Stability and Projective Synchronization in Multiple Delay Rössler System

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1 ISSN (print), (online) International Journal of Nonlinear Science Vol.7(29) No.2,pp Stability and Projective Synchronization in Multiple Delay Rössler System Dibakar Ghosh Department of Mathematics,Dinabandhu Andrews College Garia, Calcutta- 7 84, India (Received 7 August 28, accepted 1 November 28) Abstract: In this paper, the issues of projective synchronization of unidirectional coupled multiple delay systems via linear observer is investigated. A detail stability analysis of multiple delay Rössler system is derived by D-subdivision method. In the previous study, whereas projective synchronization is investigated by nonlinear observer approach, here we can achieve projective synchronization via linear observer approach. The proposed technique has been applied to synchronize multi delay Rössler system by numerical simulation. Key words: multidelay Rossler system; D-subdivision method; projective synchronization ; linear observer 1 Introduction Chaos theory has been extensively studied in various research fields after Lorenz s observation of the chaotic phenomenon [1]. Over the last decade, great attention has been paid to the dynamics of delay differential equation(dde). It has been found that the time delays not only make the systems retarded, but also are often sources of complex behaviors, like, limit cycle, loss of stability, bifurcation and chaos. So there is a growing need to depend our understanding on their complexity, particularly when multiple time delays is present. The systems with single time delay have been intensively studied but investigation of DDE s with multiple time delays has been paid little attention too. Some work has been done relative to stability analysis [2], but more complex dynamics, such as limit cycle, bifurcation and chaos in DDEs with multiple time delays, are still unclear. Chaos synchronization has received increasing attention in the last years. The phenomena of synchronization in coupled systems have been extensively studied in the context of laser dynamics, electronics circuits, chemical and biological systems [3, 4]. Over the past decade, following complete synchronization(cs) [3, 5], several new types of synchronization have been found in interesting chaotic systems, such as generalized synchronization(gs) [6], phase synchronization(ps) [7], lag synchronization(ls) [8], anticipatory synchronization(as) [9], anti-phase synchronization(aps) [1] and multiplexing synchronization [11] etc. Amongst all kinds of chaos synchronization, projective synchronization, characterized by a scaling factor that two systems synchronize proportionally, is one of the most interesting problems. This type of synchronization was first reported by Mainieri and Rehacek [12] in 1999, where they declared that the two identical systems could be synchronized up to a scaling factor α. The scaling factor is a constant transformation between the synchronized variables of the master and slave systems. Projective synchronization is not in the category of GS because the slave system of projective synchronization is not asymptotically stable. CS and APS are special cases of projective synchronization where α = 1 and α = 1 respectively. The response system attractor possesses the same topological characteristic (such as Lyapunov exponents and fractal dimensions) as the slave system attractor [13]. Projective synchronization is interesting because of its proportionality between the synchronized dynamical states. In applications to secure communications, this feature can be used to M-nary digital communication for achieving fast communication. Recently, Corresponding author. address: drghosh math@yahoo.co.in Copyright c World Academic Press, World Academic Union IJNS /219

2 28 International Journal of Nonlinear Science,Vol.7(29),No.2,pp Li et al.[14] proposed generalized projective synchronization between two different chaotic systems using combination of active control and backstepping method. Very recently, Grassi and Miller [15] introduced projective synchronization of time-delay, continuous time and discrete-time system via linear observer. In Ref. [16], Grassi and Mascolo proposed nonlinear observer design to synchronize hyperchaotic systems. An observer [16] is a dynamic system designed to be driven by the output of another dynamic system (plant) and having the property that the state of the observer converges to the state of the plant. In this paper, we discuss the details stability analysis of multiple delay Rössler system. Here we consider Rössler system with two linear time delays which occur in active sensing problems where a signal is transmitted and received at a later time. Then projective synchronization of unidirectionally coupled multidelay Rössler system is investigated with the help of linear observer(not nonlinear observer approach). The general theory for any multidelay system are investigated analytically. The effectiveness of the proposed method is verified numerically, which can also be used for other time-delayed systems. This synchronization is very much applicable in cryptography for fast communication. Recently, chaotic time delay system has been suggested as a good candidates for secure communication [17, 18]. It is proved that low dimensional chaotic systems do not ensure a sufficient level of security for communications, as the associated chaotic attractors can be reconstructed with some effort and the hidden message can be retrieved by an attacker. In this regards, time-delayed system received a lot of attention. With increases time-delay, the system is more complex and number of positive Lyapunov exponents increases and the system eventually transits to hyperchaos. The rest of this paper is organized as follows. In section 2, we will discuss stability analysis of multiple delay Rössler system. In section 3, a general theory for projective synchronization is derived by linear observer approach and verify by numerically. Finally, conclusions are made in the last section. 2 Mathematical model and bifurcation analysis A double delayed Rössler system [19] is written as ẋ = y z + α 1 x(t τ 1 ) + α 2 x(t τ 2 ), ẏ = x + β 1 y, ż = β 2 + z(x γ) (2.1) where delay parameters are τ 1 and τ 2 (τ 1 τ 2 ) and the geometric factors are α 1, α 2 while β 1, β 2, γ are the usual parameters of a standard Rössler system [2]. To start with we determine the fixed points of the system, written as; E i = (x i, y i, z i ) i)e 1 = ii)e 2 = ( ) β 2 β 1 X +, X +, β 1 X + + γ ( ) β 2 β 1 X, X, β 1 X + γ where X ± = γa± γ 2 A 2 4Aβ 1 β 2 2Aβ 1 and provided γ 2 A 2 4Aβ 1 β 2 along with A = 1 + (α 1 + α 2 )β 1. The characteristic equation of the linearized equation [19] is given by (α 1 e λτ 1 + α 2 e λτ 2 λ){λ 2 + λ(γ x β 1 ) β 1 (γ x )} = λ(1 + z ) + (γ x β 1 z ) (2.2) In[19] we have studied the special case either α = or α 2 =. To find the explicit expressions of the critical boundaries, let λ = a + ib be the root of the characteristic equation (2.2) where a and b are real. Without any loss of generality, one can assume that b >, since the roots of equation (2.2) always appear in complex conjugate pairs. If τ 1 = τ 2 = τ, then the equation (2.2) becomes {(α 1 + α 2 )e λτ λ}{λ 2 + λ(γ x β 1 ) β 1 (γ x )} = λ(1 + z ) + (γ x β 1 z ) (2.3) IJNS for contribution: editor@nonlinearscience.org.uk

3 Dibakar Ghosh: Stability and projective synchronization in multiple 29 Substituting λ = a + ib into equation (2.3) gives D(a, b, τ) = {(α 1 + α 2 )e (a+ib)τ a ib}{(a + ib) 2 + (a + ib)(γ x β 1 ) β 1 (γ x )} Separating real and imaginary part of the critical condition D(a, b, τ) = yields, and {e aτ (α 1 + α 2 )cos(bτ) a}{a 2 b 2 + a(γ x β 1 ) β 1 (γ x )} (a + ib)(1 + z ) (γ x β 1 z ) +{e aτ (α 1 + α 2 )sin(bτ) + b}{2ab + b(γ x β 1 )} a(1 + z ) (γ x β 1 z ) = {e aτ (α 1 + α 2 )cos(bτ) a}{2ab + b(γ x β 1 )} {e aτ (α 1 + α 2 )sin(bτ) + b}{a 2 b 2 + a(γ x β 1 ) β 1 (γ x )} b(1 + z ) = According to D-subdivision method by setting a = in (2.4) and (2.5), the stability boundary are determine either by the points that yields either a root λ = or a pair of purely imaginary roots of the following equation (α 1 + α 2 ){b 2 + β 1 (γ x )}cos(bτ) b(γ x β 1 ){(α 1 + α 2 )sin(bτ) + b} + (γ x β 1 z ) = b(α 1 + α 2 )(γ x β 1 )cos(bτ) + {(α 1 + α 2 )sin(bτ) + b}{b 2 + β 1 (γ x )} b(1 + z ) = (2.6) when b =, i.e. λ = then above equation yields where Eliminating τ from (2.6) gives Let µ = b 2 and (2.4) (2.5) (γ x )(α 1 + α 2 )β 1 + γ x β 1 z = (2.7) {(γ x β 1 z ) b 2 (γ x β 1 )} 2 + {b(1 + z β 1 γ + β 1 x ) b 3 } 3 = (α 1 + α 2 ) 2 {(b 2 + β 1 (γ x )) 2 + b 2 (γ x β 1 ) 2 } (2.8) µ 3 + A 1 µ 2 + A 2 µ + A 3 = (2.9) A 1 = (x γ) 2 2(1 + z ) + β 2 1 (α 1 + α 2 ) 2 A 2 = (x γ) 2 {β 2 1 (α 1 + α 2 ) 2 2} + (1 + z ) 2 2β 2 1z β 2 1(α 1 + α 2 ) 2 A 3 = (x γ) 2 {1 (α 1 + α 2 )β 2 1} + β 2 1z 2 + 2β 1 z (x γ) In the following, we need to seek conditions under which equation (2.9) has at least one positive root. Denote h(µ) = µ 3 + A 1 µ 2 + A 2 µ + A 3 (2.1) Since h(µ) as µ, we conclude that if A 3 <, then equation (2.9) has at least one positive root. From (2.1) we have dh(µ) dµ = 3µ2 + 2A 1 µ + A 2 Clearly, if = A 2 1 3A 2, then the function h(µ) is monotone increasing in µ [, ). Thus, when A 3 and, equation (2.1) has no positive real roots. On the other hand, when A and >, the following equation 3µ 2 + 2A 1 µ + A 2 = (2.11) has two real roots µ 1 = A and µ 2 = A 1 3 (2.12) Obviously, h (µ 1 ) = 2 > and h (µ 2 ) = 2 <. It follows that µ 1 and µ 2 are the local minimum and the local maximum of h(µ), respectively. Hence, we have the following simple property. IJNS homepage:

4 21 International Journal of Nonlinear Science,Vol.7(29),No.2,pp Property 1 Suppose that A 1 and >, then linear equation of equation (2.9) has positive roots if and only if µ 1 > and h(µ 1 ). Summarizing the above discussion, we have the following simple lemma. Lemma 1 For the polynomial equation (2.9), we have the following results i) if A 3 < then equation (2.9) has at least one positive root. ii) if A 3 and = A 2 1 3A 2, then equation (2.9) has no positive roots. iii) if A 3 and = A 2 1 3A 2 >, then equation (2.9) has positive roots if and only if µ 1 = 1 3 ( A 1 + ) > and h(µ 1 ). Suppose that equation (2.9) has positive roots. Without loss of generality, we assume that it has three positive roots, defined by µ 1, µ 2 and µ 3 respectively. Then equation (2.8) has three positive roots From (2.6), we have b 1 = µ 1, b 2 = µ 2 and b 3 = µ 3 cos(bτ) = b2 (γ x β 1 )[ 1 z + 2b 2 + 2β 1 (γ x )] (b 2 + β 1 γ β 1 x )(γ x β 1 z ) (α 1 + α 2 )[{b 2 + β 1 (γ x )} 2 + b 2 (γ x β 1 ) 2 ] Thus if we denote [ [ ] ] τ (j) k = 1 b k cos 1 b 2 k (γ x β 1 ){ 1 z +2b 2 k +2β 1(γ x )} (b 2 k +β 1γ β 1 x )(γ x β 1 z ) + 2jπ (α 1 +α 2 )[{b 2 k +β 1(γ x )} 2 +b 2 k (γ x β 1 ) 2 ] (2.13) where k = 1, 2, 3; j =, 1, 2,..., then ±ib k is a pair of purely imaginary roots of equation (2.2) with τ (j) k. Define τ = τ () k = min k (1,2,3) {τ () k }, b = b k (2.14) Note that when τ 1 = τ 2 =, then equation (2.2) becomes λ 3 + B 1 λ 2 + B 2 λ + B 3 = (2.15) where B 1 = γ x β 1 α 1 α 2, B 2 = β 1 (γ x ) + (1 + z ) (α 1 + α 2 )(γ x β 1 ) and B 3 = β 1 (α 1 + α 2 )(γ x ) + (γ x β 1 z ). Therefore, applying the above results we obtain the following lemma: Lemma 2 For the third degree transcendental equation (2.2), we have i) if A 3 and = A 2 1 3A 2, then all roots with positive real parts of equation (2.2) has the same sum of those of the polynomial equation (2.15) for all τ. ii) if either A 3 < or A 3, = A 2 1 3A 2 >, µ 1 > and h(µ 1 ), then all roots with positive real parts of equation (2.2) has the same sum to those of the polynomial equation (2.15) for τ [, τ ). From equation (2.2), one may find where ( ) dλ 1 = σ 1 σ 3 dτ λ=ib k σ 2 σ 4 σ 1 = { 3b 2 k β 1(γ x )+1+z }{(γ x β 1 )b 2 k (γ x β 1 z )}+2b 2 k (γ x β 1 ){b 2 k 1 z +β 1 (γ x )} σ 2 = [(γ x β 1 )b 2 k (γ x β 1 z )] 2 +b 2 k [b2 k 1 z +β 1 (γ x )] 2 σ 3 = (γ x β 1 ) 2 +2{b 2 k +β 1(γ x )} σ 4 = (γ x β 1 ) 2 b 2 k +{b2 k +β 1(γ x )} 2 In the following, we shall investigate the stability of equilibria of system (2.1) in a theorem. Theorem 3 If the conditions of Lemma 1 and 2 hold, then the equilibrium points E 1 or E 2 is unstable for any τ and at sequence of critical values of τ given by (2.13), system (2.1) can undergo a Hopf bifurcation near these equilibrium point provided real part of ( ) dλ 1 dτ. λ=ib k IJNS for contribution: editor@nonlinearscience.org.uk

5 Dibakar Ghosh: Stability and projective synchronization in multiple (a).3 (b) x 1 (t), x 2 (t) e = x 2 (t) α x 1 (t) time time Figure 1: a) Projective synchronization between x 1 (t) and x 2 (t), (b) error dynamics of projective synchronization for α = Projective synchronization via linear observer 3.1 General formula In this section we shall use linear observer design to projective synchronization in time delay system. Consider the coupled time delay system as ẋ = f(x, x τ1 ) (3.1) ẏ = f(y, y τ1 ) + u(x, y) (3.2) where x, y R n, f : R n R n is a nonlinear vector field and u(x, y) is the control term. A definition of projective synchronization is presented below. Definition 4 Systems (3.1) and (3.2) are said to be projective synchronization if the dynamical behavior in which the amplitude of the masters state variable and that of the slave s synchronizes up to a constant scaling factor α, i.e. y αx as t. and The synchronization manifold of systems (3.1) and (3.2) is We consider the dynamic (3.1) in the form y = αx (3.3) ẋ = Ax + Bf(x) + Cg(x τ1 ) + D (3.4) We defined the response system and synchronization signal respectively as ẏ = Ay + (αz ky) + Dα (3.5) z = Bf(x) + Cg(x τ1 ) + kx (3.6) where k is the coupling strength. Then the dynamic system (3.5) is said to be linear projective observer of system (3.1) if its state y αx as t. Moreover system (3.5) is said to be global projective linear observer of (3.1) if y αx as t for any initial condition x() and y(). Let e = y αx be the projective synchronization error. Then the error dynamics is ė = ẏ αẋ = (A k)e (3.7) IJNS homepage:

6 212 International Journal of Nonlinear Science,Vol.7(29),No.2,pp (a) 1 (b) 2.8 x 1 (t), x 2 (t) time e=x 2 (t) + α x 1 (t) time Figure 2: a) anti-phase projective synchronization between x 1 (t) and x 2 (t), (b) error dynamics of anti-phase projective synchronization for α = 2.. Therefore, if the controllability matrix [k Ak A 2 k... A n 1 k] is full rank, the drive system (3.1) and the linear observer (3.5) synchronize for any scaling factor α. There are two different situation arises depending upon the sign of α. For α = 1, identical synchronization occur, whereas for α = 1, anti-phase synchronization is obtain. For α > (α 1), we get projective synchronization and for α < (α 1), one obtain anti-phase projective synchronization. The main advantage of this method is that it does not require computation of any Lyapunov exponents. In the next part we numerically check the effectiveness of the above approach. 3.2 Numerical simulation To test the effectiveness of the above approach for projective synchronization, we consider the unidirectional coupled multidelay Rössler system as x 1 = y 1 z 1 + α 1 x 1 (t τ 1 ) + α 2 x 1 (t τ 2 ) y 1 = x 1 + β 1 y 1 (3.8) z 1 = β 2 + z 1 (x 1 γ) and x 2 = y 2 z 2 + α[α 1 x 1 (t τ 1 ) + α 2 x 1 (t τ 2 )] + k(αx 1 x 2 ) y 2 = x 2 + β 1 y 2 + k(αy 1 y 2 ) (3.9) z 2 = αβ 2 γz 2 + αx 1 z 1 + k(αz 1 z 2 ) The drive system (3.8) is in chaotic state[19] for the set of parameter values α 1 =.2, α 2 =.5, β 1 = β 2 =.2, γ = 5.7, τ 1 = 1. and τ 2 = 2.. For k =.5, identical and anti-phase synchronization is obtained for α = 1 and α = 1 respectively. Figure 1(a) shows the time series of the drive system x 1 (t) (solid line) and the response system x 2 (t)(dotted line) for k =.5 and α = 2.. The time series of the error system is shown in figure 1(b). It is implied that all the state variables tend to be synchronized in a proportional relation for α = 2., i.e. x 2 (t) = 2.x 1 (t). Similarly for other relations are satisfied, i.e. y 2 (t) = 2.y 1 (t) and z 2 (t) = 2.z 1 (t). The time series of the drive system x 1 (t) (solid line) and the response system x 2 (t)(dotted line) for k =.5 and α = 2. are shown in Figure 2(a). The time series of the error system is shown in figure 2(b). It is shown from figure 2(a) that the difference of phase angles of the synchronized trajectories is π, i.e. anti-phase pattern. Relation between drive and response systems is x 2 (t) = 2.x 1 (t). In figure 3(a), the chaotic attractor of drive system (3.8) is reported. For α = 2., projective synchronization occurs and the attractor of the response system (3.9) has been scaled by twice than that the attractor IJNS for contribution: editor@nonlinearscience.org.uk

7 Dibakar Ghosh: Stability and projective synchronization in multiple (a) (b) (c) y y 2 1 y x x x 2 5 Figure 3: a) Chaotic attractors of a) drive system (3.8) and response system (3.9) for (b) α = 2. and (c) α = 2.. of drive system (3.8) (figure 3(b)), synchronized chaotic attractors with an identical phase pattern where the phase angle between the synchronized trajectories is zero. Anti-phase projective synchronization with twice scale for α = 2. is shown in figure 3(c). From this figure it is observed that the synchronized chaotic attractors are in anti-phase pattern and the phase angle between the synchronized trajectories is π. Similar results are obtained for other values of scaling factor α. 4 Conclusions Chaos phenomena exist in multiple time delay Rössler system. A new technique of projective synchronization for a class of multiple time delay systems via linear observer has been developed. The proposed approach can be successfully applied to several well-known time delay systems and it does not require computation of any Lyapunov exponent. This projective synchronization in multiple time delay system are very much applicable in cryptography for fast communication. References [1] Sparrow C : Bifurcation in the Lorenz equation: Lecture notes in Applied Mathematics. (Springer- Verlag)(1982) [2] Sun Z, Xu W, Yang X and Fang T : Effects of time delays on bifurcation and chaos in a non-autonomous system with multiple time delays. Chaos, Solitons and Fractals. 31: 39-53(27) [3] Pecora L M and Carroll T L : Synchronization in chaotic systems. Phys. Rev. Lett. 64: (199) [4] CHAOS, Special issue on chaos synchronization 7,N4 (1997) edited by Ditto W L and Showalter K, Chen G and Dong X : From Chaos to Order.Methodologies, Perspectives and Applications (World Scientific, Singapore,1998), Handbook of Chaos Control, Ed. Schuster H G (Wiley-VCH, Weinheim,1999) [5] Fujisaka H and Yamada T : Stability Theory of Synchronized Motion in Coupled-Oscillator Systems. Prog. Theor. Phys. 69(1), 32-47(1983), Ghosh D and Roy Chowdhury A : Various Types of Chaos Synchronization- a Comparative Study of Empirical Mode Decomposition and Wavelet Approach. Int. J. Nonlinear Sci. 4(1): 52-66(27) [6] Ghosh D, Ray D and Roy Chowdhury A : Generalized and phase synchronization between two different time-delayed systems Mod. Phys. Letts. B (Accepted 28), Shahverdiev E M and Shore K A : Generalized synchronization in time-delayed systems. Phys. Rev. E 71:1621(25) IJNS homepage:

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