Jestr Journal of Engineering Science and Technology Review 6 (4) (0) 5-65 Special Issue on Recent Advances in Nonlinear Circuits: Theory and Applications Research Article JOURNAL OF Engineering Science and Technology Review www.jestr.org Analysis and Adaptive Synchronization of Two Novel Chaotic Systems with Hyperbolic Sinusoidal and Cosinusoidal Nonlinearity and Unknown Parameters S. Vaidyanathan * R & D Centre, Vel Tech University, 4, Avadi-Alamathi Road, Chennai-600 06, INDIA Received 0 May 0; Revised 4 September 0; Accepted 5 September 0 Abstract This research work describes the modelling of two novel -D chaotic systems, the first with a hyperbolic sinusoidal nonlinearity and two quadratic nonlinearities (denoted as system (A)) and the second with a hyperbolic cosinusoidal nonlinearity and two quadratic nonlinearities (denoted as system (B)). In this work, a detailed qualitative analysis of the novel chaotic systems (A) and (B) has been presented, and the Lyapunov exponents and Kaplan-Yorke dimension of these chaotic systems have been obtained. It is found that the maximal Lyapunov exponent (MLE) for the novel chaotic systems (A) and (B) has a large value, viz. for the system (A) and for the system (B). Thus, both the novel chaotic systems (A) and (B) display strong chaotic behaviour. This research work also discusses the problem of finding adaptive controllers for the global chaos synchronization of identical chaotic systems (A), identical chaotic systems (B) and nonidentical chaotic systems (A) and (B) with unknown system parameters. The adaptive controllers for achieving global chaos synchronization of the novel chaotic systems (A) and (B) have been derived using adaptive control theory and Lyapunov stability theory. MATLAB simulations have been shown to illustrate the novel chaotic systems (A) and (B), and also the adaptive synchronization results derived for the novel chaotic systems (A) and (B). Keywords: Chaos, chaotic attractors, Lyapunov exponents, synchronization, adaptive control.. Introduction Chaotic systems are deterministic nonlinear dynamical systems that are highly sensitive to initial conditions (known as butterfly effect ) and unpredictable on the long term. The characterizing properties of a chaotic system can be enumerated as follows.. Strong dependence of the system behaviour on initial conditions.. Sensitivity of the system to the changes in the parameters. Presence of strong harmonics in the signals 4. Fractional dimension of the state space trajectories 5. Presence of a stretch direction, characterized by a positive Lyapunov exponent. Chaotic systems are also defined as nonlinear dynamical systems having at least one positive Lyapunov exponent. Lorenz discovered the first chaotic system in 96 [], while he was studying weather patterns with a -D nonlinear dynamical system. Lorenz experimentally observed that his -D model is highly sensitive to even small changes in the initial conditions. Subsequent to Lorenz system [], many chaotic systems were found in the literature. A few important systems can be cited as Rössler system [], Rabinovich system [], Arneodo-Coullet system [4], Shimizu-Morioka system [5], Colpitt s oscillator [6], Shaw system [7], Chua circuit [8], Ruckildge system [9], Sprott systems [0], Chen system [], Lü-Chen system [], Chen-Lee system [], * E-mail address: sundarvtu@gmail.com ISSN: 79-77 0 Kavala Institute of Technology. All rights reserved. Tigan system [4], Cai system [5], Li system [6], Wang system [7], Harb system [8], Sundarapandian system [9], Gissinger system [0], etc. Most of the known chaotic systems in the literature are polynomial systems. Recently, there has been some good interest in the literature in finding novel chaotic systems with exponential nonlinearity such as Wei-Yang system [], Yu- Wang system [], Yu-Wang-Wan-Hu system [], etc. In a related work [4], Yu and Wang have modelled autonomous novel chaotic systems with hyperbolic sinusoidal nonlinearity and a quadratic nonlinearity. This research work describes two novel -D chaotic systems, the first with a hyperbolic sinusoidal nonlinearity and two quadratic nonlinearities (denoted as system (A)) and the second with a hyperbolic cosinusoidal nonlinearity and two quadratic nonlinearities (denoted as system (B)). This research work provides a detailed qualitative analysis of the novel -D chaotic systems (A) and (B). We also obtain the Lyapunov exponents and Kaplan-Yorke dimension of the novel chaotic systems (A) and (B). It is found that the maximal Lyapunov exponent (MLE) for the novel chaotic systems (A) and (B) has a large value. Explicitly, the MLE for the novel chaotic system (A) has been found as L =.794. Also, the MLE for the novel chaotic system (B) has been found as L = 5.. Thus, the novel chaotic systems (A) and (B) show strong chaotic behaviour. Chaos theory has been applied to many branches of science and engineering. A few important applications can be cited as atom optics [5, 6], optoelectronic devices [7, 8],
S. Vaidyanathan/Journal of Engineering Science and Technology Review 6 (4) (0) 5-65 chemical reactions [9, 0], ecology [, ], cell biology [, 4], forecasting [5, 6], robotics [7, 8], communications [9, 40], cryptosystems [4, 4], neural networks [4, 44], finance [45], etc. Synchronization of chaotic systems occurs when a chaotic system called as master system drives another chaotic system called as slave system. Because of the butterfly effect which causes exponential divergence of the trajectories of identical chaotic systems with nearly the same initial conditions, global synchronization of chaotic systems is a challenging problem in the literature. Chaos synchronization problem of two chaotic systems called as master and slave systems is basically to derive a feedback control law so that the states of the master and slave systems are synchronized or made equal asymptotically with time. In 990, a seminal paper on synchronizing two identical chaotic systems was proposed by Pecora and Carroll [46]. This was followed by many different techniques for chaos synchronization in the literature such as active control [47-50], adaptive control [5-54], sliding mode control [55-58], backstepping control [59-6], sampled-data feedback control [6-64], etc. Active control method is used when the system parameters are available for measurement. When the system parameters are not known, adaptive control method is used and feedback control laws are devised using estimates of the unknown system parameters and parameter update laws are derived using Lyapunov stability theory [65]. After the description and analysis of the novel chaotic systems (A) and (B), this research work deals with the following problems: The novel system (A) is defined by the -D dynamics x = a (x ) + x x = bx cx = d + sinh(x x ) where x, x, are states and abcd,,, are constant, positive, parameters of the system (). We note that the system (A) has a quadratic nonlinearity in each of the first two equations and a hyperbolic sinusoidal nonlinearity in the last equation. The system () exhibits a strange chaotic attractor for the parameter values a= 0, b= 9, c=, d = 0 () For numerical simulations, we have used classical fourthorder Runge-Kutta method (MATLAB) for solving the system (A) when the initial conditions are taken as x (0) = 0.4, x (0) = 0.6, x (0) = 0. Fig. depicts the strange attractor of the novel system (A) in -D view, while Figs., and 4 depict the -D projection of this system (A) in ( x, x), ( x, x) and ( x, x) plane, respectively. (). Adaptive synchronization of identical novel chaotic systems (A).. Adaptive synchronization of identical novel chaotic systems (B).. Adaptive synchronization of (non-identical) novel chaotic systems (A) and (B). This research work is organized as follows. Section describes the novel chaotic systems (A) and (B). The phase portraits of the novel chaotic systems (A) and (B) are depicted in this section. Section describes a detailed qualitative analysis and properties of the novel chaotic system (A). Section 4 describes a detailed qualitative analysis and properties of the novel chaotic system (B). Section 5 describes the adaptive synchronization design of identical chaotic systems (A). Section 6 describes the adaptive synchronization design of identical chaotic systems (B). Section 7 describes the adaptive synchronization design of non-identical chaotic systems (A) and (B). Numerical simulations illustrating the adaptive synchronization designs of identical and non-identical chaotic systems (A) and (B) have been described at the end of Sections 5 to 7. Section 8 concludes this research work with a summary of the main results. Fig.. Strange chaotic attractor of the novel system (A).. Two Novel -D Chaotic Systems (A) and (B) In this section, we describe the two novel -D chaotic systems, the first with a hyperbolic sinusoidal nonlinearity and two quadratic nonlinearities (denoted as system (A)) and the second with a hyperbolic cosinusoidal nonlinearity and two quadratic nonlinearities (denoted as system (B)). Fig.. A -D projection of the novel system (A) in the ( x, x) plane. 54
S. Vaidyanathan/Journal of Engineering Science and Technology Review 6 (4) (0) 5-65 Fig.. A -D projection of the novel system (A) in the ( x, x) plane. Fig. 5. Strange chaotic attractor of the novel system (B). Fig. 4. A -D projection of the novel system (A) in the ( x, x) plane. Fig. 6. A -D projection of the novel system (B) in the ( x, x) plane. The novel system (B) is defined by the -D dynamics: x = α(x ) + x x = βx γ x = δ + cosh(x x ) () where x, x, are states and αβγδ,,, are constant, positive, parameters of the system (). We note that the system (B) has a quadratic nonlinearity in each of the first two equations and a hyperbolic cosinusoidal nonlinearity in the last equation. The system () exhibits a strange chaotic attractor for the parameter values Fig. 7. A -D projection of the novel system (B) in the ( x, x) plane. α = 0, β = 98, γ =, δ = 0 (4) For numerical simulations, we have used classical fourthorder Runge-Kutta method (MATLAB) for solving the system (B) when the initial conditions are taken as x (0) = 0.4, x (0) = 0.6, x (0) = 0.7 Fig. 5 depicts the strange attractor of the novel system (B) in -D view, while Figs. 6, 7 and 8 depict the -D projection of this system (B) in ( x, x), ( x, x) and ( x, x) plane, respectively. Fig. 8. A -D Projection of the Novel System (B) in the ( x, x) plane 55
S. Vaidyanathan/Journal of Engineering Science and Technology Review 6 (4) (0) 5-65. Analysis of the Novel Chaotic System (A) with Hyperbolic Sinusoidal Nonlinearity A. Symmetry and Invariance We note that the novel chaotic system (A) is invariant under the coordinates transformation ( x, x, x ) ( x, x, x ). Thus, the novel chaotic system (A) has rotation symmetry about the axis. B. Dissipativity In vector notation, the novel chaotic system (A) can be expressed as x = f (x ) = f (x ) $ & f (x ) & & f (x ) & The divergence of the vector field f on R is given by () (A) by solving the nonlinear system of equations: f( x ) = 0 (9) Assume that abcd,,, > 0 and db > c. Solving the equations (9), we get three equilibrium points of the novel chaotic system (A), which are described as follows: E 0 : (0,0,0) ac + b ac b E : ln( m), ln( m), + ac ac b c ac + b ac b E : ln( m), ln( m), + ac ac b c where m= p+ + p and p = db. c We take the parameter values as in the chaotic case, viz. a= 0, b= 9, c=, d = 0. (0) f( x) f( x) f( x) f = div( f) = + + x x x (4) Then the three equilibrium points of the novel system (A) are obtained as The divergence of the vector field f measures the rate at which the volumes change under the flow Φ t of f. Let D be any given region in R with a smooth boundary. Let Dt () =Φ t ( D ). Let Vt () denote the volume of Dt (). By Liouville s theorem, we have dv () t = ( f ) dx dy dz dt Dt () Using the equation () of the novel system (A), we find that (5) E 0 :(0,0,0) E :(6.89,.09, 46) E :( 6.89,.09,46) () Using the first method of Lyapunov, it is easy to see that E is a saddle point and E 0, E are saddle-foci. Hence, all the equilibrium points of the novel system (A) are unstable. D. Lyapunov Exponents We take the initial state as: x(0) = 0.4, x(0) =.0, (0) = 0.6 () f f f f = + + = ( a+ d) < 0 x x x since a and d are positive constants. By substituting the value of f in Eq. (5), we get (6) Also, we take the parameters as given in (0). Then the Lyapunov exponents of the novel chaotic system (A) are obtained numerically using MATLAB as L=.794, L = 0, L =.87 () dv () t = ( a+ d) dx dy dz = ( a+ d) V( t) dt Dt () Solving the linear ODE (7), we get the solution ( ) Vt () = V(0)exp ( a+ dt ) (8) From Eq. (8), it follows that any volume Vt () must shrink to zero exponentially as t. Thus, the novel chaotic system (A) is a dissipative chaotic system. Hence, the asymptotic motion of the novel system (A) settles onto a strange attractor of the novel system (A). C. Equilibrium Points We obtain the equilibrium points of the novel chaotic system (7) This shows mathematically that the novel system (A) is indeed chaotic. Note that the maximal Lyapunov exponent (MLE) of the novel system (A) is L =.794, which is a large value. Thus, the novel system (A) depicts strong chaotic behaviour. E. Kaplan-Yorke Dimension The Kaplan-Yorke dimension of the system (A) is + j L L i j+ L i= DKY = j+ L = + =.707 L (4) Eq. (4) shows that the system (A) is a dissipative system and the Kaplan-Yorke dimension of the system (A) is fractional. The dynamics of the Lyapunov exponents of the novel 56
S. Vaidyanathan/Journal of Engineering Science and Technology Review 6 (4) (0) 5-65 system (A) is shown in Fig. 9. get: dv () t = ( α + δ) dx dy dz = ( α + δ ) V ( t) (9) dt Dt () Solving the linear ODE (9), we get the solution: ( ) Vt () = V(0)exp ( α+ δ ) t (0) Fig. 9. Dynamics of the Lyapunov exponents of the novel system (A). 4. Analysis of the Novel Chaotic System (B) with Hyperbolic Cosinusoidal Nonlinearity A. Symmetry and Invariance We note that the novel chaotic system (B) is invariant under the coordinates transformation ( x, x, x ) ( x, x, x ). Thus, the novel chaotic system (B) has rotation symmetry about the axis. B. Dissipativity In vector notation, the novel chaotic system (B) can be expressed as x = f (x ) = f (x ) $ & f (x ) & & f (x ) & The divergence of the vector field f on R is given by f( x) f( x) f( x) f = div( f) = + + x x x (5) (6) The divergence of the vector field f measures the rate at which the volumes change under the flow Φ t of f. Let D be any given region in R with a smooth boundary. Let Dt () =Φ t ( D ). Let Vt () denote the volume of Vt () By Liouville s theorem, we have dv () t = ( f ) dx dy dz dt Dt () Using the equation () of the novel system (B), we find that (7) f f f f = + + = ( α + δ ) < 0 (8) x x x since a and δ are positive constants. By substituting the value of f f in Eq. (8), we From Eq. (0), it follows that any volume Vt () must shrink to zero exponentially as t. Thus, the novel chaotic system (B) is a dissipative chaotic system. Hence, the asymptotic motion of the novel system (B) settles onto a strange attractor of the novel system (B). C. Equilibrium Points We obtain the equilibrium points of the novel chaotic system (B) by solving the nonlinear system of equations f( x ) = 0 () Assume that αβγδ,,, > 0 and δβ > γ. Solving the equations (), we get three equilibrium points of the novel chaotic system (B), which are described as follows: E 0 : (0,0,0) αγ + β αγ β E : ln( n), ln( n), + αγ αγ β γ αγ + β αγ β E : ln( n), ln( n), + αγ αγ β γ δβ where n= q+ + q and q =. γ We take the parameter values as in the chaotic case, viz. α = 0, β = 98, γ =, δ = 0. () Then the three equilibrium points of the novel system (B) are obtained as: E 0 :(0,0,0) E :(6.747,.0805, 49) E :( 6.747,.0805,49) () Using the first method of Lyapunov, it is easy to see that E0 is a saddle point and E, Eare saddle-foci. Hence, all the equilibrium points of the novel system (B) are unstable. D. Lyapunov Exponents We take the initial state as x(0) = 0.4, x(0) =.0, (0) = 0.6 (4) Also, we take the parameters as given in (). Then the Lyapunov exponents of the novel chaotic system (B) are obtained numerically using MATLAB as: L= 5., L = 0, L = 4.440 (5) 57
S. Vaidyanathan/Journal of Engineering Science and Technology Review 6 (4) (0) 5-65 This shows mathematically that the novel system (B) is indeed chaotic. Note that the maximal Lyapunov exponent (MLE) of the novel system (B) is L = 5., which is a large value. Thus, the novel system (B) depicts strong chaotic behaviour. E. Kaplan-Yorke Dimension The Kaplan-Yorke dimension of the system (B) is + j L L i j+ L i= DKY = j+ L = + =.488 L (6) Eq. (6) shows that the system (B) is a dissipative system and the Kaplan-Yorke dimension of the system (B) is fractional. The dynamics of the Lyapunov exponents of the novel system (B) is shown in Fig. 0. y = a ( y y ) + y y +u y = by cy y +u (8) y = dy + sinh( y y ) +u where y, y, y are the states and u, u, u are adaptive controllers to be designed. We define the chaos synchronization error between the systems (7) and (8) as: e = y x e = y x e = y x The synchronization error dynamics is easily obtained as: e& = a( e e ) + y y x x + u e& = be c( y y x x ) + u e& = de + sinh( y y ) sinh( x x ) + u (9) (0) Next, we introduce the nonlinear controller defined by u = aˆ( t)( e e ) y y + x x k e u = bˆ () t e + cˆ ()( t y y x x ) k e u = dˆ( t ) e sinh( y y ) + sinh( x x ) k e () where at ˆ(), bt ˆ(), ct ˆ(), dt ˆ() are estimates of the unknown system parameters abcd,,, respectively, and k, k, k are positive gains. By substituting the control law () into (0), we get the closed-loop error dynamics as: Fig. 0. Dynamics of the Lyapunov exponents of the novel system (B). 5. Adaptive Synchronization of the Novel Chaotic Systems (A) with Unknown Parameters e& = ( a aˆ ( t))( e e ) k e e& = ( b bˆ ( t)) e ( c cˆ ( t))( y y x x ) k e e& = ( d dˆ ( t)) e k e We define the errors in estimating system parameters as () In this section, we shall discuss the adaptive synchronization of the novel chaotic systems (A) with unknown system parameters. Using adaptive control method, we design an adaptive synchronizer, which makes use of estimates of the unknown system parameters, and the error convergence is established for all initial conditions using Lyapunov stability theory [65]. As the master (or drive) system, we consider the novel system (A), which is described by x = a (x ) + x x = bx cx = d + sinh(x x ) (7) where x, x, x are the states and abcd,,, are unknown system parameters. As the slave (or response) system, we consider the controlled novel system (A), which is described by e () t = a aˆ () t a e () t = b bˆ () t b e () t = c cˆ () t c e () t = d dˆ () t d Differentiating () with respect to t, we obtain e a (t ) = a ˆ (t ) e b (t ) = b ˆ (t ) e c (t ) = ˆ c (t ) e d (t ) = d ˆ (t ) The error dynamics () can be simplified by using () as: () (4) e = e a (e e ) k e e = e b e e c ( y y ) k e (5) e = e d e k e 58
S. Vaidyanathan/Journal of Engineering Science and Technology Review 6 (4) (0) 5-65 Next, we derive an update law for the parameter estimates using Lyapunov stability theory. So, we take a candidate Lyapunov function defined by ( a b c d) V = e + e + e + e + e + e + e, (6) which is a quadratic and positive-definite function on R Next, we calculate the time-derivative of V along the trajectories of (4) and (5). We obtain V = k e k e k e +e a e (e e ) aˆ $ +e b e e b & ˆ $ ' +e c e ( y y ) ˆ c$ +e d e d & ˆ $ ' (7) To guarantee global exponential stability of the systems (4) and (5), we need to choose parameter updates in such a way that V & is a quadratic, negative definite function on R Thus, we choose the parameter update law as follows: ˆ a = e (e e ) + k 4 e a ˆ b = e e + k 5 e b ˆ c = e ( y y ) + k 6 e c ˆ d = e + k 7 e d (8) The values of the system parameters for the systems (7) and (8) are taken as in the chaotic case, viz. a= 0, b= 9, c=, d = 0 The initial state of the master system (7) is taken as: x (0) =., x (0) =.6, x (0) = 5.4 The initial state of the slave system (8) is taken as: y (0) = 4., y (0) =.7, y (0) =.6 The initial values of the parameter estimates are taken as: aˆ(0) =.9, bˆ(0) = 4.5, cˆ(0) = 6.8, d ˆ(0) = 5. Fig. depicts the complete chaos synchronization of the identical novel chaotic systems (A) described by the equations (7) and (8), while Fig. depicts the time history of the synchronization errors e(), t e(), t e(). t Also, Fig. depicts the time history of the parameter estimation errors ea(), t eb(), t ec(), t ed(). t where k4, k5, k6, k7are positive constants. Next, we prove the main result of this section. Theorem. The adaptive controller defined by () and the parameter update law defined by (8) globally and exponentially synchronize the novel chaotic systems (A) with unknown parameters described by the equations (7) and (8), where ki,( i =, K,7) are positive constants. The parameter estimation errors ea(), t eb(), t ec(), t ed() t exponentially converge to zero with time. Proof. The assertions are established using Lyapunov stability theory [65]. We have already noted that the Lyapunov function V defined by (6) is quadratic and positive definite on R Next, we substitute the parameter update law defined by (8) into the dynamics (7). This simplifies the dynamics (7) as Fig.. Complete synchronization of the novel chaotic systems (A). V = k e k e k e k 4 e a k 5 e b k 6 e c k 7 e d (9) which is a quadratic and negative definite function on R Hence, by Lyapunov stability theory [65], the synchronization and parameter estimation errors globally and exponentially converge to zero with time. This completes the proof. For numerical simulations, the classical fourth order Runge-Kutta method with step size 8 h =0 has been used with MATLAB to solve the chaotic systems (7) and (8) when the adaptive controller () and the parameter update law (8) are applied. We take the positive gains k i as ki = 5,( i =, K,7). Fig.. Time history of the synchronization errors e(), t e(), t e() t. 59
S. Vaidyanathan/Journal of Engineering Science and Technology Review 6 (4) (0) 5-65 where ˆ α(), t ˆ β(), t ˆ γ(), t ˆ δ () t are estimates of the unknown system parameters αβγδ,,, respectively, and k, k, k are positive gains. By substituting the control law (44) into (4), we get the closed-loop error dynamics as: e = (α ˆα(t ))(e e ) k e e = (β ˆβ(t ))e (γ ˆγ(t ))( y y ) k e (45) e = (δ ˆδ(t ))e k e We define the errors in estimating system parameters as: Fig.. Time history of the parameter estimation errors ea(), t eb(), t ec(), t ed() t. 6. Adaptive Synchronization of the Novel Chaotic Systems (B) with Unknown Parameters e () t = α ˆ α() t α e () t = β ˆ β() t β e () t = γ ˆ γ() t γ e () t = δ ˆ δ() t δ (46) In this section, we shall discuss the adaptive synchronization of the novel chaotic systems (B) with unknown system parameters. As the master (or drive) system, we consider the novel system (B), which is described by x = α(x ) + x x = βx γ x = δ + cosh(x x ) (40) where x, x, x are the states and αβγδ,,, are unknown system parameters. As the slave (or response) system, we consider the controlled novel system (B), which is described by y = α( y y ) + y y +u y = β y γ y y +u (4) y = δ y + cosh( y y ) +u where y, y, y are the states and u, u, u are adaptive controllers to be designed. We define the chaos synchronization error between the systems (40) and (4) as: e = y x e = y x e = y x The synchronization error dynamics is easily obtained as: (4) e = α(e e ) + y y x +u e = βe γ( y y ) +u (4) e = δe + cosh( y y ) cosh(x x ) +u Next, we introduce the nonlinear controller defined by Differentiating (46) with respect to t, we obtain e α (t ) = ˆα(t ) e β (t ) = ˆβ(t ) e γ (t ) = ˆγ(t ) e δ (t ) = ˆδ(t ) The error dynamics (45) can be simplified by using (46) as: (47) e = e α (e e ) k e e = e β e e γ ( y y ) k e (48) e = e δ e k e Next, we derive an update law for the parameter estimates using Lyapunov stability theory. So, we take a candidate Lyapunov function defined by ( ) V = e + e + e + eα + eβ + eγ + eδ, (49) which is a quadratic and positive-definite function on R Next, we calculate the time-derivative of V along the trajectories of (47) and (48). We obtain V = k e k e k e +e α e (e e ) ˆα $ +e β & e e ˆβ $ ' (50) +e γ e ( y y ) ˆγ $ +e δ e ˆδ $ & ' To guarantee global exponential stability of the systems (47) and (48), we need to choose parameter updates in such a way that V is a quadratic, negative definite function on R Thus, we choose the parameter update law as follows: u = ˆ( α t)( e e ) y y + x x k e u = ˆ β() t e + ˆ γ()( t y y x x ) k e u = ˆ( δ t ) e cosh( y y ) + cosh( x x ) k e (44) 60
S. Vaidyanathan/Journal of Engineering Science and Technology Review 6 (4) (0) 5-65 α ˆ = e (e e ) + k 4 e α ˆ β = e e + k 5 e β ˆ γ = e ( y y ) + k 6 e γ ˆ δ = e + k 7 e δ where k4, k5, k6, k7are positive constants. Next, we prove the main result of this section. (5) Fig. 4 depicts the complete chaos synchronization of the identical novel chaotic systems (B) described by the equations (40) and (4), while Fig. 5 depicts the time history of the synchronization errors e(), t e(), t e(). t Also, Fig. 6 depicts the time history of the parameter estimation errors eα(), t eβ(), t eγ(), t eδ (). t Theorem. The adaptive controller defined by (44) and the parameter update law defined by (5) globally and exponentially synchronize the novel chaotic systems (B) with unknown parameters described by the equations (40) and (4), where ki,( i =, K,7) are positive constants. The parameter estimation errors eα(), t eβ(), t eγ(), t eδ () t exponentially converge to zero with time. Proof. The assertions are established using Lyapunov stability theory [65]. We have already noted that the Lyapunov function V defined by (49) is quadratic and positive definite on R Next, we substitute the parameter update law defined by (5) into the dynamics (50). This simplifies the dynamics (50) as Fig. 4. Complete synchronization of the novel chaotic systems (B). V = k e k e k e k 4 e α k 5 e β k 6 e γ k 7 e δ (5) which is a quadratic and negative definite function on R Hence, by Lyapunov stability theory [65], the synchronization and parameter estimation errors globally and exponentially converge to zero with time. This completes the proof. For numerical simulations, the classical fourth order Runge-Kutta method with step size 8 h =0 has been used with MATLAB to solve the chaotic systems (40) and (4) when the adaptive controller (44) and the parameter update law (5) are applied. We take the positive gains ki as ki = 5,( i =, K,7). The values of the system parameters for the systems (40) and (4) are taken as in the chaotic case, viz. Fig. 5. Time history of the synchronization errors e(), t e(), t e() t. α = 0, β = 98, γ =, δ = 0 α = 0, β = 98, γ =, δ = 0 The initial state of the master system (40) is taken as: x (0) =., x (0) =.9, x (0) = 0.8 The initial state of the slave system (4) is taken as: y (0) =., y (0) =.7, y (0) = 5. The initial values of the parameter estimates are taken as: ˆ α(0) =.6, ˆ β(0) = 5., ˆ γ(0) = 8., ˆ δ (0) = 7.4 Fig. 6. Time history of the parameter estimation errors eα(), t eβ(), t eγ(), t eδ () t. 6
S. Vaidyanathan/Journal of Engineering Science and Technology Review 6 (4) (0) 5-65 7. Adaptive Synchronization of the Novel Chaotic Systems (A) and (B) with Unknown Parameters ea (t ) = a aˆ (t ), eb (t ) = b bˆ(t ), ec (t ) = c cˆ(t ), ed (t ) = d dˆ(t ) e (t ) = α αˆ (t ), e (t ) = β βˆ (t ), e (t ) = γ γˆ(t ), e (t ) = δ δˆ(t ) In this section, we shall discuss the adaptive synchronization of the novel chaotic systems (A) and (B) with unknown system parameters. As the master (or drive) system, we consider the novel system (A), which is described by Differentiating (59) with respect to t, we obtain x = a ( x x ) + x ea (t ) = aˆ (t ), eb (t ) = bˆ(t ), ec (t ) = cˆ(t ), ed (t ) = dˆ (t ) ˆ (t ), eβ (t ) = βˆ (t ), eγ (t ) = γˆ(t ), eδ (t ) = δˆ(t ) eα (t ) = α(60) x = bx cx (5) α β γ (59) δ x = d + sinh( xx ) The error dynamics (58) can be simplified by using (59) as: where x, x, x are the states and a, b, c, d are unknown system parameters. As the slave (or response) system, we consider the controlled novel system (B), which is described by e = eα ( y y ) ea ( x x ) ke y = α ( y y ) + y y + u y = β y γ y y + u (54) y = δ y + cosh( y y ) + u where y, y, y are the states, α, β, γ, δ are unknown system parameters, and u, u, u are adaptive controllers to be designed. We define the chaos synchronization error between the systems (5) and (54) as: e = y x (55) e = y x e = y x The synchronization error dynamics is easily obtained as: e = α ( y y ) a ( x x ) + y y x + u e = β y bx γ y y + cx + u (56) e = δ y + d + cosh( y y ) sinh( xx ) + u Next, we introduce the nonlinear controller defined by u = αˆ (t )( y y ) aˆ (t )( x x ) y y + x x ke u = βˆ (t ) y + bˆ(t ) x + γˆ (t ) y y cˆ(t ) x x k e (57) u = δˆ(t ) y dˆ (t ) x cosh( y y ) + sinh( x x ) ke where aˆ (t ), bˆ(t ), cˆ(t ), dˆ (t ), αˆ (t ), estimates of the unknown a, b, c, d, α, β, γ, δ, respectively, positive gains. are system parameters and k, k, k are 58) We define the errors in estimating system parameters as: Next, we derive an update law for the parameter estimates using Lyapunov stability theory. So, we take a candidate Lyapunov function defined by V= e + e + e + ea + eb + ec + ed + eα + eβ + eγ + eδ ( ( ) (6) which is a quadratic and positive-definite function on R. Next, we calculate the time-derivative of V along the trajectories of (60) and (6). We obtain $ V = ke k e ke + ea e ( x x ) aˆ $ + eb & e x bˆ' $ + ec e x cˆ$ + ed &e dˆ ' + eα e ( y y ) αˆ $ $ $ ˆ + e β &e y β ' + eγ e y y γˆ$ + eδ & e y δˆ' (6) To guarantee global exponential stability of the systems (60) and (6), we need to choose parameter updates in such a way that V&is a quadratic, negative definite function on R. Thus, we choose the parameter update law as follows: aˆ = e ( x x ) + k 4ea bˆ = e x + k5eb cˆ = e x x + k e 6 c dˆ = e + k 7ed αˆ = e ( y y ) + k8eα βˆ = e y + k 9e β γˆ = e y y + k e By substituting the control law (57) into (56), we get the closed-loop error dynamics as: (6) e = eδ y + ed ke βˆ (t ), γˆ(t ), δˆ(t ) e = (α α (t ))( y y ) (a aˆ (t ))( x x ) ke e = ( β β (t )) y (b bˆ(t )) x (γ γˆ(t )) y y + (c cˆ(t )) x k e e = (δ δ (t )) y + (d dˆ (t )) x k e e = e β y eb x eγ y y + ec x k e (64) 0 γ δˆ = e y + keδ where ki,(i = 4,K,) are positive constants. Next, we prove the main result of this section. Theorem. The adaptive controller defined by (57) and the parameter update law defined by (64) globally and exponentially synchronize the novel chaotic systems (A) and (B) with unknown parameters described by the equations (5) 6
S. Vaidyanathan/Journal of Engineering Science and Technology Review 6 (4) (0) 5-65 and (54), where ki,( i=, K,) are positive constants. All the parameter estimation errors exponentially converge to zero with time. Proof. The assertions are established using Lyapunov stability theory [65]. We have already noted that the Lyapunov function V defined by (6) is quadratic and positive definite on R Next, we substitute the parameter update law defined by (64) into the dynamics (6). This simplifies the dynamics (6) as: V = k e k e k e k 4 e a k 5 e b k 6 e c k 7 e d k 8 e α k 9 e β k 0 e γ k e δ (65) Fig. 7. Complete synchronization of the novel chaotic systems (A) and (B). which is a quadratic and negative definite function on R. Hence, by Lyapunov stability theory [65], the synchronization and parameter estimation errors globally and exponentially converge to zero with time. This completes the proof. For numerical simulations, the classical fourth order Runge-Kutta method with step size 8 h =0 has been used with MATLAB to solve the chaotic systems (5) and (54) when the adaptive controller (57) and the parameter update law (64) are applied. We take the positive gains k as ki = 5,( i =, K,). The values of the system parameters for the systems (5) and (54) are taken as in the chaotic case, viz. i a= 0, b= 9, c=, d = 0, α = 0, β = 98, γ =, δ = 0 Fig. 8. Time history of the synchronization errors e(), t e(), t e() t. The initial state of the master system (5) is taken as: x (0) = 0.8, x (0) =., x (0) =.7 The initial state of the slave system (54) is taken as: y (0) =.4, y (0) = 0., y (0) =.5 The initial values of the parameter estimates are taken as: aˆ(0) =.8, bˆ(0) = 4., cˆ(0) = 6.7, dˆ(0) = 5.6, ˆ α(0) =.4, ˆ β(0) =.8, ˆ γ(0) = 5.4, ˆ δ(0) = 6. Fig. 7 depicts the complete chaos synchronization of the identical novel chaotic systems (A) and (B), which are described by the equations (5) and (54) respectively, while Fig. 8 depicts the time history of the synchronization errors e(), t e(), t e(). t Also, Fig. 9 depicts the time history of the parameter estimation errors ea(), t eb(), t ec(), t ed(). t Finally, Fig. 0 depicts the time history of the parameter estimation errors e (), t e (), t e (), t e (). t α β γ δ Fig. 9. Time history of the parameter estimation errors ea(), t eb(), t ec(), t ed() t. 6
S. Vaidyanathan/Journal of Engineering Science and Technology Review 6 (4) (0) 5-65 8. Conclusion Fig. 0. Time history of the parameter estimation errors eα(), t eβ(), t eγ(), t eδ () t. In this research work, we have derived two novel - dimensional chaotic systems, the first with a hyperbolic sinusoidal nonlinearity and two quadratic nonlinearities (denoted as system (A)) and the second with a hyperbolic cosinusoidal nonlinearity and two quadratic nonlinearities (denoted as system (B)). First, we provided a detailed qualitative analysis of the two novel chaotic systems (A) and (B). We also calculated the Lyapunov exponents and Kaplan- Yorke dimensions of the chaotic systems (A) and (B). It was found that the maximal Lyapunov exponent (MLE) for the novel chaotic systems (A) and (B) has a large value, viz. L =.794for system (A) and L = 5.for system (B). 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