JOURNAL OF Journal of Engineering Science and Technology Review 8 (2) (2015) 106 - 115 Engineering Science and Jestr Special Issue on Synchronization and : Theory, Technology Review Methods and Applications www.jestr.org Research Article

A 3-D Novel Highly Chaotic System with Four Quadratic Nonlinearities, its Adaptive Control and Anti-Synchronization with Unknown Parameters

S. Vaidyanathan*,1

1Research and Development Centre, Vel Tech University, Avadi, Chennai-600062, Tamil Nadu, India.

Received 21 September 2014; Revised 28 October 2014; Accepted 13 November 2014 ______

Abstract

This research work proposes a seven-term 3-D novel dissipative chaotic system with four quadratic nonlinearities. The

Lyapunov exponents of the 3-D novel chaotic system are obtained as L1 = 11.36204, L2 = 0 and L3 = –47.80208. Since the sum of the Lyapunov exponents is negative, the 3-D novel chaotic system is dissipative. Also, the Kaplan-Yorke dimension

of the 3-D novel chaotic system is obtained as DKY = 2.23769. The maximal (MLE) of the novel chaotic system is L1 = 11.36204, which is a large value for a polynomial chaotic system. Thus, the proposed 3-D novel chaotic system is highly chaotic. The phase portraits of the novel chaotic system simulated using MATLAB depict the highly chaotic of the novel system. This research work also discusses other qualitative properties of the system. Next, an adaptive controller is designed to stabilize the 3-D novel chaotic system with unknown parameters. Also, an adaptive synchronizer is designed to achieve anti-synchronization of the identical 3-D novel chaotic systems with unknown parameters. The adaptive results derived in this work are established using Lyapunov . MATLAB simulations have been shown to illustrate and validate all the main results derived in this work.

Keywords: Chaos, chaotic systems, dissipative systems, adaptive control, anti-synchronization. ______

1. Introduction [24], Vaidyanathan systems [25-30], Vaidyanathan- Madhavan system [31], Pehlivan-Moroz-Vaidyanathan Chaotic systems are defined as nonlinear dynamical systems system [32], Jafari system [33], Pham system [34], etc. which are very sensitive to initial conditions, topologically We note that the chaotic systems [2-34] are dissipative and also with dense periodic orbits [1]. systems, in which the system limit sets are ultimately The sensitivity to initial conditions of a chaotic system is confined into a specific of zero volume and the indicated by a positive Lyapunov exponent. A dissipative asymptotic motion of the chaotic system settles onto a chaotic system is characterized by the condition that the sum strange attractor of the system. of the Lyapunov exponents of the chaotic system is negative. has many important applications in Since Lorenz discovered a 3-D chaotic system of a science and engineering such as vibration control [35-37], weather model [2], great interest has been shown in the oscillators [38-40], lasers [41-43], robotics [44-47], chemical chaos literature in the analysis and modelling of many 3-D reactors [48-50], biology [51,52], ecology [53,54], chaotic systems such as Rössler system [3], Rabinovich cardiology [55], memristors [56-59], neural networks [60- system [4], ACT system [5], Sprott systems [6], Chen 62], secure communications [63-66], cryptosystems [67-70], system [7], Lü system [8], Shaw system [9], Feeny system network design [71, 72], economics [73-76], market [10], Shimizu system [11], Liu-Chen system [12], Cai forecasting [77], etc. system [13], Tigan system [14], Colpitt’s oscillator [15], Chaos control and chaos synchronization are important WINDMI system [16], Zhou system [17], etc. research problems in the chaos theory. In the last three Recently, many 3-D chaotic systems have been decades, many mathematical methods have been developed discovered such as Li system [18], Elhadj system [19], Pan successfully to address these research problems. system [20], Sundarapandian system [21], Yu-Wang system The study of control of a chaotic system investigates [22], Sundarapandian-Pehlivan system [23], Zhu system methods for designing feedback control laws that globally or locally asymptotically stabilize or regulate the outputs of a ______chaotic system. * E-mail address: [email protected] Many methods have been developed for the control and ISSN: 1791-2377 © 2015 Kavala Institute of Technology. All tracking of chaotic systems such as active control [78-82], rights reserved.

S. Vaidyanathan /Journal of Engineering Science and Technology Review 8 (2) (2015) 106 -115 adaptive control [83-89], backstepping control [90-92], We take the initial conditions as: sliding mode control [93, 94], etc. Chaos synchronization problem deals with the �! 0 = 0.8, �! 0 = 0.5, �! 0 = 0.7 (3) synchronization of a couple of systems called the master or drive system and the slave or response system. To solve this The 3-D portrait of the strange chaotic attractor (1) for problem, control laws are designed so that the output of the the parameter values (2) and the initial conditions (3) is slave system tracks the output of the master system depicted in Fig. 1, and the 2-D portraits (projections on the asymptotically with time. three coordinate planes) are depicted in Figs. 2-4. In the chaos anti-synchronization problem, control laws are designed so that the sum of the outputs of the master and slave systems is driven to zero asymptotically, i.e. the outputs of the two systems are asymptotically equal in magnitude but opposite in phase. Because of the , both synchronization and anti-synchronization of chaotic systems are challenging problems even when the initial conditions of the master and slave systems are nearly identical because of the exponential divergence of the outputs of the two systems in the absence of any control. The synchronization of chaotic systems has applications in secure communications [95-97], cryptosystems [98, 99], encryption [100-104], etc. In the chaos literature, many different methodologies have been also proposed for the synchronization and anti- synchronization of chaotic systems such as PC method [103], active control [104-114], time-delayed feedback control [115,116], adaptive control [117-126], sampled-data ! feedback control [127-130], backstepping control [131-137], Fig. 1. The chaotic attractor of the novel chaotic system in � . sliding mode control [138-143], etc. In this research work, a seven-term 3-D novel dissipative chaotic system with four quadratic nonlinearities is proposed. The Lyapunov exponents of the 3-D novel chaotic system are found as L1 = 11.36204, L2 = 0 and L3 = –47.80208. The Kaplan-Yorke of the 3-D novel chaotic system is found as DKY = 2.23769. Since the maximal Lyapunov exponent (MLE) of the novel chaotic system has a large value, viz. L1 = 11.36204, it is noted that the 3-D novel chaotic system is highly chaotic. In Section 2, we describe the equations and phase portraits of the novel chaotic system. In Section 3, we derive the qualitative properties of the novel chaotic system. In Section 4, we derive an adaptive controller for the stabilization of 3-D novel chaotic system with unknown parameters. In Section 5, we derive an adaptive synchronizer for the anti-synchronization of identical 3-D novel chaotic systems with unknown parameters. Section 6 concludes this Fig. 2. The 2-D projection of the chaotic attractor on the (x1, x2) plane. research work with a summary of main results.

2. A Seven-Term 3-D Novel Chaotic System with Four Quadratic Nonlinearities

The dynamics of the seven-term 3-D novel chaotic system is described by

!! ! = � � − � + � � !" ! ! ! ! !! ! = −�� � + ��! (1) !" ! ! ! !! ! = −� + � � !" ! ! where x1, x2, x3 are the states and a, b, c, p are positive parameters. The nonlinear system (1) depicts a chaotic attractor when the parameter values are taken as: Fig. 3. The 2-D projection of the chaotic attractor on the (x2, x3) plane. � = 40, � = 26, � = 160, � = 0.3 (2)

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Hence, the 3-D system (1) is dissipative and the asymptotic motion of the 3-D system (1) settles exponentially onto a set of measure zero, i.e. a strange attractor.

3.2. Equilibrium Points

The equilibrium points of the novel chaotic system (1) are obtained by solving the following system of equations with the parameter values as in the chaotic case (2):

� �! − �! + �!�! = 0 ! −��!�! + ��! = 0 (10) −� + �!�! = 0

Solving (10), we obtain two equilibrium points of the Fig. 4. The 2-D projection of the chaotic attractor on the (x1, x3) plane. system (1), viz.

3. Analysis of the 3-D Novel Chaotic System 5.0996 −5.0984 � = , � = (11) ! 5.0984 ! −5.0996 0.0096 −0.0096 In this section, qualitative properties of the 3-D novel chaotic system are detailed. The Jacobian matrix of the system (1) is obtained as

−� � + �! �! 3.1. Dissipativity � � = −�� 2�� −�� (12) ! ! ! �! �! 0 In vector notation, we may express the system (1) as:

Thus, the Jacobian matrix at E1 is obtained as: �!(�!, �!, �!) !� = � � = �!(�!, �!, �!) (4) !" −40.0000 40.0096 5.0984 �!(�!, �!, �!) � �! = −1.5360 3.0590 −815.9360 (13) 5.0984 5.0996 0 where which has the eigenvalues

�! �!, �!, �! = � �! − �! + �!�! ! �! = −60.5268, �!,! = 11.7929 ± 73.2294� (14) �! �!, �!, �! = −��!�! + ��! (5) � � , � , � = −� + � � ! ! ! ! ! ! This shows that the equilibrium E is a saddle-focus. 1 Also, the Jacobian matrix at E is obtained as: We take the parameter values as in the chaotic case (2). 2

Let Ω be any region in �! with a smooth boundary and −40.0000 39.9904 −5.0996 also Ω(t) = Φ (Ω), where Φ ιs the flow of f. t t � � = 1.5360 −3.0598 815.7440 (15) Furthermore, let V(t) denote the volume of Ω(t). ! −5.0996 −5.0984 0 By Liouville’s theorem, we have which has the eigenvalues !" = ∇ ∙ � �� �� �� (6) !" !(!) ! ! ! �! = −61.9776, �!,! = 9.4589 ± 72.6427� (16) The divergence of the novel chaotic system (1) is easily found as: This shows that the equilibrium E2 is also a saddle-focus. Hence, both equilibrium points E1 and E2 are unstable. !! !! !! ∇ ∙ � = ! + ! + ! = −� < 0 (7) !!! !!! !!! 3.4. Lyapunov Exponents and Kaplan-Yorke Dimension Substituting (7) into (6), we obtain the first order ODE. For the chosen parameter values (2), the Lyapunov !" = −� �� �� �� = −�� (8) exponents of the novel chaotic system (1) are obtained using !" !(!) ! ! ! MATLAB as:

Integrating (8), we obtain the unique solution as: �! = 11.36204, �! = 0, �! = −47.80208 (17)

� � = exp −�� �(0) for all � ≥ 0 (9) Since the spectrum of Lyapunov exponents (17) has a positive term L1, the system (1) is chaotic. From (9), we find that �(�) shrinks to zero exponentially Since the sum of the Lyapunov exponents is zero, the as t → ∞. novel chaotic system (1) is dissipative.

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!! The maximal Lyapunov exponent (MLE) of the novel ! = � − � � � − � − � � !" ! ! ! ! chaotic system (1) is L1 =11.36204, which is a large value. !!! ! This shows that the 3-D novel system (1) is a highly = − � − � � �!�! + � − � � �! !" (21) chaotic system. −�!�! Also, the Kaplan-Yorke dimension of the novel chaotic !! ! = − � − � � − � � system (1) is calculated as: !" ! !

!!!!! The parameter estimation errors are defined by �!" = 2 + = 2.23769 (18) !!

�!(�) = � − �(�) Fig. 5 depicts the dynamics of the Lyapunov exponents � (�) = � − �(�) of the novel chaotic system (1). ! (22) �!(�) = � − �(�)

�!(�) = � − �(�)

Substituting (22) into the state dynamics (21), we get

!! ! = � � − � − � � !" ! ! ! ! ! !! ! = −� � � + � �! − � � (23) !" ! ! ! ! ! ! ! !! ! = −� − � � !" ! ! !

Differentiating (22) with respect to t, we get

!! !! ! = − !" !" !! !! ! = − !" !" (24) !! !! ! = − !" !"

!! !! Fig. 5. Dynamics of the Lyapunov exponents of the novel system. ! = − !" !"

4. Adaptive Control of the 3-D Novel Chaotic System Next, we use theory for finding an update law for the parameter estimates. In this section, we construct an adaptive controller for Consider the quadratic Lyapunov function defined by globally stabilizing the unstable 3-D novel chaotic system ! with unknown parameters. The adaptive controller design is � = �! + �! + �! + �! + �! + �! + �! , (25) ! ! ! ! ! ! ! ! carried out using Lyapunov stability theory. We consider the controlled chaotic system which is positive definite on �!. Differentiating � along the trajectories of (23) and (24), !! ! = � � − � + � � + � we get !" ! ! ! ! ! !! ! = −�� � + ��! + � (19) !" ! ! ! ! !" ! ! ! = −�!�! − �!�! − �!�! !!! !" = −� + �!�! + �! !! !" +� � (� − � ) − ! ! ! ! !" !! !! (26) where x1, x2, x3 are state variables and a, b, c, p are unknown, +� −� − + � −� � � − ! ! !" ! ! ! ! !" constant, parameters and u1, u2, u3 are adaptive controls to be !! +� �! − designed using estimates � � , � � , � � , �(�) of the ! ! !" unknown parameters a, b, c, p, respectively. We consider the adaptive controller defined by In view of (26), we take the parameter update law as:

!! �! = −� � �! − �! − �!�! − �!�! = � (� − � ) ! !" ! ! ! �! = � � �!�! − � � �! − �!�! (20) !! �! = �(�) − �!�! − �!�! = −�! !" (27) !! = −�!�!�! where k1, k2, k3 are positive gain constants, and � � , � � , !" !! � � , �(�) are estimates of the unknown parameters a, b, c, = �! p, respectively. !" ! Substituting (20) into (19), we get the closed-loop system as: Next, we state and prove the main result of this section.

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Theorem 1. The novel chaotic system (19) is globally and 5. Adaptive Anti-Synchronization of Identical 3-D Novel exponentially stabilized by the adaptive control law (20) and Chaotic Systems the parameter update law (27) for all initial conditions, where k1, k2, k3 are positive constants. In this section, we construct an adaptive synchronizer for global anti-synchronization of identical 3-D novel chaotic Proof. We prove this result using Lyapunov stability theory. systems. The adaptive synchronizer design is carried out For this purpose, we consider the quadratic Lyapunov using Lyapunov stability theory. function � defined by (25), which is positive definite on �!. As the master system, we take the novel chaotic system Substituting the parameter update law (27) into (26), we !! ! = � � − � + � � obtain the time derivative of V as: !" ! ! ! ! !!! ! !" = −��!�! + ��! (31) = −� �! − � �! − � �! (28) !" ! ! ! ! ! ! !! !" ! = −� + � � !" ! ! which is a negative semi-definite function on �!. Thus, we can conclude that the state vector x(t) and the where x1, x2, x3 are state variables and a, b, c, p are unknown, parameter estimation error are globally bounded. constant, parameters. As the slave system, we take the novel chaotic system We define k = min{k1, k2, k3}. Then we get

!!! !" ! ! !" = � � − � + � � + � ≤ −� � or � � ≤ − (29) !" ! ! ! ! ! !" !" !! ! = −�� � + ��! + � (32) !" ! ! ! ! Integrating the inequality (29) from 0 to t, we get !! ! = −� + � � + � !" ! ! ! ! � �(�) !�� ≤ � 0 − �(�) (30) ! where y1, y2, y3 are state variables and u1, u2, u3 are adaptive controls to be designed using estimates � � , � � , � � , �(�) From (30), it follows that x(t) ∈ �!. Using (23), we can of the unknown parameters a, b, c, p, respectively. conclude that � ∈ �!. The anti-synchronization error between the systems (31) Thus, using Barbalat’s lemma [145], we conclude that and (32) is defined as: x(t) → 0 exponentially as t → ∞ for all initial conditions ! x(0) ∈ � . �! = �! + �! This completes the proof. n �! = �! + �! (33) �! = �! + �! For numerical simulations, the parameter values of the novel chaotic system (19) are taken as in the chaotic case, The error dynamics is easily obtained as: viz. a = 40, b = 26, c = 160 and p = 0.3. We take the gain !! constants as k = 6, k = 6 and k = 30. ! = � � − � + � � + � � + � 1 2 3 !" ! ! ! ! ! ! ! The initial conditions of the chaotic system (19) are !!! ! ! = −� �!�! + �!�! + � �! + �! + �! (34) taken as x1(0) = 1.3, x2(0) = 2.7 and x3(0) = –3.5. !" !! The initial conditions of the parameter estimates are ! = −2� + � � + � � + � !" ! ! ! ! ! taken as � 0 = 21, � 0 = 30, � 0 = 25 and � 0 =3. Fig. 6 describes the time-history of the state x(t) . We consider the adaptive controller defined by

�! = −� � �! − �! − �!�! − �!�! − �!�! � = � � � � + � � − � � �! + �! ! ! ! ! ! ! ! (35) −�!�!

�! = 2�(�) − �!�! − �!�! − �!�!

where k1, k2, k3 are positive gain constants and � � , � � , � � , �(�) are estimates of the unknown parameters a, b, c, p, respectively. Substituting (35) into (34), we get the closed-loop error dynamics as:

!! ! = � − � � � − � − � � !" ! ! ! ! !!! = − � − � � �!�! + �!�! !" (36) ! ! + � − � � �! + �! − �!�! !! Fig. 6. Time-history of the controlled states x , x , x of the chaotic ! = −2 � − � � − � � 1 2 3 !" ! ! system

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!" The parameter estimation errors are defined by Since is a negative semi-definite function on �!, we !" can conclude that the anti-synchronization vector e(t) and � (�) = � − �(�) ! the parameter estimation error are globally bounded. � (�) = � − �(�) ! (37) We define k = min{k1, k2, k3}. Then we get �!(�) = � − �(�) !" !" �!(�) = � − �(�) ≤ −� � ! or � � ! ≤ − (44) !" !"

Substituting (37) into the error dynamics (36), we get Integrating the inequality (44) from 0 to t, we get

!! ! = � � − � − � � ! !" ! ! ! ! ! ! (45) � ! �(�) �� ≤ � 0 − �(�) !! ! = −� � � + � � + � �! + �! − � � (38) !" ! ! ! ! ! ! ! ! ! ! !!! From (45), it follows that e(t) ∈ �!. Using (38), we can = −2�! − �!�! !" conclude that � ∈ �!. Thus, using Barbalat’s lemma [145], we conclude that Differentiating (37) with respect to �, we get e(t) → 0 exponentially as t → ∞ for all initial conditions e(0) ∈ �!. !! !! ! = − This completes the proof. n !" !" !!! !! = − For numerical simulations, the parameter values of the !" !" (39) !! !! novel chaotic systems are taken as in the chaotic case, viz. ! = − !" !" a = 40, b = 26, c = 160 and p = 0.3. We take the gain !!! !! = − constants as as k1 = 20, k2 = 20 and k3 = 30. !" !" The initial conditions of the master system (31) are taken Consider the quadratic Lyapunov function defined by as x1(0) = 0.4, x2(0) = 2.3 and x3(0) = –0.5 The initial conditions of the slave system (32) are taken ! as y (0) = 1.7, y (0) = 1.2 and y (0) = –2.8. � = �! + �! + �! + �! + �! + �! + �! , (40) 1 2 3 ! ! ! ! ! ! ! ! The initial conditions of the parameter estimates are taken as � 0 = 15, � 0 = 22, � 0 = 11 and � 0 = 4. ! which is positive definite on � . Figure 7 describes the time-history of the anti- Differentiating V along the trajectories of (38) and (39), synchronization error e(t). we get

!" = −� �! − � �! − � �! !" ! ! ! ! ! ! !! +� � (� − � ) − ! ! ! ! !" !! +� −2� − (41) ! ! !" !! +� −� � � + � � − ! ! ! ! ! ! !" !! +� � �! + �! − ! ! ! ! !"

In view of (41), we take the parameter update law as:

!! = � (� − � ) !" ! ! ! !! = −2�! !" (42) !! = −�! �!�! + �!�! !" Fig. 7. Time-history of the anti-synchronization error. !! = � �! + �! !" ! ! !

Theorem 2. The novel chaotic systems (31) and (32) are 6. Conclusion globally and exponentially anti-synchronized by the adaptive control law (35) and the parameter update law (42) for all In this research work, we have proposed a seven-term 3-D initial conditions, where k1, k2, k3 are positive constants. novel chaotic system with four quadratic nonlinearities. The Lyapunov exponents of the 3-D novel chaotic system have Proof. We prove this result using Lyapunov stability theory. been found as L1 = 11.36204, L2 = 0 and L3 = –47.80208. For this purpose, we consider the quadratic Lyapunov The Kaplan-Yorke of the 3-D novel chaotic system has been ! function V defined by (40), which is positive definite on � . found as DKY = 2.23769. Since the maximal Lyapunov Substituting the parameter update law (42) into (41), we exponent (MLE) of the novel chaotic system has a large obtain the time derivative of V as: value, viz. L1 = 11.36204, the 3-D novel chaotic system is a highly chaotic system and it has potential applications in !" = −� �! − � �! − � �! (43) encryption and secure communication systems. We have !" ! ! ! ! ! !

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S. Vaidyanathan /Journal of Engineering Science and Technology Review 8 (2) (2015) 106 -115 also designed control laws for adaptive stabilization and Lyapunov stability theory. MATLAB simulations have been adaptive anti-synchronization of the 3-D novel chaotic shown to illustrate the phase portraits of the highly chaotic system with unknown system parameters. The main adaptive system and also the adaptive stabilization and anti- results derived in this work were established using synchronization results derived in this work.

______References

1. S.H. Strogatz, Nonlinear dynamics and chaos: With applications 25. S. Vaidyanathan, A new six-term 3-D chaotic system with an to physics, biology, chemistry, and engineering, Perseus Books, exponential nonlinearity, Far East Journal of Mathematical Massachusetts, US (1994). Sciences, vol. 79, pp. 135-143 (2013). 2. E.N. Lorenz, Deterministic nonperiodic flow, Journal of the 26. S. Vaidyanathan, Analysis and adaptive synchronization of two Atmospheric Sciences, vol. 20, pp. 130-141 (1963). novel chaotic systems with hyperbolic sinusoidal and 3. O.E. Rössler, An equation for continuous chaos, Physics Letters cosinusoidal nonlinearity and unknown parameters, Journal of A, vol. 57, pp. 397-398 (1976). Engineering Science and Technology Review, vol. 6, pp. 53-65 4. M.I. Rabinovich and A.L. Fabrikant, Stochastic self-modulation (2013). of waves in nonequilibrium media, Sov. Phys. JETP, vol. 50, pp. 27. S. Vaidyanathan, A new eight-term 3-D polynomial chaotic 311-317 (1979) . system with three quadratic nonlinearities, Far East Journal of 5. A. Arneodo, P. Coullet, and C. Tresser, Possible new strange Mathematical Sciences, vol. 84, pp. 219-226 (2014). with spiral structure, Communications in Mathematical 28. S. Vaidyanathan, Analysis, control and synchronization of a six- Physics, vol. 79, pp. 573-579 (1981). term novel chaotic system with three quadratic nonlinearities, 6. J.C. Sprott, Some simple chaotic flows, Physical Review E, vol. International Journal of Modelling, Identification and Control, 50, pp. 647-650 (1994). vol. 22, pp. 41-53 (2014). 7. G. Chen and T. Ueta, Yet another chaotic oscillator, 29. S. Vaidyanathan, Analysis and adaptive synchronization of International Journal of Bifurcation and Chaos, vol. 9, pp. 1465- eight-term 3-D polynomial chaotic systems with three quadratic 1466 (1999). nonlinearities, European Physical Journal: Special Topics, vol. 8. J. Lü and G. Chen, A new chaotic attractor coined, International 223, pp. 1519-1529 (2014). Journal of Bifurcation and Chaos, vol. 12, pp. 659-661 (2002). 30. S. Vaidyanathan, Ch. Volos, V.T. Pham, K. Madhavan, and B.A. 9. R. Shaw, Strange attractors, chaotic behaviour and information Idowu, Adaptive backstepping control, synchronization and flow, Zeitschrift für Naturforschung, vol. 36, pp. 80-112 (1981). circuit simulation of a 3-D novel jerk chaotic system with two 10. B. Feeny and F.C. Moon, Chaos in a forced dry-friction hyperbolic sinusoidal nonlinearities, Archives of Control oscillator: Experiments and numerical modeling, Journal of Sciences, vol. 24(3), pp. 257-285 (2014). Sound and Vibration, vol. 170, pp. 303-323 (1994). 31. S. Vaidyanathan and K. Madhavan, Analysis, adaptive control 11. T. Shimizu and N. Moroika, On the bifurcation of a symmetric and synchronization of a seven-term novel 3-D chaotic system, limit cycle to an asymmetric one in a simple model, Physics International Journal of Control Theory and Applications, vol. 6, Letters A, vol. 76, pp. 201-204 (1980). pp. 121-137 (2013). 12. W. Liu and G. Chen, A new chaotic system and its generation, 32. I. Pehlivan, I.M. Moroz, and S. Vaidyanathan, Analysis, International Journal of Bifurcation and Chaos, vol. 13, pp. 261- synchronization and circuit design of a novel butterfly attractor, 267 (2003). Journal of Sound and Vibration, vol. 333, pp. 5077-5096 (2014). 13. G. Cai and Z. Tan, Chaos synchronization of a new chaotic 33. S. Jafari and J.C. Sprott, Simple chaotic flows with a line system via nonlinear control, Journal of Uncertain Systems, vol. equilibrium, Chaos, Solitons and , vol. 57, pp. 79-84 1, pp. 235-240 (2007). (2013). 14. G. Tigan and D. Opris, Analysis of a 3D chaotic system, Chaos, 34. V.T. Pham, C. Volos, S. Jafari, Z. Wei and X. Wang, Solitons and Fractals, vol. 36, pp. 1315-1319 (2008). Constructing a novel no-equilibrium chaotic system, 15. G.P. Kennedy, Chaos in the Colpitts oscillator, IEEE International Journal of Bifurcation and Chaos, vol. 24, 1450073 Transactions on Circuits and Systems-I, vol. 41, pp. 771-774 (2014). (1994). 35. Z. Wang and K.T. Chau, Control of chaotic vibration in 16. J. Wang, D. Lu and L. Tian, Global synchronization for time automotive wiper systems, Chaos, Solitons and Fractals, vol. 39, delay of WINDMI system, Chaos, Solitons and Fractals, vol. 30, no. 1, pp. 168-181 (2009). pp. 629-635 (2006). 36. J. Shi, F. Zhao, X. Shen and X. Wang, Chaotic operation and 17. W. Zhou, Y. Xu, H. Lu, and L. Pan, On dynamics analysis of chaos control of travelling wave ultrasonic motor, Ultrasonics, a new chaotic attractor, Physics Letters A, vol. 372, pp. 5773- vol. 53, no. 6, pp. 1112-1123 (2013). 5777 (2008). 37. D. Yang and J. Zhou, Connections among several chaos 18. D. Li, A three-scroll chaotic attractor, Physics Letters A, vol. feedback control approaches and chaotic vibration control of 372, pp. 387-393 (2008). mechanical systems, Communications in Nonlinear Science and 19. Z. Elhadj, Dynamical analysis of a 3-D chaotic system with only Numerical Simulation, vol. 19, no. 11, pp. 3954-3968 (2014). two quadratic nonlinearities, Journal of Systems Science and 38. H.B. Fotsin and J. Daafouz, Adaptive synchronization of , vol. 21, pp. 67-75 (2008). uncertain Colpitts oscillators based on parameter identification, 20. L. Pan, D. Xu and W. Zhou, Controlling a novel chaotic attractor Physics Letters A, vol. 339, pp. 304-315 (2005). using linear feedback, Journal of Information and Computing 39. G.H. Li, S.P. Zhou and K. Yang, Controlling chaos in Colpitts Science, vol. 5, pp. 117-124 (2010). oscillator, Chaos, Solitons and Fractals, vol. 33, pp. 582-587 21. V. Sundarapandian, Analysis and anti-synchronization of a novel (2007). chaotic system via active and adaptive controllers, Journal of 40. A. Sharma, V. Patidar, G. Purohit, and K.K. Sud, Effects on the Engineering Science and Technology Review, vol. 6, pp. 45-52 bifurcation and chaos in forced Duffing oscillator due to (2013). nonlinear damping, Communications in Nonlinear Science and 22. F. Yu, C. Wang, Q. Wan, and Y. Hu, Complete switched Numerical Simulation, vol. 17, pp. 2254-2269 (2012). modified function projective synchronization of a five-term 41. S. Donati and S.K. Hwang, Chaos and high-level dynamics in chaotic system with uncertain parameters and disturbances, coupled lasers and their applications, Progress in Quantum Pramana, vol. 80, pp. 223-235 (2013). Electronics, vol. 36, pp. 293-341 (2012). 23. V. Sundarapandian and I. Pehlivan, Analysis, control, 42. I.V. Ermakov, S.T. Kingni, V.Z. Tronciu, and J. Danckaert, synchronization and circuit design of a novel chaotic system, Chaotic semiconductor ring lasers subject to optical feedback: Mathematical and Computer Modelling, vol. 55, pp. 1904-1915 Applications to chaos-based communications, Optics (2012). Communications, vol. 286, pp. 265-272 (2013). 24. C. Zhu, Y. Liu, and Y. Guo, Theoretical and numerical study of 43. N. Li, W. Pan, L. Yan, B. Luo, and X. Zou, Enhanced chaos a new chaotic system, Intelligent Information Management, vol. synchronization and communication in cascade-coupled 2, pp. 104-109 (2010). semiconductor ring lasers, Communications in Nonlinear 112

S. Vaidyanathan /Journal of Engineering Science and Technology Review 8 (2) (2015) 106 -115

Science and Numerical Simulation, vol. 19, pp. 1874-1883 Remarkable stability to noise, Physics Letters A, vol. 374, pp. (2014). 2925-2931 (2010). 44. U. Nehmzow and K. Walker, Quantitative description of robot- 66. A. Abdullah, Synchronization and secure communication of environment interaction using chaos theory, Robotics and uncertain chaotic systems based on full-order and reduced-order Autonomous Systems, vol. 53, pp. 177-193 (2005). output-affine observers, Applied Mathematics and Computation, 45. S. Iqbal, X. Zang, Y. Zhu, and J. Zhao, Bifurcations and chaos in vol. 219, pp. 10000-10011 (2013). passive dynamic walking: A review, Robotics and Autonomous 67. N. Smaoui and A. Kanso, Cryptography with chaos and Systems, vol. 62, pp. 889-909 (2014). shadowing, Chaos, Solitons and Fractals, vol. 42, pp. 2312-2321 46. Ch.K. Volos, I.M. Kyprianidis, and I.N. Stouboulos, Α chaotic (2009). path planning generator for autonomous mobile robots, Robotics 68. R. Rhouma and S. Belghith, Cryptanalysis of a chaos-based and Autonomous Systems, vol. 60, pp. 651-656 (2012). cryptosystem on DSP, Communications in Nonlinear Science 47. Ch.K. Volos, I.M. Kyprianidis, and I.N. Stouboulos, and Numerical Simulation, vol. 16, pp. 876-884 (2011). Experimental investigation on coverage performance of a 69. Ch K. Volos, I.M. Kyprianidis, and I.N. Stouboulos, Text chaotic autonomous mobile robot, Robotics and Autonomous encryption scheme realized with a chaotic pseudo-random bit Systems, vol. 61(12), pp. 1314-1322 (2013). generator, Journal of Engineering Science and Technology 48. J.C. Roux, Chaos in experimental chemical systems: two Review, vol. 6(4), pp. 9-14 (2013). examples, North-Holland Mathematics Studies, vol. 103, pp. 70. Ch.K. Volos, I.M. Kyprianidis, and I.N. Stouboulos, Image 345-352 (1985). encryption process based on chaotic synchronization 49. Y.N. Li, L. Chen, Z.S. Cai, and X.Z. Zhao, Study on chaos phenomena, Signal Processing, vol. 93(5), pp. 1328-1340 synchronization in the Belousov-Zhabotinsky chemical system, (2013). Chaos, Solitons and Fractals, vol. 17, pp. 699-707 (2003). 71. J. Yang, D. Xiao and T. Xiang, Cryptanalysis of a chaos block 50. Y. Gong, Y. Xie, X. Lin, Y. Hao, and X. Ma, Ordering chaos cipher for wireless sensor network, Communications in and synchronization transitions by chemical delay and coupling Nonlinear Analysis and Numerical Simulation, vol. 16(2), pp. on scale-free neuronal networks, Chaos, Solitons and Fractals, 844-850 (2011). vol. 43, pp. 96-103 (2010). 72. N. Bezzo, P.J. Cruz, F. Sorrentino and R. Fierro, Decentralized 51. M. Kyriazis, Applications of chaos theory to the molecular identification and control of networks of coupled mobile biology of aging, Experimental Gerontology, vol. 26, pp. 569- platforms through adaptive synchronization of chaos, Physica D: 572 (1991). Nonlinear Phenomena, vol. 267, pp. 94-103 (2014). 52. G. Böhm, Protein folding and deterministic chaos: Limits of 73. D. Guégan, Chaos in economics and finance, Annual Reviews in protein folding simulations and calculations, Chaos, Solitons and Control, vol. 33, pp. 89-93 (2009). Fractals, vol. 1, pp. 375-382 (1991). 74. Ch.K. Volos, I.M. Kyprianidis, and I.N. Stouboulos, 53. J.C. Sprott, J.A. Vano, J.C. Wildenberg, M.B. Anderson, and Synchronization phenomena in coupled nonlinear systems J.K. Noel, Coexistence and chaos in complex ecologies, Physics applied in economic cycles, WSEAS Trans. Systems, vol. Letters A, vol. 335, pp. 207-212 (2005). 11(12), pp. 681-690 (2012). 54. B. Sahoo and S. Poria, The chaos and control of a food chain 75. Ch. K. Volos, I. M. Kyprianidis, S. G. Stavrinides, I. N. model supplying additional food to top-predator, Chaos, Solitons Stouboulos, I. Magafas, and A. N. Anagnostopoulos, Nonlinear and Fractals, vol. 58, pp. 52-64 (2014). dynamics of a financial system from an engineer’s point of view, 55. T.A. Denton, G.A. Diamond, R.H. Helfant, S. Khan, and H. 76. Journal of Engineering Science and Technology Review, vol. Karagueuzian, Fascinating rhythm: A primer on chaos theory 4(3), pp. 281-285 (2011). and its applications to cardiology, American Heart Journal, vol. 77. P. Caraiani, Testing for nonlinearity and chaos in economic time 120, pp. 1419-1440 (1990). series with noise titration, Economics Letters, vol. 120, pp. 192- 56. A. Wu, S. Wen, and Z. Zeng, Synchronization control of a class 194 (2013). of memristor-based neural networks, Information Sciences, vol. 78. A. Kazem, E. Sharifi, F.K. Hussain, M. Saberi and O.K. 183, pp. 106-116, (2012). Hussain, Support vector regression with chaos-based firefly 57. S. Wen, Z. Zeng, and T. Huang, Adaptive synchronization of algorithm for stock market price forecasting, Applied Soft memristor-based Chua’s circuits, Physics Letters A, vol. 376, pp. Computing, vol. 13(2), pp. 947-958 (2013). 2775-2780, (2012). 79. V. Sundarapandian, Output regulation of the Lorenz attractor, 58. A. Wu, and Z. Zeng, Anti-synchronization control of a class of Far East Journal of Mathematical Sciences, vol. 42, pp. 289-299 memristive recurrent neural networks, Communications in (2010). Nonlinear Science and Numerical Simulation, vol. 18, pp. 373- 80. C.A. Kitio Kwuimy and B.R. Nana Nbendjo, Active control of 385, (2013). horseshoes chaos in a driven Rayleigh oscillator with fractional 59. Ch. K. Volos, I. M. Kyprianidis, and I. N. Stouboulos, order deflection, Physics Letters A, vol. 375 (39), pp. 3442 -3449 Chaotic dynamics from a nonlinear circuit based on memristor (2011). with a cubic nonlinearity, In Proc. of 7th International 81. S. Vaidyanathan, Output regulation of Arneodo-Coullet chaotic Conference of the Balkan Physical Union, (BPU 2009), AIP system, Communications in Computer and Information Science, Conference Proceedings, vol. 1203, pp. 626-631 (2009), vol. 131, pp. 585-593 (2011). Alexandroupolis, Greece. 82. S. Vaidyanathan, Output regulation of the unified chaotic 60. G. He, Z. Cao, P. Zhu, and H. Ogura, Controlling chaos in a system, Communications in Computer and Information Science, chaotic neural network, Neural Networks, vol. 16, pp. 1195- vol. 198, pp. 1-9 (2011). 1200 (2003). 83. S. Vaidyanathan, Output regulation of the Liu chaotic system, 61. E. Kaslik and S. Sivasundaram, Nonlinear dynamics and chaos Applied Mechanics and Materials, vols. 110-116, pp. 3982-3989 in fractional-order neural networks, Neural Networks, vol. 32, (2012). pp. 245-256 (2012). 84. G. Chen, A simple adaptive feedback control method for chaos 62. I.M. Kyprianidis and A.T. Makri, Complex dynamics of and hyper-chaos control, Applied Mathematics and FitzHugh-Nagumo type neurons coupled with gap junction Computation, vol. 217, pp. 7258-7264 (2011). under external voltage stimulation, Journal of Engineering 85. J. Zheng, A simple universal adaptive feedback controller for Science and Technology Review, vol. 6(4), pp. 104-114 (2013). chaos and hyperchaos control, Computers & Mathematics with 63. K. Suzuki and Y. Imai, Decryption characteristics in message Applications, vol. 61, pp. 2000-2004 (2011). modulation type chaos secure communication system using 86. S. Vaidyanathan, Adaptive controller and synchronizer design optical fiber ring resonators, Optics Communications, vol. 259, for the Qi-Chen chaotic system, Lecture Notes of the Institute pp. 88-93 (2006). for Computer Sciences, Social-Informatics and 64. X.Y. Wang and Y.F. Gao, A switch-modulated method for chaos Telecommunications Engineering, vol. 85, pp. 124-133 (2012). digital secure communication based on user-defined protocol, 87. V. Sundarapandian, Adaptive control and synchronization Communications in Nonlinear Science and Numerical design for the Lu-Xiao chaotic system, Lecture Notes in Simulation, vol. 15, pp. 99-104 (2010). Electrical Engineering, vol. 131, pp. 319-327 (2013). 65. O.I. Moskalenko, A.A. Koronovskii, and A.E. Hramov, 88. S. Vaidyanathan, A ten-term novel 4-D hyperchaotic system Generalized synchronization of chaos for secure communication: with three quadratic nonlinearities and its control, International

113

S. Vaidyanathan /Journal of Engineering Science and Technology Review 8 (2) (2015) 106 -115

Journal of Control Theory and Applications, vol. 6, pp. 97-109 active nonlinear control, Communications in Computer and (2013). Information Science, vol. 204, pp. 84-93 (2011). 89. S. Vaidyanathan, Analysis, control and synchronization of 112. S. Vaidyanathan, Hybrid chaos synchronization of Liu and Lü hyperchaotic Zhou system via adaptive control, Advances in systems by active nonlinear control, Communications in Intelligent Systems and Computing, vol. 177, pp. 1-10 (2013). Computer and Information Science, vol. 204, pp. 1-10 (2011). 90. D. Yang and J. Zhou, Connections among several chaos 113. P. Sarasu and V. Sundarapandian, Active controller design for feedback control approaches and chaotic vibration control of generalized projective synchronization of four-scroll chaotic mechanical systems, Communications in Nonlinear Science and systems, International Journal of Systems Signal Control and Numerical Simulation, vol. 19, pp. 3954-3968 (2014). Engineering Application, vol. 4, pp. 26-33 (2011). 91. M.T. Yassen, Chaos control of chaotic dynamical systems using 114. S. Vaidyanathan and K. Rajagopal, Hybrid synchronization of backstepping design, Chaos, Solitons and Fractals, vol. 27, pp. hyperchaotic Wang-Chen and hyperchaotic Lorenz systems by 537-548 (2006). active non-linear control, International Journal of Systems 92. J.A. Laoye, U.E. Vincent, and S.O. Kareem, Chaos control of Signal Control and Engineering Application, vol. 4, pp. 55-61 4D chaotic systems using recursive backstepping nonlinear (2011). controller, Chaos, Solitons and Fractals, vol. 39, pp. 356-362 115. S. Pakiriswamy and S. Vaidyanathan, Generalized projective (2009). synchronization of three-scroll chaotic systems via active 93. D. Lin, X. Wang, F. Nian, and Y. Zhang, Dynamic fuzzy neural control, Lecture Notes of the Institute for Computer Sciences, networks modeling and adaptive backstepping tracking control Social-Informatics and Telecommunications Engineering, vol. of uncertain chaotic systems, Neurocomputing, vol. 73, pp. 85, pp. 146-155 (2012). 2873-2881 (2010). 116. V. Sundarapandian and R. Karthikeyan, Hybrid synchronization 94. S. Vaidyanathan, Sliding mode control based global chaos of hyperchaotic Lorenz and hyperchaotic Chen systems via control of Liu-Liu-Liu-Su chaotic system, International Journal active control, Journal of Engineering and Applied Sciences, vol. of Control Theory and Applications, vol. 5, pp. 15-20 (2012). 7, pp. 254-264 (2012). 95. S. Vaidyanathan, Global chaos control of hyperchaotic Liu 117. R. Karthikeyan and V. Sundarapandian, Hybrid chaos system via sliding mode control, vol. 5, pp. 117-123 (2012). synchronization of four-scroll systems via active control, Journal 96. L. Kocarev and U. Parlitz, General approach for chaos of Electrical Engineering, vol. 65, pp. 97-103 (2014). synchronization with applications to communications, Physical 118. E.M. Shahverdiev and K.A. Shore, Impact of modulated Review Letters, vol. 74, pp. 5028-5030 (1995). multiple optical feedback time delays on laser diode chaos 97. K. Murali and M. Lakshmanan, Secure communication using a synchronization, Optics Communications, vol. 282, pp. 3568- compound signal using sampled-data feedback, Applied 3572 (2009). Mathematics and Mechanics, vol. 11, pp. 1309-1315, (1995). 119. T. Botmart, P. Niamsup, and X. Liu, Synchronization of non- 98. M. Feki, An adaptive chaos synchronization scheme applied to autonomous chaotic systems with time-varying delay via secure communication, Chaos, Solitons and Fractals, vol. 18, pp. delayed feedback control, Communications in Nonlinear Science 141-148 (2003). and Numerical Simulation, vol. 17, pp. 1894-1907 (2012). 99. J. Yang and F. Zhu, Synchronization for chaotic systems and 120. S. Bowong, Adaptive synchronization between two different chaos-based secure communications via both reduced-order and chaotic dynamical systems, Communications in Nonlinear step-by-step sliding mode observers, Communications in Science and Numerical Simulation, vol. 12, pp. 976-985 (2007). Nonlinear Science and Numerical Simulation, vol. 18, pp. 926- 121. W. Lin, Adaptive chaos control and synchronization in only 937 (2013). locally Lipschitz systems, Physics Letters A, vol. 372, pp. 3195- 100. L. Kocarev, Chaos-based cryptography: a brief overview, IEEE 3200 (2008). Circuits and Systems, vol. 1, pp. 6-21 (2001). 122. H. Salarieh and A. Alasty, Adaptive chaos synchronization in 101. H. Gao, Y. Zhang, S. Liang and D. Li, A new chaotic algorithm Chua’s systems with noisy parameters, Mathematics and for image encryption, Chaos, Solitons and Fractals, vol. 29, pp. Computers in Simulation, vol. 79, pp. 233-241 (2008). 393-399 (2006). 123. H. Salarieh and A. Alasty, Adaptive synchronization of two 102. Y. Wang, K.W. Wang, X. Liao and G. Chen, A new chaos-based chaotic systems with stochastic unknown parameters, fast image encryption, Applied Soft Computing, vol. 11, pp. Communications in Nonlinear Science and Numerical 514-522, (2011). Simulation, vol. 14, pp. 508-519 (2009). 103. Ch. K. Volos, I. M. Kyprianidis, and I. N. Stouboulos, Image 124. S. Vaidyanathan and K. Rajagopal, Global chaos encryption process based on chaotic synchronization synchronization of Lü and Pan systems by adaptive nonlinear phenomena, Signal Processing, vol. 93(5), pp. 1328-1340 control, Communications in Computer and Information Science, (2013). vol. 205, pp. 193-202 (2011). 104. Ch. K. Volos, Image encryption using the coexistence of chaotic 125. V. Sundarapandian and R. Karthikeyan, Anti-synchronization of synchronization phenomena, Journal of Applied Mathematics Lü and Pan chaotic systems by adaptive nonlinear control, and Bioinformatics, vol. 3(1), pp. 123-149 (2013). European Journal of Scientific Research, vol. 64, pp. 94-106 105. Y. Xu, H. Wang, Y. Li and B. Pei, Image encryption based on (2011). synchronization of fractional chaotic systems, Communications 126. V. Sundarapandian and R. Karthikeyan, Anti-synchronization of in Nonlinear Science and Numerical Simulation, vol. 19(10), pp. hyperchaotic Lorenz and hyperchaotic Chen systems by adaptive 3735-3744, (2014). control, International Journal of Systems Signal Control and 106. L.M. Pecora and T.L. Carroll, Synchronization in chaotic Engineering Application, vol. 4, pp. 18-25 (2011). systems, vol. 64, pp. 821-825 (1990). 127. V. Sundarapandian and R. Karthikeyan, Adaptive anti- 107. S. Vaidyanathan and S. Rasappan, New results on the global synchronization of uncertain Tigan and Li systems, Journal of chaos synchronization for Liu-Chen-Liu and Lü chaotic systems, Engineering and Applied Sciences, vol. 7, pp. 45-52 (2012). Communications in Computer and Information Science, vol. 128. P. Sarasu and V. Sundarapandian, Generalized projective 102, pp. 20-27 (2010). synchronization of three-scroll chaotic systems via adaptive 108. S. Vaidyanathan and S. Rasappan, Hybrid synchronization of control, European Journal of Scientific Research, vol. 72, pp. hyperchaotic Qi and Lü systems by nonlinear control, 504-522 (2012). Communications in Computer and Information Science, vol. 129. P. Sarasu and V. Sundarapandian, Generalized projective 131, pp. 585-593 (2011). synchronization of two-scroll systems via adaptive control, 109. S. Vaidyanathan and K. Rajagopal, Anti-synchronization of Li International Journal of Soft Computing, vol. 7, pp. 146-156 and T chaotic systems by active nonlinear control, (2012). Communications in Computer and Information Science, vol. 130. P. Sarasu and V. Sundarapandian, Adaptive controller design for 198, pp. 175-184 (2011). the generalized projective synchronization of 4-scroll systems, 110. S. Vaidyanathan and S. Rasappan, Global chaos synchronization International Journal of Systems Signal Control and Engineering of hyperchaotic Bao and Xu systems by active nonlinear control, Application, vol. 5, pp. 21-30 (2012). Communications in Computer and Information Science, vol. 131. S. Vaidyanathan and K. Rajagopal, Global chaos 198, pp. 10-17 (2011). synchronization of hyperchaotic Pang and hyperchaotic Wang 111. S. Vaidyanathan and K. Rajagopal, Global chaos systems via adaptive control, International Journal of Soft synchronization of hyperchaotic Pang and Wang systems by Computing, vol. 7, pp. 28-37 (2012).

114

S. Vaidyanathan /Journal of Engineering Science and Technology Review 8 (2) (2015) 106 -115

132. S.H. Lee, V. Kapila, M. Porfiri, and A. Panda, Master-slave 141. S. Vaidyanathan and S. Rasappan, Global chaos synchronization synchronization of continuously and intermittently coupled of n-scroll Chua circuit and Lur’e system using backstepping sampled-data chaotic oscillators, Communications in Nonlinear control design with recursive feedback, Arabian Journal for Science and Numerical Simulation, vol. 15, pp. 4100-4113 Science and Engineering, vol. 39, pp. 3351-3364 (2014). (2010). 142. S. Rasappan and S. Vaidyanathan, Global chaos synchronization 133. X.Z. Jin and J.H. Park, Adaptive synchronization for a class of of WINDMI and Coullet chaotic systems using adaptive faulty and sampling coupled networks with its circuit implement, backstepping control design, Kyungpook Mathematical Journal, Journal of the Franklin Institute, vol. 351, pp. 4317-4333 (2014). vol. 54, pp. 293-320 (2014). 134. C.K. Zhang, L. Jiang, Y. He, Q.H. Wu and M. Wu, 143. H.T. Yau, Chaos synchronization of two uncertain chaotic Asymptotical synchronization for chaotic Lur’e systems using nonlinear gyros using fuzzy sliding mode control, Mechanical sampled-data control, Communications in Nonlinear Science and Systems and Signal Processing, vol. 22, pp. 408-418 (2008). Numerical Simulation, vol. 18, pp. 2743-2751 (2013). 144. H. Li, X. Liao, C. Li, and C. Li, Chaos control and 135. X. Xiao, L. Zhou, and Z. Zhang, Synchronization of chaotic synchronization via a novel chatter free sliding mode control Lur’e systems with quantized sampled-data controller, strategy, Neurocomputing, vol. 74, pp. 3212-3222 (2012). Communications in Nonlinear Science and Numerical 145. S. Vaidyanathan and S. Sampath, Global chaos synchronization Simulation, vol. 19, pp. 2039-2047 (2014). of hyperchaotic Lorenz systems by sliding mode control, 136. S. Rasappan and S. Vaidyanathan, Global chaos synchronization Communications in Computer and Information Science, vol. of WINDMI and Coullet chaotic systems by backstepping 205, pp. 156-164 (2011). control, Far East Journal of Mathematical Sciences, vol. 67, pp. 146. V. Sundarapandian and S. Sivaperumal, Sliding controller 265-287 (2012). design of hybrid synchronization of four-wing chaotic systems, 137. S. Rasappan and S. Vaidyanathan, Synchronization of International Journal of Soft Computing, vol. 6, pp. 224-231 hyperchaotic Liu system via backstepping control with recursive (2011). feedback, Communications in Computer and Information 147. S. Vaidyanathan and S. Sampath, Anti-synchronization of four- Science, vol. 305, pp. 212-221 (2012). wing chaotic systems via sliding mode control, International 138. S. Rasappan and S. Vaidyanathan, Hybrid synchronization of n- Journal of Automation and Computing, vol. 9, pp. 274-279 scroll Chua and Lur’e chaotic systems via backstepping control (2012). with novel feedback, Archives of Control Sciences, vol. 22, pp. 148. H.T. Yau, Chaos synchronization of two uncertain chaotic 343-365 (2012). nonlinear gyros using fuzzy sliding mode control, Mechanical 139. R. Suresh and V. Sundarapandian, Global chaos synchronization Systems and Signal Processing, vol. 22(2), pp. 408-418 (2013). of a family of n-scroll hyperchaotic Chua circuits using 149. S. Vaidyanathan, Global chaos synchronisation of identical Li- backstepping control with recursive feedback, Far East Journal Wu chaotic systems via sliding mode control, International of Mathematical Sciences, vol. 73, pp. 73-95 (2013). Journal of Modelling, Identification and Control, vol. 22, no. 2, pp. 170-177 (2014). 140. S. Rasappan and S. Vaidyanathan, Hybrid synchronization of 150. H.K. Khalil, Nonlinear System, 3rd ed., Prentice Hall, New n-scroll chaotic Chua circuits using adaptive backstepping Jersey, USA (2002). control design with recursive feedback, Malaysian Journal of Mathematical Sciences, vol. 7, pp. 219-246 (2013).

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