t) (5) express Eq. (8) as y˙s(t)=f21(d2(t),ys,t)+c(t) − where the dynamics of xn and yn are now rep- e˙(t)= λe + φ(t)ξ(t) , (9) resented by the functions d and d respectively, 1 2 ∗ which we assume act as disturbances. Then we can where φ(t)=k −k(t) is the parameter error. We formulate the error dynamics for the system such then need to find an expression for k(t)whichstabi- that lizes the system such that φ(t) → 0ast→∞.This we can achieve by choosing a Lyapunov function of − e˙(t)=f21(d2(t),ys,t) f12(d1(t),xs,t) the form − e2 φφT +c(t) g(u, t)(6)V (t)= + , (10) 2 2γ where the error, e = ys − xs.Thiscanthenbe expressed as where γ is the controller gain. Then the derivative of V with respect to time is e˙(t)=∆f(t)+c(t)−g(u, t)(7) − 1 where ∆f(t)=f21 f12. For effective performance V˙ (t)=e(−λe + φ(t)ξ(t)) + φφ˙T , (11) of the controller, we require that the equilibrium, γ e = 0 is stable.

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