Identical synchronization in chaotic jerk dynamical systems

Transcription

1 EJTP 3, No. 11 (2006) Electronic Journal of Theoretical Physics Identical synchronization in chaotic jerk dynamical systems Vinod Patidar 1 and K. K. Sud 2 1 Department of Physics Banasthali Vidyapith Deemed University Banasthali , Rajasthan, INDIA 2 Department of Physics College of Science Campus M. L. S. University, Udaipur , INDIA Received 23 December 2005, Accepted 24 February 2006, Published 25 June 2006 Abstract: It has been recently investigated that the jerk dynamical systems are the simplest ever systems, which possess variety of dynamical behaviours including chaotic motion. Interestingly, the jerk dynamical systems also describe various phenomena in physics and engineering such as electrical circuits, mechanical oscillators, laser physics, solar wind driven magnetosphere ionosphere (WINDMI) model, damped harmonic oscillator driven by nonlinear memory term, biological systems etc. In many practical situations chaos is undesirable phenomenon, which may lead to irregular operations in physical systems. Thus from a practical point of view, one would like to convert chaotic solutions into periodic limit cycle or fixed point solutions. On the other hand, there has been growing interest to use chaos profitably by synchronizing chaotic systems due to its potential applications in secure communication. In this paper, we have made a thorough investigation of synchronization of identical chaotic jerk dynamical systems by implementing three well-known techniques: (i) Pecora- Carroll (PC) technique, (ii) Feedback (FB) technique and (iii) Active Passive decomposition (APD). We have given a detailed review of these techniques followed by the results of our investigations of identical synchronization of chaos in jerk dynamical systems. The stability of identical synchronization in all the aforesaid methods has also been discussed through the transversal stability analysis. Our extensive numerical calculation results reveal that in PC and FB techniques the x-drive configuration is able to produce the stable identical synchronization in all the chaotic jerk dynamical systems considered by us (except for a few cases), however y-drive and z-drive configurations do not lead to the stable identical synchronization. For the APD approach, we have suggested a generalized active passive decomposition, which leads to the stable identical synchronization without being bothered about the specific form of the jerk dynamical system. Several other active passive decompositions have also been listed with their corresponding conditional Lyapunov exponents to achieve the stable identical synchronization in various chaotic jerk dynamical systems. c Electronic Journal of Theoretical Physics. All rights reserved. Keywords: Chaos, Jerk dynamical systems, Identical synchronization of chaos, synchronized chaos PACS (2006): b; j; a vinod r patidar@yahoo.co.in

2 34 Electronic Journal of Theoretical Physics 3, No. 11 (2006) Introduction Various studies of nonlinear dynamical systems in the last four decades have significantly extended the notion of oscillations in these systems. It has been shown that the post-transient oscillations in dynamical systems can be associated not only with the regular behavior such as periodic or quasiperiodic oscillations, but also with chaotic behavior [1-5]. Chaos has long-term unpredictable behavior, which is usually couched mathematically as sensitivity to initial conditions i.e., where the system s dynamics takes it, is hard to predict from the starting point. One way to demonstrate this is to run two identical chaotic systems side by side, starting both at very close, but not exactly equal initial conditions. The systems soon diverge from each other, but both retain the same attractor pattern. An interesting question to ask is: Can we force the two chaotic systems to follow the same path on the attractor? i.e., Can chaotic systems be made synchronized? The affirmative answer is possible to this question. It has been shown that some of the ideas of synchronization can also be extended for the description of particular type of behaviour in coupled systems possessing chaotic dynamics. For example, Fujisaka and Yamada [6-8] have demonstrated that two identical systems with chaotic individual dynamics can change their behaviour from uncorrelated chaotic oscillations to identical chaotic oscillations as the strength of the coupling between the systems is increased. Synchronization of chaotic systems is important as the noise-like behavior of chaotic systems suggests us that such behavior might be useful in some type of private communications. One glance at the Fourier spectrum from a chaotic system suggests the same. There are typically no dominant peaks, no special frequencies i.e., the spectrum is broadband. To use a chaotic signal in communications we are immediately led to the requirement that somehow the receiver must have a duplicate of the transmitter s chaotic signal or, better yet, synchronize with the transmitter. In fact, synchronization is a requirement of many types of communication systems, not only chaotic ones. Unfortunately, if we look at how other signals are synchronized, we will get very little insight as to how to do it with chaos. New methods are therefore required. The organization of rest of the paper is as follows: In Section 2, we introduce the jerk dynamical systems, which are emerged as a sub class of dynamical systems with rich variety of dynamical behaviour including the chaotic motion. We also summarize in a tabular form the various classes of jerk dynamical systems, which we have used for studying the identical synchronization of chaos in these systems. In Section 3, we investigate the identical synchronization of chaos in jerk dynamical system using various algorithms along with their brief descriptions. Particularly in subsections 3.1, 3.2 and 3.3, we investigate the identical synchronization of various jerk dynamical systems using Pecora-Carroll (PC) technique, feedback (FB) technique and active-passive decomposition (APD) respectively. We also investigate the stability of identical synchronization in the aforesaid methods using extensive transversal stability analysis (i.e., extensive conditional Lyapunov exponents (CLE s) calculation) as well as Lyapunov function con-

3 Electronic Journal of Theoretical Physics 3, No. 11 (2006) struction criterion. Finally in Section 4, we summarize the major conclusions drawn from our study of identical synchronization of chaotic jerk dynamical systems. 2. Chaotic Jerk Dynamical Systems In this section, we briefly discuss how the jerk dynamical systems came into existence in the studies of chaotic dynamics. In recent years, there has been a growing interest to find three dimensional dynamical systems which are functionally as simple as possible but nevertheless chaotic [10-22]. Using extensive computer search, Sprott [9] found 19 distinct chaotic models (one of them is conservative and remaining are dissipative, referred as Model A to S) with three-dimensional vector fields that consist of five terms including two nonlinearities or of six terms with one quadratic nonlinearity. Subsequently, Hoover [10] pointed out that the only conservative system (Model A) found by Sprott is an already known special case of Nose-Hoover thermostat dynamical system, which exhibits time reversible Hamiltonian chaos [11]. Except this the other models B to S were apparently unknown. On the other hand, in 1996 Gottlieb [12] reported that Sprott s Model A can be recast into an explicit third order form x = J(x, ẋ, ẍ), which he called jerk function. Because it involves a third derivative of x, which in a mechanical system is a rate of change of acceleration and is called jerk [13]. Since it is known that any explicit ordinary differential equation can be recast in the form of a system of coupled first order differential equations but the converse does not hold in general. Even if one may reduce the dynamical system to a jerk form for each of the phase space variables, the resulting differential equation may look quite different i.e. there may be different possible jerk forms of a single dynamical system. With this, Gottlieb [12] asked a provocative question What is the simplest jerk function that gives chaos? By following the study of Gottlieb, Linz [14] reported that the original Rössler model, Lorenz model and the Sprott s model R can be reduced to jerk form and further showed that the jerk form of Rössler and Lorenz models are rather complicated and are not suitable candidates for the Gottlieb s simplest jerk function. However, the jerk form of Sprott s model R demonstrates the existence of a much simpler form of jerk function that exhibits chaos. By following the Linz s study [14], in 1998 Eichhorn et. al. [15] used the method of Görbner bases and showed that the fifteen of the Sprott s chaotic flows [9] can be recast into jerk form. They also showed that these fifteen models, the Rössler toroidal model [16] and Sprott s minimal chaotic flow [17] can be arranged into seven classes (referred as JD1 to JD7) of jerky dynamics as a hierarchy of quadratic jerk equations with increasingly many terms. Table 1 summarizes the classification simple polynomial chaotic flows suggested by Eichhorn et al. [15]. The alphabets in the first column are the labels assigned by Sprott [9], except for TR and SJ which are the abbereviations for Toroidal Rössler and Simplest Jerk systems respectively. Such a classification provides a simple mean to compare the functional complexity of different systems and also demonstrate the equivalence of cases not otherwise apparent. These seven different classes of chaotic jerk dynamical systems, we have used to study the identical synchronization in section 3.

4 36 Electronic Journal of Theoretical Physics 3, No. 11 (2006) In a subsequent study Eichhorn et. al. [18] examined the simple cases of JD1 and JD2 in more detail and identified the regions of parameter space over which they exhibit chaos. Patidar and Sud [19] have studied in detail, the dynamical behaviour of a special family of jerk dynamical system x = Aẍ ẋ + G(x), where A is a system parameter, G(x)is nonlinear function containing one nonlinearity, one system parameter and a constant term. Table 1: Basic classes of chaotic dissipative jerky dynamics Model Basic classes and parameter values for which there is chaotic behavior JD 1... x = k 1 ẍ + k 2 x + xẋ + k 3 I k 1 = 1 k 2 = 2α = 0.4 k 3 = 2α 2 = 0.08 J k 1 = β = 2 k 2 = αβ = 4 k 3 = αβ (1 α ) = 4 L k 1 = 1 k 2 = α = 3.9 k 3 = α (2βγ α ) = 8.19 N k 1 = γ = 2 k 2 = αγ = 4 k 3 = α (2β α γ) = 4 R k 1 = 1 k 2 = α = 0.9 k 3 = β = 0.4 SJ k 1 = a = k 2 = 1 k 3 = 0... JD 2 x = k 1 ẍ + k 2 ẋ + x 2 + k 3 M k 1 = 1 k 2 = β = 1.7 k 3 = α β2 4 = Q k 1 = β 1 = 0.5 k 2 = β α = 2.6 k 3 = α2 4 = S k 1 = 1 k 2 = α = 4 k 3 = α 2 β = 16 TR k 1 = β = 0.2 k 2 = 1... JD 3 x = k 1 ẍ + k 2 ẋ + k 3 x 2 + xẋ + k 4 k 3 = 1 4 (α + β ) F k 1 = α 1 = 0.5 k 2 = α 1 α 1 = 2.5 k 3 = α 2 = 0.25 k 4 = 1 2α = 1 G k 1 = α 1 = 0.6 k 2 = α 1 2α 1 = 1.85 k 3 = α = 0.4 k 4 = 1 4α = k 1 = α 1 = 0.5 k 2 = α 1 H α 1 = 2.5 k 3 = α 2 = 0.25 k 4 = 1 2α = 1... JD 4 x = k 1 ẍ + k 2 ẋ + k 3 x 2 + xẍ + k 4 k 1 = 1 k O 2 2 = 1 α = 1.7 k 3 = 1 k 4 = JD 5 x = k 1 ẍ + k 2 x 2 + k 3 ẋ 2 + xẍ D k 1 = 1 k 2 = 1 k 3 = α = 3... JD 6 x = k 1 ẍ + k 2 ẋ + k 3 x 2 + k 4 ẋ 2 + xẍ + k 5 k 1 = 1 k 2 = 1 α = 1.7 k 3 = 1 P 2 k 4 = 1 k 5 = 1/2... JD 7 x = k 1 ẍ + k 2 ẋ + k 3 x 2 + k 4 ẋ 2 + k 5 xẋ + xẍ + k 6 K k 1 = α 1 2α k 2 = α 1 α k 3 = α = 0.3 k 4 = 1 k 5 = 2 α = 1.7 k 6 = 1 4α 0.83 They have identified the regions of parameter space, where different type of long time dynamical behaviour dominates, using some analytical methods as well as extensive Lyapunov spectra calculation in complete parameter space by considering five different forms of nonlinear function G(x) comprising of absolute, quadratic, cubic, quartic and quintic nonlinearities. They observed that only systems having absolute and quadratic nonlinearities in G(x) exibit chaos. As a result they made an important conclusion for these jerk dynamical systems that a certain amount of nonlinearity is sufficient for exhibiting chaotic behaviour but more is not necessarily better. Apart from the above studies on the dynamics of jerk dynamical system, Patidar et. al. [20] and Patidar and Sud [21] respectively, have attempted the problems of controlling chaos and synchronization of

5 Electronic Journal of Theoretical Physics 3, No. 11 (2006) chaos in jerk dynamical systems. In the next section, we investigate the identical synchronization of chaos in jerk dynamical system using various algorithms along with their brief descriptions. 3. Identical Synchronization In Chaotic Jerk Dynamical Systems Early work on synchronization of coupled chaotic systems was done by Fujisaka and Yamada [6-8]. In that work, some sense of how the dynamics might change was brought out by a study of the Lyapunov exponents of synchronized, coupled systems. Although Fujisaka and Yamada [6,-8] were the first to exploit local analysis for the study of synchronized chaos, their papers went relatively unnoticed. Later, a now-famous paper by Afraimovich et. al. [22] exposed many of the concepts necessary for analyzing synchronous chaos, although it was not until many years later that wide-spread study of synchronized, chaotic systems took hold. As already mentioned, chaotic systems are dynamical systems that defy synchronization due to their essential feature of displaying high sensitivity to initial conditions. As a result, two identical chaotic systems starting at nearly the same initial points in phase space develop onto trajectories which become uncorrelated in the course of time. However, it has been shown that it is possible to synchronize these kind of systems, to make them evolving on the same chaotic trajectory [6, 22, 23, 24, 25]. When we deal with coupled identical systems, synchronization appears as the equality of the state variables while evolving in time. We refer to this type of synchronization as identical synchronization (IS). Other names in literature for this notion are: complete synchronization or conventional synchronization [26]. In this section we will discuss main properties of this kind of synchronization. The appearance of this kind of synchronization have been estabilished by means of several different coupling mechanisms, such as Pecora and Carroll method [24, 25, 27], the negative feedback [28], the sporadic driving [29], the active-passive decomposition [30, 31], the diffusive coupling and some hybrid method of Guemez and Matias [32] etc. In this section, we will concentrate our attention to explain the properties and stability of synchronized motion in three particular coupling schemes namely Pecora and Carroll (PC) method, feedback technique and active passive decomposition (APD). 3.1 Pecora Carroll (PC) Technique Pecora and Carroll [24, 25] discovered a way to achieve identical synchronization. They take a complete chaotic system and choose a subsystem within it. Then they make a replica of this subsystem. The original system is called drive (master) system and the duplicate subsystem is called response (slave). The response is just like the drive except it is missing one or more variables. The missing variables are sent from the drive to the response, inputting the variable wherever it is needed in the response subsystem. If a stable response subsystem has been chosen, then the response s dynamic variables will

6 38 Electronic Journal of Theoretical Physics 3, No. 11 (2006) converge to their counterparts in the drive and will remain synchronized with them as long as the drive continues to supply the missing variables to the response subsystem i.e. the key idea for identical synchronization is to choose a stable subsystem. Consider an autonomous n-dimensional dynamical system Ẋ = F (X), (1) here X = (x 1, x 2, x 3,..., x n ) T and F (X) = (f 1 (X), f 2 (X),..., f n (X)) T. Divide the system into two parts in an arbitrary way as: X = (v, w), v = g(v, w), ẇ = h(v, w), (2) where v = (x 1, x 2,..., x m ), g = (f 1 (X), f 2 (X),..., f m (X)), w = (x m+1, x m+2,..., x n ) and h = (f m+1 (X), f m+2 (X),..., f n (X)). Now create a new subsystem w identical to w system and, substitute the set of variables v for the corresponding v in the function h i.e., ẇ = h(v, w )and v = v i.e. the new subsystem will be drived by the original system ẇ = h(v, w ). (3) In such a way we have the following compound system: v = g(v, w), ẇ = h(v, w), ẇ = h(v, w ) (4) Now we examine the difference w = w w, Under the right condition (i.e. if the chosen subsystem is stable) both the system will synchronize as time grows i.e. w 0 as t. In order to study the dynamics of difference w, subtract Eq. (3) from Eq. (2), we get an equation for the dynamics of w as: ẇ ẇ = d dt ( w) = D wh(v, w) w, (5) where D w his the matrix of derivatives of hwith respect to w i.e. the Jacobian of the w subsystem vector field with respect to w only. The technique described above is also known as complete replacement technique because in this technique we completely replace all the v variables of response subsystem by their counterparts (v) in the original system. It is clear that in the phase space of the compound system described by Eq. (4), there exists a manifold w = w such that if the initial conditions lie in this manifold, the consequent evolution of the system will take place in this manifold as well. In other words, the manifold w = w is an invariant manifold and is called synchronization manifold. In the above analysis, we have said that both systems will show identical chaotic oscillations if a stable subsystem is chosen. In order to define which subsystem is stable,

7 Electronic Journal of Theoretical Physics 3, No. 11 (2006) one needs to study the stability of synchronized oscillations. Since in our case the synchronized oscillations are chaotic, these trajectories are always unstable. However in the analysis of synchronized chaotic motions one has to distinguish between the instability for perturbation tangent to synchronization manifold and transverse to it. The regime of identical chaotic oscillations is stable, when the synchronized trajectories are stable for the perturbations in the transverse direction to the synchronization manifold. This analysis is called transversal stability analysis. The Eq. (5) is a basic equation for much of the discussion on synchronized chaotic systems. Three most frequent criteria for the stability of synchronized chaotic motions are: (1) The criterion based on eigenvalues of the Jacobian matrix corresponding to the flow evaluated on synchronization manifold (i.e. Jacobian matrix defined on difference system) has been introduced by Fujisaka and Yamada [6, 7], which requires that for the stable synchronization the largest eigenvalue of the aforesaid Jacobian matrix should be negative. For the case of Pecora Carroll technique discussed above, this criterion states that the largest eigenvalue of the Jacobian matrix D w hmust be negative for onset of synchronization. If the D w h is constant over the attractor i.e. response is linear then the calculation of eigenvalues (λ 1, λ 2,..., λ n m ) is straightforward and hence one can easily conclude whether the synchronization is possible or not. But the complications arise; when the Jacobian is not constant over the attractor i.e. response system is nonlinear in nature (In case of PC method the response means the subsystem, but for the other coupling mechanisms, where subsystem is not required, we will consider the difference system instead of response system i.e. the basic equation (Eq. (5)), which describe the dynamics of difference of drive and response systems). (2) He and Vaidya [27] developed a criterion for chaos synchronization based on the notion of asymptotic stability of dynamical systems, which refers to the condition for a given chaotic system with drive-response (master-slave) configuration to reach the same eventual state at a fixed time irrespective of the choice of initial conditions. One of the practical way to establish the asymptotic stability of the response subsystem is to find a suitable Lyapunov function L( w)(where w represents the difference) that satisfy the following properties: a) L( w) 0in a certain region containing the chaotic set of synchronized motion. b) L(0) = 0, In other words, L( )is zero everywhere on the synchronization manifold (i.e.w = w ). c) dl 0 everywhere in a certain region containing the chaotic set of synchronized dt motion and dl = 0 on the synchronization manifold. dt If one succeeds in construction of such Lyapunov function for a particular choice of response subsystem then one can conclude that the subsystem is stable and the chaotic set of synchronized oscillations is transversally stable therefore the composite system evolves towards the regime of identical synchronized chaotic oscillations. The criterion based on construction of Lyapunov function for the vector field

8 40 Electronic Journal of Theoretical Physics 3, No. 11 (2006) of perturbation transverse to the synchronization manifold enables one to prove that all trajectories in the phase space of the coupled systems are attracted by the synchronization manifold. Despite the fact that it is a very reliable criterion, which guarantees the onset of chaotic synchronization, it has two serious disadvantages. First of all, there is no generic method for construction of Lyapunov function for an arbitrary system. As a result the search for it is time consuming or practically impossible even though these systems are known to be synchronized. Thus this method is not general. The second disadvantage is that the estimate for the minimum coupling strength necessary to synchronize two systems provided by this method is usually not accurate. It is found that the systems can be synchronized by much weaker coupling than predicted by the Lyapunov function method. (The second disadvantage is not applicable for the PC method as there is no coupling parameter involved in PC technique. However it is applicable for feedback methods, where we introduce the coupling between the drive and response systems rather than creating a new subsystem, as we did in PC method. These methods will be clear in the subsequent sections). (3) In contrast with the Lyapunov function criterion, the analysis of transversal Lyapunov exponents is quite straight forward and can be easily employed, even for rather complicated systems. This can be done by calculating the Lyapunov exponents of Eq. (5) by using methods similar to ones used for the computation of conventional Lyapunov exponents for any dynamical systems [33-36]. When all the (n m) Lyapunov exponents are negative, then the compound system will move towards the synchronization manifold. Note that the full system (1) has a set of n Lyapunov exponents (at least one will be positive because we are considering the chaotic system), but the Lyapunov exponents for Eq. (5) do not form a subset of the Lyapunov exponents of full system because they depend upon the drive variables (v) so called conditional Lyapunov exponents (CLE s). Using Pecora and Carroll approach described above, identical synchronization of chaos has been demonstrated numerically as well as experimentally on several chaotic systems such as Lorenz system [24, 25, 27, 37-32], Rossler system [24], the hysteretic circuit [24] Chua s circuit [43], driven Chua s circuit [44], DVP oscillator [45-47], phase-locked loops [48-50] etc. Now, we demonstrate Pecora and Carroll approach for identical synchronization by taking chaotic jerk dynamical systems as prototypical examples. First of all we consider the two chaotic jerk dynamical systems studied by Patidar and Sud [19], which are members of x = Aẍ ẋ + G(x)(for G(x) = B (x 2 + C) and G(x) = B ( x + C)), then we will extend the results to the jerky representations of the Sprott s simple chaotic flows by considering their classification according to the hierarchy of quadratic jerk functions with increasingly many more terms (see Table 1 for detailed classification). We consider the jerk equation of the form: x + A ẍ + ẋ = G(x) (6)

9 Electronic Journal of Theoretical Physics 3, No. 11 (2006) The dynamical system representation of Eq. (6) in terms of ODE s by using the transformation andwill be: ż = Az y + G(x) There are three possibilities to implement the PC method in this system. (i) x-drive configuration: If we consider the x as drive variable then the drive and response systems for jerk dynamical system (Eq. (7)) will be as ż = Az y + G(x) and ż = A z y + G(x) and the difference system for y = y y and z = z z in matrix form is [ d ( y) ] [ ] [ ] dt 0 1 y d ( z) = 1 A z dt Eq. (9) is the basic equation for describing the stability of the perturbation transverse to the synchronization manifold under x-drive configuration for jerk dynamical system (Eq. (7)). Since in this equation the coefficient matrix (Jacobian matrix) is a constant matrix and it is very easy to calculate the eigenvalues of this matrix. The eigenvalues for this matrix are: A ± A 2 4, for A = 0.6(chaotic case) the eigenvalues are 0.3 ± i 0.95, 2 which give the following solution for Eq. (9) y = z = e 0.3t (K 1 cos (0.95t) + K 2 sin (0.95t)) (10) here K 1 and K 2 are constants of integration. It is clear that as t, y = z = 0, hence drive and response systems synchronize. If we numerically calculate the Lyapunov exponents of the subsystem (subjected to x-drive condition) i.e. conditional Lyapunov exponents (CLE s), these come out to E-1, E-1 (for both the cases (i) G(x) = B(x 2 + C) with B = 0.58 and C = 1 (ii) G(x) = B( x + C) with B = 1.0 and C = 2.0). As all the CLE s are negative and hence synchronization is possible. In the above paragraph, we have analyzed the possibility of identical synchronization in chaotic jerk dynamical system (Eq. (7)) for x-drive configuration using eigenvalues of the Jacobian matrix and CLE s. Now we analyze the possibility of synchronization using the Lyapunov function construction method. If we consider the Lyapunov function L( y, z) = 1 [( ) 1 + A (( y) 2 + ( z) 2) ] + ( y + z) 2 + A ( y) 2, (11) 2 A here y = y y and z = z z. Lyapunov function L ( y, z)is positive definite and L(0, 0) = 0 i.e., it is zero everywhere on the synchronization manifold (y = y and z = z ). Now the time rate of change of Lyapunov function is given by dl ( y, z) = ( 1 + A dt A ) ( ( y) d dt ( y) + ( z) d dt ( z) ) + ( y + z) ( d dt ( y) + d dt ( z)) + A ( y) d dt ( y) (7) (8) (9)

10 42 Electronic Journal of Theoretical Physics 3, No. 11 (2006) = ( 2A ( z) 2 + ( y) 2) (12) for A = 0.6 > 0, dl 0and also dl = 0 on the synchronization manifold, hence the dt dt synchronization is possible i.e. y = z = 0 as t. From the analysis given above, we found that all the three criteria guarantee the identical synchronization of chaotic jerk dynamical systems described by Eq. (7) using the x-drive configuration. Fig. 1 Identical synchronization of chaos in jerk dynamical system having quadratic nonlinearity using Pecora-Carroll technique (i.e., Eq. (8) with G(x) = B(x 2 + C)) for A = 0.6, B = 0.58 and C = 1.0: (a) time history of y-variables of drive and response systems (b) the difference between y-variables of drive and response as a function of time (c) time history of z-variables of drive and response systems (d) the difference between z-variables of drive and response as a function of time. We have also numerically solved the drive-response system of jerk dynamical system for x-drive configuration i.e., Eq. (8) by using fourth-order Runge-kutta integrator with the step size The results of the numerical calculations are shown in Figures 1 and 2 for G(x) = B(x 2 + C)with A=0.6, B=0.58 and C = 1. Particularly in Figures 1(a) and 1(b), we have shown the y variables of both drive and response systems and difference between y variables of both drive and response systems respectively. It is clear that ast increases the y-variables of drive and response systems synchronize i.e., difference becomes zero. Similar behaviour is discussed in Figures 1(c) and 1(d) for z-variables. There are more ways to visualize identical synchronization between the drive and response systems. Some of them we have shown in Figure 2. Particularly in Figure 2(a), we have plotted the y-variable of drive versus y-variable of response while in Figure 2(b) z-variable of drive versus z-variable of response (after some transient time die out). Both plots are straight lines inclined with an angle of 45 o and passing through the origin (0,0) which also show the synchronized behaviour of drive and response systems. In Figure 2(c) we have shown the trajectories of the drive and response systems in the xy-plane. We see that the drive and response start with different initial conditions but after some time both converge to the same trajectory. A better view has been shown in Figure 2(d) in which we have plotted the trajectory in the y z plane (i.e. difference

11 Electronic Journal of Theoretical Physics 3, No. 11 (2006) Fig. 2 Identical synchronization of chaos in jerk dynamical system having quadratic nonlinearity using Pecora-Carroll technique (i.e., Eq. (8) with G(x) = B(x 2 + C)) for A = 0.6, B = 0.58 and C = 1.0: (a) y-variable of drive versus y-variable of response system (b) z-variable of drive versus z-variable of response system (c) the trajectory of drive and response systems in yz-plane (d) the trajectory of the difference of y-variables and z-variables of drive and response systems in difference plane. Fig. 3 Identical synchronization of chaos in jerk dynamical system having absolute nonlinearity using Pecora-Carroll technique (i.e., Eq. (8) with G(x) = B( x +C)) for A = 0.6, B = 1.0 and C = 2.0: (a) time history of y-variables of drive and response systems (b) the difference between y-variables of drive and response as a function of time (c) time history of z-variables of drive and response systems (d) the difference between z-variables of drive and response as a function of time. plane), it is a spiral ending at a fixed point (0,0). Results of the similar numerical calculations for Eq. (8) with G(x) = B( x + C) for A = 0.6, B = 1.0 and C = -2.0 are shown in Figures 3 and 4. (ii) y-drive configuration: If we consider y as a drive variable then the drive and response systems for jerk dynamical system (Eq. (7)) will be as

12 44 Electronic Journal of Theoretical Physics 3, No. 11 (2006) Fig. 4 Identical synchronization of chaos in jerk dynamical system having absolute nonlinearity using Pecora-Carroll technique (i.e., Eq. (8) with G(x) = B( x +C)) for A = 0.6, B = 1.0 and C = 2.0: (a) y-variable of drive versus y-variable of response system (b) z-variable of drive versus z-variable of response system (c) the trajectory of drive and response systems in yz-plane (d) the trajectory of the difference of y-variables and z-variables of drive and response systems in difference plane. ż = Az y + G(x) and ẋ = y ż = A z y + G(x ) and the difference system for x = x x and z = z z will be as: d ( x) = y y = 0 dt d ( z) = A( z) + G(x) dt G(x ) Eq. (14) is the basic equation for describing the stability of the perturbation transverse to the synchronization manifold under y-drive configuration for jerk dynamical system (Eq. (7)). Since in this equation the coefficient matrix (Jacobian matrix) is not a constant matrix as in case of x-drive configuration. So it is not easy to calculate the eigenvalues of this matrix because these will depend on the time evolution of x and z variables of both drive and response systems. Instead of finding the eigenvalues, we numerically calculate the conditional Lyapunov exponents and which are E-4 ( 0) & E-1 (for G(x) = B(x 2 + C);A = 0.6, B = 0.58 and C = 1.0) and E-4 ( 0) & E-1 (for G(x) = B( x +C);A = 0.6, B = 1.0 and C = 2.0). Since one of them (in both cases) is approximately zero or positive definite and hence identical synchronization is not possible. We have also numerically solved the drive-response system of jerk dynamical system for y-drive configuration i.e., Eq. (13) by using fourth-order Runge-kutta integrator with the step size The results of the numerical calculations are shown in Figure 5 (for (13) (14)

13 Electronic Journal of Theoretical Physics 3, No. 11 (2006) Fig.5 Identical synchronization of chaos in jerk dynamical system having quadratic nonlinearity using Pecora-Carroll technique (Eq. (13) with G(x) = B(x 2 + C)) for A = 0.6, B = 0.58 and C = 1.0: (a) the difference between x-variables of drive and response as a function of time (b) the difference between z- variables of drive and response as a function of time (c) ) x-variable of drive versus x-variable of response system (d) z-variable of drive versus z-variable of response system. Fig.6 Identical synchronization of chaos in jerk dynamical system having absolute nonlinearity using Pecora-Carroll technique (Eq. (13) with G(x) = B( x +C)) for A = 0.6, B = 1.0 and C = 2.0: (a) the difference between x- variables of drive and response as a function of time (b) the difference between z-variables of drive and response as a function of time (c) ) x-variable of drive versus x-variable of response system (d) z-variable of drive versus z-variable of response system. G(x) = B(x 2 + C); A = 0.6, B = 0.58 and C = 1.0) and in Figure 6 (for G(x) = B( x + C);A = 0.6, B = 1.0 and C = 2.0). Particularly in Frames (a) of both figures, we have shown the difference between x-variable of the drive and response systems while in Frames (b) the difference between the z-variables of the drive and response systems. We observe that the difference between the x-variables of drive and response systems is constant and equal to the initial difference. It is so because one of the CLE s is equal to zero for y-drive configuration, which suggests that neither the initially nearby trajectories diverge nor converge. However the difference between the z-variables of the drive and response systems is large and oscillatory and hence the drive and response systems do not synchronize for y-drive configuration. In Frames (c) of both the figures, we have plotted the x-variables of drive versus the x-variable of response system. It is a straight line passing through the point (0, 16) not through origin, which suggests that x-variables of drive and response systems do not synchronize (x x ) but have a linear relationship

14 46 Electronic Journal of Theoretical Physics 3, No. 11 (2006) (x = x + 16). However the plot of z versus z (see Frames (d) of both the Figures) is a complicated geometrical structure, which shows the non-synchronized behaviour of z variables of drive and response systems and also suggests that zand z have a nonlinear relationship as evident from a complicated attractor type structure in Frames (d). (iii) z-drive configuration: If we consider z as the drive variable then the drive and response systems for jerk dynamical system (Eq. (7)) will be as: ż = Az y + G(x) and ẏ = z (15) and the difference system for x = x x and y = y y in matrix form is [ d ( x) ] [ ] [ ] dt 0 1 x d ( y) = (16) 0 0 y dt Eq. (16) is the basic equation for describing the stability of the perturbation transverse to the synchronization manifold under z-drive configuration for jerk dynamical system (Eq. (7)). Since in this equation the coefficient matrix (Jacobian matrix) is a constant matrix and it is very easy to calculate the eigenvalues of this matrix. The eigenvalues for this matrix are: 0, 0 for any value ofa and hence the trajectories for x(t)&x (t)and y(t) & y (t)respectively, will remain apart by a constant distances x(0) x (0) and z(0) z (0), in this case also synchronization is not possible. If we numerically calculate the conditional Lyapunov exponents, these turn out to be (0.0, 0.0), which are consistent with the above results. From the analysis of identical synchronization using PC technique for the jerk dynamical system (Eq. (7)), we can summarize the results in the Table 2. Table 2: Conditional Lyapunov exponents (CLE s) for different combination of drive and response systems for jerk dynamical system having quadratic and absolute nonlinearities.... x + Aẍ + ẋ = B(x 2 + C ); A = 0.6; B = 0.58; C = 1 Drive Variable Response Subsystem Conditional Lyapunov Exponents Whether IS is possible or not? x (y, z) E-1, E-1 Yes y (x, z) E-4, E-1 No z (x, y) E-0, E-0 No... x + Aẍ + ẋ = B( x + C); A = 0.6; B = 1; C = 2 x (y, z) E-1, E-1 Yes y (x, z) E-4, E-1 No z (x, y) E-0, E-0 No In the above analysis, we have discussed in detail the possibilities of identical synchronization of chaos in jerk dynamical systems having quadratic and absolute nonlinearities using the Pecora-Carroll approach. Now we extend our calculations for the identical synchronization of chaos using PC approach to the jerk dynamical representations of the Sprott s simple chaotic flows [9] by considering their classification [15] according to the hierarchy of quadratic jerk functions with increasing many more terms. In Table 3, we have summarized all these results. We observe from the Table 3 that the x-drive configuration for all the jerk dynamical systems guarantees the identical synchronization except

15 Electronic Journal of Theoretical Physics 3, No. 11 (2006) for the models L, R and D, where no subsystem is stable. However the y-drive and z-drive configurations are unstable for all the jerk dynamical systems.

16 48 Electronic Journal of Theoretical Physics 3, No. 11 (2006) Feedback (FB) Technique In Section 3.1, we observed that Pecora and Carroll method of chaos synchronization works fairly good for the jerk dynamical systems. However it requires dividing the original

17 Electronic Journal of Theoretical Physics 3, No. 11 (2006) system into two stable subsystems. In this section, we address the same problem of synchronization but in different way by considering the following question: Can one make a chaotic trajectory of one system to synchronize with a chaotic trajectory of other identical chaotic system, starting with different initial conditions without dividing the original system into two stable subsystems? It is possible by means of feedback method or one-way coupling method [45, 47, 51-53]. By one-way coupling we mean that the behaviour of second chaotic system (response) is dependent on the behaviour of the first identical system (drive) but the first one (drive) is not influenced by the behaviour of second (response) system. The one way coupling method is also called as feedback method as in this we choose a drive variable from the drive system and feedback control is applied to the response system. The feedback control is proportional to the difference of the chosen variable from drive and its counterpart from response system. Under suitable conditions, as time elapses, the amount of feedback decreases and soon both the drive and response systems achieve complete synchronization by following the same trajectory and afterward the feedback amount becomes zero and the identical synchronization persists. The similar feedback method has been used by Singer et. al. [54], Pyragas [55], Chen and Dong [56], Pyragas [57], Kapitaniak and Chua [58], Kapitaniak et. al. [59] for the controlling of chaos, where the drive system exhibits the periodic motion of desired periodicity while the response system is chaotic and the aim of the feedback control is to bring the response system to the periodicity of drive system. In the following, we discuss the formulation and stability of the feedback method for identical synchronization of chaos. Consider an n-dimensional dynamical system, which is chaotic, as Ẋ = F (X), (17) where X = (x 1, x 2, x 3,..., x n ) T and F (X) = (f 1 (X), f 2 (X),..., f n (X)) T. Now choose a dynamical variable as drive variable from it e.g. x i (1 i n). Consider another chaotic system identical to Eq. (17) but starting from different initial conditions (i.e. with different variables), as Ẋ = F (X ), (18) where X = (x 1, x 2,..., x n ) T and F (X ) = (f 1 (X ), f 2 (X ),..., f n (X )) T. Now the feedback control, which is proportional to the difference of the drive variable x i and its counterpart x i in the response, is applied to response system. Hence the response system looks as ẋ 1 = f 1 (x 1, x 2,..., x n). ẋ i = f i (x 1, x 2,..., x n) c(x i x i ), (19).. ẋ n = f n (x 1, x 2,..., x n) where c is a constant and termed as feedback constant or coupling strength. The pair of drive and response dynamical systems (i.e. Eqs. (17) and (19)) synchronize if the

18 50 Electronic Journal of Theoretical Physics 3, No. 11 (2006) dynamical system describing the evolution of the difference, ė 1... ė i ė n ẋ 1 ẋ 1 f 1 (x 1, x 2,..., x n) f 1 (x 1, x 2,..., x n ).. ẋ = i ẋ i f = i (x 1, x 2,..., x n) f i (x 1, x 2,..., x n ) c(x i x i ), (20).... ẋ n ẋ n f n (x 1, x 2,..., x n) f n (x 1, x 2,..., x n ) possesses a stable fixed point at the origin E = 0,where E = (e 1, e 2,..., e n ) T and e i = x i x i for i = 1 to n. The dynamics of the error system can be understood by studying the following linearized system for small E: Ė = DF E, (21) where DF is the Jacobian (matrix of derivatives ) of response system with respect to Ei.e. f 1 f e 1. 1 f e i.. 1 e n f i f DF = e 1. i f e i c.. i e n......, (22) f n f e 1. n e i.. All the derivatives in the Jacobian matrix are evaluated at E = 0i.e., on the synchronization manifold. It is clear that the coupling strength c also appears in equation of error system and hence it also affects the stability of synchronization. Now the question arises: for which selection of drive variable and coupling strength the identical synchronization will be stable. To study the stability of identical synchronization in this case, one can use the three basic criteria introduced in the Section 3.1: (i) If the largest eigenvalue of the Jacobian matrix DF appeared in Eq. (21) is negative then drive and response systems will possess identical synchronization [6-8]. (ii) If a suitable Lyapunov function [27] L(E)exists, which satisfies: (a)l(e) 0, (b) L(0) = 0, (c) dl(e) / dt 0 and (d) dl(0) /dt = 0. (iii) If all the Lyapunov exponent of the system described by Eq. (21) (conditional Lyapunov exponents) are negative, then the identical synchronization between drive and response systems will be stable [24, 25]. Now we use the feedback technique described above to demonstrate the identical synchronization of chaos in jerk dynamical systems. First of all we consider the two jerk dynamical systems, which have been studied in detail by Patidar and Sud [19], then we will extend the results to the jerky representations of the Sprott s simple chaotic flows by considering their classification (as suggested by Eichhorn et al. [15]) according to the hierarchy of quadratic jerk functions with increasing many more terms [see Table 1 for detailed classification] First, we consider the following family of jerk dynamical system: f n e n x = A ẍ ẋ + G(x), (23)

19 Electronic Journal of Theoretical Physics 3, No. 11 (2006) where A is a system parameter and G(x) is a nonlinear function. The dynamical system representation of Eq. (23) in terms of first order ODE s by using the transformations and will be as: ż = Az y + G(x). (24) Now choose a dynamical variable x as drive variable and applied the feedback control as explained in the previous subsection. The response system after inclusion of feedback term will look like as: c(x x) ż = Az y + G(x ), (25) where c is the coupling strength between the drive and response system. The dynamical system for the evolution of difference between drive and response systems described by Eqs. (24) and (25) will be as: Ė = DF E, (26) where E = (x x, y y, z z) T and DF = [ ] c (27) G/ x 1 A The derivative in the Jacobian matrix DF is evaluated ate = 0. It is clear that for a nonlinear functiong(x), the Jacobian matrix given above is not a constant matrix and it is not easy to calculate its eigenvalues. For analyzing the stability of identical synchronization in this case, we have performed the numerical calculation of the Lyapunov exponents of Eq. (26) (i.e. conditional Lyapunov exponents). First of all we have numerically integrated the coupled systems described by Eqs. (24) and (25) with A = 0.6, G(x) = 0.58 (x 2 1) and for different values of coupling strength (c). In Figure 7, We have shown the solutions for two specific values of coupling strength c = 0and c = 0.8. Particularly in Frames (a), (b) and (c) respectively, we have shown the solution of coupled systems in xx -plane, yy -plane and zz -plane for c = 0 i.e when both the systems are evolving independently from different initial conditions. It is clear that these figures correspond to an unsynchronized motion. While in Frames (d), (e) and (f) respectively, we have shown the solution of coupled systems in xx -plane, yy -plane and zz -plane for c = 0.8. In all these frames ((d), (e) and (f)), we observe a line inclined at an angle of 45 o and passing through the origin, which suggests the equality of all three dynamical variable of drive with its counterparts in response systems i.e. identical synchronization. For both cases, we have also calculated the Lyapunov exponents of the response system. For c = 0, the drive and response systems are chaotic and evolve independently and hence both have the same set of Lyapunov exponents i.e.(0.6832e- 01, 0.0, ). For c = 0.8, the response system is dependent on the behaviour of drive system however the drive system is not influenced by the behaviour of response system. Hence the drive system possesses the same Lyapunov exponents as for c = 0,

20 52 Electronic Journal of Theoretical Physics 3, No. 11 (2006) Fig. 5 Identical synchronization of chaos in jerk dynamical system using feedback technique (Eqs. (24) and (25)) for A = 0.6and G(x)=0.58(x 2 1): Frames (a), (b) and (c) respectively, show the unsynchronized motion of x, y and z variables of drive and response systems, when both systems are uncoupled i.e. c = 0.0 while Frames (d), (e) and (f) respectively, show the synchronized motion of x, y and z variables of drive and response systems, when both systems are coupled through coupling strength c = 0.8. while the values of Lyapunov exponents of response system i.e. conditional Lyapunov exponents (CLE s) are ( E-1, E-0, E+1). From the values of CLE s it is clear that all the CLE s are negative or the largest CLE is negative and hence identical synchronization is stable for c = 0.8. For getting the complete information for the value of coupling strength, at which the identical synchronization is stable or unstable, we have calculated the conditional Lyapunov exponents for a certain range of coupling strength (c) in step of In Figure 8, we have shown the plot of largest CLE as a function of coupling strength. We observe from this figure that when c is greater than 0.65 then the largest CLE is negative and hence identical synchronization is stable while for c less than 0.65, the largest CLE is positive and identical synchronization is unstable. We have also numerically integrated the coupled systems described by Eqs. (24) and (25) with A = 0.6, G(x) = ( x 2) and for different values of coupling strength (c). In Figure 9, we have shown the solutions for two specific values of coupling strength c = 0 (unsynchronized) and c = 0.7(synchronized). For both cases, we have also calculated the Lyapunov exponents of the response system. For c = 0, when both the drive and response systems are chaotic and evolve independently hence have the same set of Lyapunov exponents i.e. ( E-01, 0.0, E-0) while for c = 0.7 the response system is dependent on the behaviour of drive system however the drive system is not influenced by the behaviour of response system hence the drive system possess the same Lyapunov exponents as for c = 0, while the values of Lyapunov exponents of response system i.e. conditional Lyapunov exponents (CLE s) are ( E-1, E-0, E-0). All

21 Electronic Journal of Theoretical Physics 3, No. 11 (2006) Fig. 6 Largest conditional Lyapunov exponent (CLE of Eq. strength (c) for A = 0.6and G(x)=0.58(x 2 1). (26)) as a function of coupling the CLE s are negative or the largest CLE is negative and hence identical synchronization is stable for c = 0.7. For getting the complete information for the value of coupling strength at which identical synchronization is stable or unstable, we have calculated the conditional Lyapunov exponents for a certain range of coupling strength (c) in step of In Figure 10, we have shown the plot of largest CLE as a function of coupling strength. We observe from this figure that when c is greater than 0.55 then the largest CLE is negative and hence identical synchronization is stable while for c less than 0.55 largest CLE is positive and identical synchronization is unstable. Finally, we extend the feedback technique for identical synchronization of chaos in jerk dynamical representations of the Sprott s simple chaotic flows [9] by considering their classification [15] according to the hierarchy of quadratic jerk functions with increasing many more terms. In Figures 11 to 14, we have shown the largest CLE as a function of coupling strength for the jerk dynamical representations of Sprott s simple chaotic flows, when x-variable is used as a drive for calculating the feedback control term. In the respective frames we have also depicted the threshold value of coupling strength beyond which the largest CLE is negative i.e. identical synchronization is possible. 3.3 Active Passive Decomposition (APD) In Sections 3.1 and 3.2, we have discussed two different techniques for synchronizing two identical chaotic systems as well as their application to various jerk dynamical systems. In Section 3.1, we observed that the drive and response technique introduced by Pecora

22 54 Electronic Journal of Theoretical Physics 3, No. 11 (2006) Fig. 7 Identical synchronization of chaos in jerk dynamical system using feedback technique (Eqs. (24) and (25)) for A = 0.6and G(x)=( x -2): Frames (a), (b) and (c) respectively, show the unsynchronized motion of x, y and z variables of drive and response systems, when both systems are uncoupled i.e. c = 0.0 while Frames (d), (e) and (f) respectively, show the synchronized motion of x, y and z variables of drive and response systems, when both systems are coupled through coupling strength c = 0.7. Fig. 8 Largest conditional Lyapunov exponent (CLE of Eq. strength (c) for A = 0.6and G(x)=( x -2). (26)) as a function of coupling and Carroll [24, 25] works fairly for synchronizing various identical jerk dynamical systems. The x-drive configuration for all jerk dynamical systems guarantees the identical synchronization except for the models L, R and D, where no subsystem is found to be