Hybrid Synchronization of Identical Hyperchaotic Systems via Novel Sliding Control with Application to Hyperchaotic Vaidyanathan- Volos System

Transcription

1 I J C T A, 9(38), 2016, pp Iteratioal Sciece Press Hybrid Sychroizatio of Idetical Hyperchaotic Systems via Novel Slidig Cotrol with Applicatio to Hyperchaotic Vaidyaatha- Volos System S. Sivaperumal* ABSTRACT First, this paper proposes a geeral procedure for the hybrid sychroizatio of idetical hyperchaotic systems usig ovel slidig mode cotrol. The geeral result derived usig ovel slidig mode cotrol method is established via Lyapuov stability theory. As a applicatio of the geeral result, the problem of hybrid sychroizatio of idetical hyperchaotic Vaidyaatha-Volos systems (2016) is studied ad a ew slidig mode cotroller is derived. Numerical simulatios with MATLAB have bee show to illustrate all the mai results derived i this work. Keywords: Hyperchaos, hyperchaos sychroizatio, hybrid sychroizatio, slidig mode cotrol. 1. INTRODUCTION A chaotic system is commoly defied as a oliear dissipative dyamical system that is highly sesitive to eve small perturbatios i its iitial coditios [1]. I other words, a chaotic system is a oliear dyamical system with at least oe positive Lyapuov expoet. Some paradigms of chaotic systems ca be listed as Areodo system [4], Sprott systems [5], Che system [6], Lü-Che system [7], Liu system [8], Cai system [9], Tiga system [10], etc. I the last two decades, may ew chaotic systems have bee also discovered like Li system [11], Sudarapadia systems [12-13], Vaidyaatha systems [14-33], Pehliva systems [34-35], Pham systems [36-37], Jafari system [38], etc. Hyperchaotic systems are the chaotic systems with more tha oe positive Lyapuov expoet. They have importat applicatios i cotrol ad commuicatio egieerig. Some recetly discovered 4-D hyperchaotic systems are hyperchaotic Vaidyaatha systems [39-40], hyperchaotic Vaidyaatha-Azar system [41], hyperchaotic Sampath system [42], etc, Chaos theory has several applicatios i a variety of fields such as oscillators [43-44], chemical reactors [45-58], biology [59-80], ecology [81-82], eural etworks [83-84], robotics [85-86], memristors [87-89], fuzzy systems [90-91], etc. The problem of cotrol of a chaotic system is to fid a state feedback cotrol law to stabilize a chaotic system aroud its ustable equilibrium [92-93]. Some popular methods for chaos cotrol are active cotrol [94-98], adaptive cotrol [99-100], slidig mode cotrol [ ], etc. If a particular chaotic system is called the master or drive system ad aother chaotic system is called the slave or respose system, the the idea of hybrid sychroizatio is to completely sychroize oe part * School of Electrical ad Computig, Vel Tech Uiversity, Avadi, Cheai, Tamil Nadu, Idia, sivaperumals@gmail.com.

2 262 S. Sivaperumal of the systems (amely, the odd states) ad ati-sychroize the other part of the systems (amely, the eve states) so that both complete ad ati-sychroizatio persist i the process of sychroizatio of the master ad slave systems. The sychroizatio of chaotic systems has applicatios i secure commuicatios [ ], cryptosystems [ ], ecryptio [ ], etc. The chaos sychroizatio problem has bee paid great attetio i the literature ad a variety of impressive approaches have bee proposed. Sice the pioeerig work by Pecora ad Carroll [ ] for the chaos sychroizatio problem, may differet methods have bee proposed i the cotrol literature such as active cotrol method [ ], adaptive cotrol method [ ], sampled-data feedback cotrol method [ ], time-delay feedback approach [152], backsteppig method [ ], slidig mode cotrol method [ ], etc. I this paper, ew results have bee derived for the hybrid sychroizatio of idetical hyperchaotic systems usig ovel slidig cotrol. The slidig mode cotrol has advatages of low sesitivity to parameter variatios i the plat ad disturbaces affectig the plat. I Sectio 2, we describe the hybrid sychroizatio of idetical hyperchaotic systems. I Sectio 3, we derive a geeral result for the hybrid sychroizatio of idetical hyperchaotic systems usig ovel slidig mode cotrol. I Sectio 4, we describe the hyperchaotic Vaidyaatha-Volos system ([42], 2016) ad its qualitative properties. The phase portraits of the hyperchaotic Vaidyaatha-Volos system are described usig MATLAB. I Sectio 5, we describe the slidig mode cotroller desig for the hybrid sychroizatio of the hyperchaotic Vaidyaatha-Volos systems usig ovel slidig mode cotrol ad its umerical simulatios usig MATLAB. Sectio 6 cotais a summary of the mai results derived i this paper. 2. HYBRID SYNCHRONIZATION OF IDENTICAL HYPERCHAOTIC SYSTEMS I this sectio, we provide a problem statemet for the hybrid sychroizatio of idetical hyperchaotic systems. As the master or drive system, we cosider the hyperchaotic system give by x Ax ( x) (1) I Eq. (1), x R deotes the state of the system, A deotes the matrix of system parameters ad cotais the oliear parts of the system. As the slave or respose system, we take the cotrolled hyperchaotic system give by I Eq. (2), y R deotes the state of the system ad y Ay ( y) u (2) u R is the cotrol. The hybrid sychroizatio error betwee the hyperchaotic systems (1) ad (2) is defied as yi xi if i is odd ei yi xi if i is eve A simple calculatio yields the error dyamics as (3) e Ae ( x, y) u (4) Thus, the hybrid sychroizatio problem for the hyperchaotic systems (1) ad (2) ca be defied as follows: Fid a cotroller u so as to reder the hybrid sychroizatio error e(t) to be globally asymptotically stable for all values of e(0) R, i.e.

3 Hybrid Sychroizatio of Idetical Hyperchaotic Systems via Novel Slidig Cotrol lim e( t) 0 for all e(0) R. (5) t 3. A NOVEL SLIDING CONTROLLER DESIGN First, we set the desig by settig the cotrol as I Eq. (6), u( t) ( x, y) Bv( t) (6) B R is chose such that (A, B) is completely cotrollable. By substitutig (6) ito (4), we get the closed-loop error dyamics e Ae Bv (7) The system (7) is a liear time-variat cotrol system with sigle iput v. We start the slidig cotroller desig by defiig the slidig variable as s( e) Ce c e c e c e, (8) where 1 C R is a costat vector to be determied. The slidig maifold is defied as the hyperplae We shall assume that a slidig motio occurs o the hyperplae S. I slidig mode, the followig equatios must be satisfied: We assume that S e R : s( e) Ce 0. (9) s 0 ad s CAe CBv 0 (10) CB 0 (11) The slidig motio is iflueced by the equivalet cotrol derived from (10) as v t CB CAe t (12) ( ) ( ) 1 ( ) eq By substitutig (12) ito (7), we obtai the equivalet error dyamics i the slidig phase as where 1 e Ae ( CB) CAe Ee, (13) 1 E [ I B( CB) C] A (14) We ote that E is idepedet of the cotrol ad has at most ( 1) ozero eigevalues, depedig o the chose switchig surface, while the associated eigevectors belog to ker(c) Sice (A, B) is cotrollable, we ca use slidig cotrol theory to choose B ad C so that E has ay desired ( 1) stable eigevalues. This shows that the dyamics i the slidig mode is globally asymptotically stable. Fially, for the slidig cotroller desig, we apply a ovel slidig cotrol law, viz. s ks qs 2 sg( s) (15) I Eq. (15), sg(.) deotes the sig fuctio ad the slidig mode cotrol costats k > 0, q > 0 are foud i such a way that the slidig coditio is satisfied ad that the slidig motio will occur.

4 264 S. Sivaperumal as By combiig equatios (10), (12) ad (15), we fially obtai the slidig mode cotrol (SMC) v(t) Next, we establish the mai result of this sectio. 1 2 v( t) ( CB) [ C( ki A) e qs sg( s)] (16) Theorem 1. The slidig mode cotroller law defied by (6) achieves global ad asymptotic hybrid sychroizatio of the idetical hyperchaotic systems (1) ad (2) for all iitial coditios x(0), y(0) R, where v is defied by the ovel slidig cotrol law (16), 1 B R is such that (A, B) is cotrollable, 1 C R is such that CB 0 ad that the matrix E defied by (14) has ( 1) stable eigevalues. Proof. Upo substitutio of the cotrol laws (6) ad (16) ito the error dyamics (4), we get the closedloop error dyamics as 1 2 e Ae B( CB) C( ki A) e qs sg( s) We shall show that the error system (17) is globally asymptotically stable by cosiderig the quadratic Lyapuov fuctio The slidig mode motio is characterized by the equatios (17) 1 2 V ( e) s ( e) (18) 2 s( e) 0 ad s ( e) 0 (19) By the choice of E, the dyamics i the slidig mode is globally asymptotically stable. Whe s( e) 0, V ( e) 0. Also, whe s( e) 0, differetiatig V alog the error dyamics (17) or the equivalet dyamics (15), we get 2 3 V ( e) ss ks qs sg( s) 0. (20) Hece, by Lyapuov stability theory [174], the error dyamics (17) is globally asymptotically stable for all e(0) R. This completes the proof. 4. HYPERCHAOTIC VAIDYANATHAN-VOLOS SYSTEM Hyperchaotic Vaidyaatha-Volos system ([42], 2016) is described by the 4-D dyamics x 1 a( x2 x1 ) x4 x bx px x x x 3 x1x2 cx3 x 4 x1 x (21) where x1, x2, x3, x 4 are state variables ad a, b, c, p are positive parameters of the system. The Vaidyaatha-Volos system (21) exhibits a hyperchaotic attractor for the values

5 Hybrid Sychroizatio of Idetical Hyperchaotic Systems via Novel Slidig Cotrol a 12, b 36, c 5, p 12 (22) The Lyapuov expoets of the system (21) are umerically obtaied with MATLAB as L , L , L 0, L (23) As there are two positive Lyapuov expoets i (23), the system (21) is hyperchaotic. Sice the sum of the Lyapuov expoets i (23) is egative, the hyperchaotic Vaidyaatha-Volos system is dissipative. The Lyapuov dimesio of the hyperchaotic Vaidyaatha-Volos system (21) is foud as D L L L L L4 (24) For simulatios, the iitial values of the Vaidyaatha-Volos system (21) are take as x (0) 1.8, x (0) 1.6, x (0) 1.2, x (0) 1.4 (25) Figures 1-4 show the 3-D view of the hyperchaotic attractor of the Vaidyaatha-Volos system (21) i ( x1, x2, x 3), ( x1, x2, x4), ( x1, x3, x4) ad ( x2, x3, x 4) spaces, respectively. Figure 1: 3-D view of hyperchaotic Vaidyaatha-Volos system i (x 1, x 2, x 3 ) space

6 266 S. Sivaperumal Figure 2: 3-D view of hyperchaotic Vaidyaatha-Volos system i (x1, x2, x4) space Figure 3: 3-D view of hyperchaotic Vaidyaatha-Volos system i (x1, x3, x4) space

7 Hybrid Sychroizatio of Idetical Hyperchaotic Systems via Novel Slidig Cotrol Figure 4: 3-D view of hyperchaotic Vaidyaatha-Volos system i (x 2, x 3, x 4 ) space 5. HYBRID SYNCHRONIZATION OF HYPERCHAOTIC VAIDYANATHAN-VOLOS SYSTEMS VIA NOVEL SLIDING CONTROL I this sectio, we describe ovel slidig mode cotrol desig for the hybrid sychroizatio of idetical hyperchaotic Vaidyaatha-Volos systems. As the master system, we cosider the hyperchaotic Vaidyaatha-Volos system give by x 1 a( x2 x1 ) x4 x bx px x x x 3 x1x2 cx3 x 4 x1 x (26) where x 1, x 2, x 3, x 4 are state variables ad a, b, c, d, p, q are positive parameters. As the slave system, we cosider the hyperchaotic Vaidyaatha-Volos system give by y 1 a( y2 y1) y4 u1 y by py y y u y 3 y1 y2 cy3 u3 y 4 y1 y2 u (27) where y 1, y 2, y 3, y 4 are state variables ad u 1, u 2, u 3, u 4 are the cotrols.

8 268 S. Sivaperumal The hybrid sychroizatio error betwee the systems (26) ad (27) is defied by e1 y1 x1 e y x e y x e y x (28) The the error dyamics is obtaied as e 1 a( e2 e1 ) e4 2ax2 2x4 u1 e be e 2 bx p( y y x x ) u e 3 ce3 y1 y2 x1 x2 u3 e 4 e1 e2 2x1 u (29) I matrix form, we ca write the error dyamics (29) as where e Ae ( x, y) u, (30) a a 0 1 2ax2 2x4 u1 b bx1 p( y1 y3 x1 x3 ) u, (, ), 2 A x y u 0 0 c 0 y1 y2 x1 x 2 u x u 1 4 (31) First, we set u as where B is selected such that (A, B) is completely cotrollable. We choose B as u( t) ( x, y) Bv( t), (32) 1 1 B 1 1 We choose the parameters of the Vaidyaatha-Volos systems as i the hyperchaotic case, viz. The slidig mode variable is selected as (33) a 12, b 36, c 5, p 12 (34) s Ce e e e e e (35) which reders the slidig motio globally asymptotically stable. Next, we take the slidig mode gais as

9 Hybrid Sychroizatio of Idetical Hyperchaotic Systems via Novel Slidig Cotrol From Eq. (16) of Sectio 3, we obtai the ovel slidig cotrol v as k 6 ad q 0.2. (36) v e e e e s s (37) sg( ) As a applicatio of Theorem 1 to the idetical hyperchaotic Vaidyaatha-Volos systems, we obtai the followig mai result of this sectio. Theorem 2. The idetical hyperchaotic Vaidyaatha-Volos systems (26) ad (27) are globally ad 4 asymptotically hybrid sychroized for all iitial coditios x(0), y(0) R with the slidig cotroller u defied by (32), where ( x, y) ad B are defied by (31) adis defied by (37). For umerical simulatio, we take the parameter values as i the hyperchaotic case, i.e. a 12, b 36, c 5, p 12 As a iitial coditio for the master system (25), we take x (0) 5.2, x (0) 2.3, x (0) 4.5, x (0) As a iitial coditio for the slave system (26), we take y (0) 1.4, y (0) 5.8, y (0) 2.6, y (0) Figures 5-8 depict the hybrid sychroizatio of the hyperchaotic Vaidyaatha-Volos systems. Figure 9 depicts the time-history of the hybrid sychroizatio errors. Figure 5: Hybrid sychroizatio of the states x 1 ad y 1

10 270 S. Sivaperumal Figure 6: Hybrid sychroizatio of the states x 2 ad y 2 Figure 7: Hybrid sychroizatio of the states x 3 ad y 3

11 Hybrid Sychroizatio of Idetical Hyperchaotic Systems via Novel Slidig Cotrol Figure 8: Hybrid sychroizatio of the states x 4 ad y 4 Figure 9: Time history of the hybrid sychroizatio errors

12 272 S. Sivaperumal 6. CONCLUSIONS I this paper, a ovel slidig mode cotroller has bee desiged for the hybrid sychroizatio of idetical hyperchaotic systems. Lyapuov stability theory has bee used to prove this mai result of the work. Next, as a applicatio of the mai result, a slidig cotroller has bee desiged for achievig hyperchaos hybrid sychroizatio of idetical hyperchaotic Vaidyaatha-Volos systems (2016). Numerical simulatios usig MATLAB have bee provided to illustrate phase portraits of the hyperchaotic Vaidyaatha-Volos system ad the ovel slidig mode cotroller for the hyperchaos hybrid sychroizatio of idetical hyperchaotic Vaidyaatha-Volos systems. REFERENCES [1] A.T. Azar ad S. Vaidyaatha, Chaos Modelig ad Cotrol Systems Desig, Spriger, New York, USA, [2] E.N. Lorez, Determiistic operiodic flow, Joural of the Atmospheric Scieces, 20, , [3] O.E. Rössler, A equatio for cotiuous chaos, Physics Letters A, 57, , [4] A. Areodo, P. Coullet ad C. Tresser, Possible ew strage attractors with spiral structure, Commuicatios i Mathematical Physics, 79, , [5] J.C. Sprott, Some simple chaotic flows, Physical Review E, 50, , [6] G. Che ad T. Ueta, Yet aother chaotic attractor, Iteratioal Joural of Bifurcatio ad Chaos, 9, , [7] J. Lü ad G. Che, A ew chaotic attractor coied, Iteratioal Joural of Bifurcatio ad Chaos, 12, , [8] C.X. Liu, T. Liu, L. Liu ad K. Liu, A ew chaotic attractor, Chaos, Solitos ad Fractals, 22, , [9] G. Cai ad Z. Ta, Chaos sychroizatio of a ew chaotic system via oliear cotrol, Joural of Ucertai Systems, 1, , [10] G. Tiga ad D. Opris, Aalysis of a 3D chaotic system, Chaos, Solitos ad Fractals, 36, , [11] D. Li, A three-scroll chaotic attractor, Physics Letters A, 372, , [12] V. Sudarapadia ad I. Pehliva, Aalysis, cotrol, sychroizatio ad circuit desig of a ovel chaotic system, Mathematical ad Computer Modellig, 55, , [13] V. Sudarapadia, Aalysis ad ati-sychroizatio of a ovel chaotic system via active ad adaptive cotrollers, Joural of Egieerig Sciece ad Techology Review, 6, 45-52, [14] S. Vaidyaatha ad K. Madhava, Aalysis, adaptive cotrol ad sychroizatio of a seve-term ovel 3-D chaotic system, Iteratioal Joural of Cotrol Theory ad Applicatios, 6, , [15] S. Vaidyaatha, A ew six-term 3-D chaotic system with a expoetial oliearity, Far East Joural of Mathematical Scieces, 79, , [16] S. Vaidyaatha, Aalysis ad adaptive sychroizatio of two ovel chaotic systems with hyperbolic siusoidal ad cosiusoidal oliearity ad ukow parameters, Joural of Egierig Sciece ad Techology Review, 6, 53-65, [17] S. Vaidyaatha, A ew eight-term 3-D polyomial chaotic system with three quadratic oliearities, Far East Joural of Mathematical Scieces, 84, , [18] S. Vaidyaatha, Aalysis, cotrol ad sychroisatio of a six-term ovel chaotic system with three quadratic oliearities, Iteratioal Joural of Modellig, Idetificatio ad Cotrol, 22, 41-53, [19] S. Vaidyaatha, C. Volos, V.-T. Pham, K. Madhava ad B.A. Idowu, Adaptive backsteppig cotrol, sychroizatio ad circuit simulatio of a 3-D ovel jerk chaotic system with two hyperbolic siusoidal oliearities, Archives of Cotrol Scieces, 24, , [20] S. Vaidyaatha, Aalysis ad adaptive sychroizatio of eight-term 3-D polyomial chaotic systems with three quadratic oliearities, Europea Physical Joural: Special Topics, 223, , [21] S. Vaidyaatha, Geeralised projective sychroisatio of ovel 3-D chaotic systems with a expoetial oliearity via active ad adaptive cotrol, Iteratioal Joural of Modellig, Idetificatio ad Cotrol, 22, , [22] S. Vaidyaatha, Qualitative aalysis, adaptive cotrol ad sychroizatio of a seve-term ovel 3-D chaotic system with a quartic oliearity, Iteratioal Joural of Cotrol Theory ad Applicatios, 7, 1-20, 2014.

13 Hybrid Sychroizatio of Idetical Hyperchaotic Systems via Novel Slidig Cotrol [23] S. Vaidyaatha, C.K. Volos ad V.-T. Pham, Global chaos cotrol of a ovel ie-term chaotic system via slidig mode cotrol, Studies i Comptuatioal Itelligece, 576, , [24] S. Vaidyaatha ad A.T. Azar, Aalysis, cotrol ad sychroizatio of a ie-term 3-D ovel chaotic system, Studies i Computatioal Itelligece, 581, 19-38, [25] S. Vaidyaatha, Aalysis, properties ad cotrol of a eight-term 3-D chaotic system with a expoetial oliearity, Iteratioal Joural of Modellig, Idetificatio ad Cotrol, 23, , [26] S. Vaidyaatha, A 3-D ovel highly chaotic system with four quadratic oliearities, its adaptive cotrol ad atisychroizatio with ukow parameters, Joural of Egieerig Sciece ad Techology Review, 8, , [27] S. Vaidyaatha, K. Rajagopal, C.K. Volos, I.M. Kypriaidis ad I.N. Stouboulos, Aalysis, adaptive cotrol ad sychroizatio of a seve-term ovel 3-D chaotic system with three quadratic oliearities ad its digital implemetatio i LabVIEW, Joural of Egieerig Sciece ad Techology Review, 8, , [28] S. Sampath, S. Vaidyaatha, C.K. Volos ad V.-T. Pham, A eight-term ovel four-scroll chaotic system with cubic oliearity ad its circuit simulatio, Joural of Egieerig Sciece ad Techology Review, 8, 1-6, [29] S. Vaidyaatha ad S. Pakiriswamy, A 3-D ovel coservative chaotic system ad its geeralized projective sychroizatio via adaptive cotrol, Joural of Egieerig Sciece ad Techology Review, 8, 52-60, [30] S. Vaidyaatha, C.K. Volos, I.M. Kypriaidis, I.N. Stouboulos ad V.-T. Pham, Aalysis, adaptive cotrol ad atisychroizatio of a six-term ovel jerk chaotic system with two expoetial oliearities ad its circuit simulatio, Joural of Egieerig Sciece ad Techology Review, 8, 24-36, [31] S. Vaidyaatha, C.K. Volos ad V.-T. Pham, Aalysis, adaptive cotrol ad adaptive sychroizatio of a ie-term ovel 3-D chaotic system with four quadratic oliearities ad its circuit simulatio, Joural of Egieerig Sciece ad Techology Review, 8, , [32] S. Vaidyaatha ad C. Volos, Aalysis ad adaptive cotrol of a ovel 3-D coservative o-equilibrium chaotic system, Archives of Cotrol Scieces, 25, , [33] S. Vaidyaatha, Aalysis, cotrol ad sychroizatio of a 3-D ovel jerk chaotic system with two quadratic oliearities, Kyugpook Mathematical Joural, 55, , [34] V. Sudarapadia ad I. Pehliva, Aalysis, cotrol, sychroizatio ad circuit desig of a ovel chaotic system, Mathematical ad Computer Modellig, 55, , [35] I. Pehliva, I.M. Moroz ad S. Vaidyaatha, Aalysis, sychroizatio ad circuit desig of a ovel butterfly attractor, Joural of Soud ad Vibratio, 333, , [36] V.-T. Pham, C.K. Volos ad S. Vaidyaatha, Multi-scroll chaotic oscillator based o a first-order delay differetial equatio, Studies i Computatioal Itelligece, 581, 59-72, [37] V.-T. Pham, S. Vaidyaatha, C.K. Volos ad S. Jafari, Hidde attractors i a chaotic system with a expoetial oliear term, Europea Physical Joural: Special Topics, 224, , [38] S. Jafari ad J.C. Sprott, Simple chaotic flows with a lie equilibrium, Chaos, Solitos ad Fractals, 57, 79-84, [39] S. Vaidyaatha, A te-term ovel 4-D hyperchaotic system with three quadratic oliearities ad its cotrol, Iteratioal Joural of Cotrol Theory ad Applicatios, 6, , [40] S. Vaidyaatha, Qualitative aalysis ad cotrol of a eleve-term ovel 4-D hyperchaotic system with two quadratic oliearities, Iteratioal Joural of Cotrol Theory ad Applicatios, 7, 35-47, [41] S. Vaidyaatha ad A.T. Azar, Aalysis ad cotrol of a 4-D ovel hyperchaotic system, Studies i Computatioal Itelligece, 581, 3-17, [42] S. Vaidyaatha, C.K. Volos ad V.T. Pham, Adaptive cotrol ad circuit simulatio of a ovel 4-D hyperchaotic system with two quadratic oliearities, Studies i Computatioal Itelligece, 636, , [43] M. Lakshmaa ad K. Murali, Chaos i Noliear Oscillators: Cotrollig ad Sychroizatio, World Scietific: Sigapore, [44] S.K. Ha, C. Kerrer ad Y. Kuramoto, Dephasig ad burstig i coupled eural oscillators, Physical Review Letters, 75, , [45] S. Vaidyaatha, Adaptive sychroizatio of chemical chaotic reactors, Iteratioal Joural of ChemTech Research, 8 (2), , [46] S. Vaidyaatha, Adaptive cotrol of a chemical chaotic reactor, Iteratioal Joural of PharmTech Research, 8 (3), , [47] S. Vaidyaatha, Dyamics ad cotrol of Brusselator chemical reactio, Iteratioal Joural of ChemTech Research, 8 (6), , 2015.

14 274 S. Sivaperumal [48] S. Vaidyaatha, Ati-sychroizatio of Brusselator chemical reactio systems via adaptive cotrol, Iteratioal Joural of ChemTech Research, 8 (6), , [49] S. Vaidyaatha, Dyamics ad cotrol of Tokamak system with symmetric ad magetically cofied plasma, Iteratioal Joural of ChemTech Research, 8 (6), , [50] S. Vaidyaatha, Sychroizatio of Tokamak systems with symmetric ad magetically cofied plasma via adaptive cotrol, Iteratioal Joural of ChemTech Research, 8 (6), , [51] S. Vaidyaatha, A ovel chemical chaotic reactor system ad its adaptive cotrol, Iteratioal Joural of ChemTech Research, 8 (7), , [52] S. Vaidyaatha, Adaptive sychroizatio of ovel 3-D chemical chaotic reactor systems, Iteratioal Joural of ChemTech Research, 8 (7), , [53] S. Vaidyaatha, Global chaos sychroizatio of chemical chaotic reactors via ovel slidig mode cotrol method, Iteratioal Joural of ChemTech Research, 8 (7), , [54] S. Vaidyaatha, Slidig mode cotrol of Rucklidge chaotic system for oliear double covectio, Iteratioal Joural of ChemTech Research, 8 (8), 25-35, [55] S. Vaidyaatha, Global chaos sychroizatio of Rucklidge chaotic systems for double covectio via slidig mode cotrol, Iteratioal Joural of ChemTech Research, 8 (8), 61-72, [56] S. Vaidyaatha, Ati-sychroizatio of chemical chaotic reactors via adaptive cotrol method, Iteratioal Joural of ChemTech Research, 8 (8), 73-85, [57] S. Vaidyaatha, Adaptive sychroizatio of Rikitake two-disk dyamo chaotic systems, Iteratioal Joural of ChemTech Research, 8 (8), , [58] S. Vaidyaatha, Adaptive cotrol of Rikitake two-disk dyamo system, Iteratioal Joural of ChemTech Research, 8 (8), , [59] S. Vaidyaatha, Adaptive backsteppig cotrol of ezymes-substrates system with ferroelectric behaviour i brai waves, Iteratioal Joural of PharmTech Research, 8 (2), , [60] S. Vaidyaatha, Adaptive biological cotrol of geeralized Lotka-Volterra three-species biological system, Iteratioal Joural of PharmTech Research, 8 (4), , [61] S. Vaidyaatha, 3-cells Cellular Neural Network (CNN) attractor ad its adaptive biological cotrol, Iteratioal Joural of PharmTech Research, 8 (4), , [62] S. Vaidyaatha, Adaptive sychroizatio of geeralized Lotka-Volterra three-species biological systems, Iteratioal Joural of PharmTech Research, 8 (5), , [63] S. Vaidyaatha, Sychroizatio of 3-cells Cellular Neural Network (CNN) attractors via adaptive cotrol method, Iteratioal Joural of PharmTech Research, 8 (5), , [64] S. Vaidyaatha, Chaos i euros ad adaptive cotrol of Birkhoff-Shaw strage chaotic attractor, Iteratioal Joural of PharmTech Research, 8 (5), , [65] S. Vaidyaatha, Adaptive chaos sychroizatio of ezymes-substrates system with ferroelectric behaviour i brai waves, Iteratioal Joural of PharmTech Research, 8 (5), , [66] S. Vaidyaatha, Lotka-Volterra populatio biology models with egative feedback ad their ecological moitorig, Iteratioal Joural of PharmTech Research, 8 (5), , [67] S. Vaidyaatha, Chaos i euros ad sychroizatio of Birkhoff-Shaw strage chaotic attractors via adaptive cotrol, Iteratioal Joural of PharmTech Research, 8 (6), 1-11, [68] S. Vaidyaatha, Lotka-Volterra two species competitive biology models ad their ecological moitorig, Iteratioal Joural of PharmTech Research, 8 (6), 32-44, [69] S. Vaidyaatha, Colema-Gomatam logarithmic competitive biology models ad their ecological moitorig, Iteratioal Joural of PharmTech Research, 8 (6), , [70] S. Vaidyaatha, Output regulatio of the forced Va der Pol chaotic oscillator via adaptive cotrol method, Iteratioal Joural of PharmTech Research, 8 (6), , [71] S. Vaidyaatha, Adaptive cotrol of the FitzHugh-Nagumo chaotic euro model, Iteratioal Joural of PharmTech Research, 8 (6), , [72] S. Vaidyaatha, Global chaos sychroizatio of the forced Va der Pol chaotic oscillators via adaptive cotrol method, Iteratioal Joural of PharmTech Research, 8 (6), , [73] S. Vaidyaatha, Adaptive sychroizatio of the idetical FitzHugh-Nagumo chaotic euro models, Iteratioal Joural of PharmTech Research, 8 (6), , 2015.

15 Hybrid Sychroizatio of Idetical Hyperchaotic Systems via Novel Slidig Cotrol [74] S. Vaidyaatha, Global chaos sychroizatio of the Lotka-Volterra biological systems with four competitive species via active cotrol, Iteratioal Joural of PharmTech Research, 8 (6), , [75] S. Vaidyaatha, Ati-sychroizatio of 3-cells cellular eural etwork attractors via adaptive cotrol method, Iteratioal Joural of PharmTech Research, 8 (7), 26-38, [76] S. Vaidyaatha, Active cotrol desig for the ati-sychroizatio of Lotka-Volterra biological systems with four competitive species, Iteratioal Joural of PharmTech Research, 8 (7), 58-70, [77] S. Vaidyaatha, Ati-sychroizatio of the FitzHugh-Nagumo chaotic euro models via adaptive cotrol method, Iteratioal Joural of PharmTech Research, 8 (7), 71-83, [78] S. Vaidyaatha, Slidig cotroller desig for the global chaos sychroizatio of ezymes-substrates systems, Iteratioal Joural of PharmTech Research, 8 (7), 89-99, [79] S. Vaidyaatha, Slidig cotroller desig for the global chaos sychroizatio of forced Va der Pol chaotic oscillators, Iteratioal Joural of PharmTech Research, 8 (7), , [80] S. Vaidyaatha, Lotka-Volterra two-species mutualistic biology models ad their ecological moitorig, Iteratioal Joural of PharmTech Research, 8 (7), , [81] B. Blasius, A. Huppert ad L. Stoe, Complex dyamics ad phase sychroizatio i spatially exteded ecological system, Nature, 399, , [82] I. Suárez, Masterig chaos i ecology, Ecological Modellig, 117, , [83] K. Aihira, T. Takabe ad M. Toyoda, Chaotic eural etworks, Physics Letters A, 144, , [84] I. Tsuda, Dyamic lik of memory chaotic memory map i oequilibrium eural etworks, Neural Networks, 5, , [85] S. Lakalapalli ad A. Ghosal, Chaos i robot cotrol equatios, Itertioal Joural of Bifurcatio ad Chaos, 7, , [86] Y. Nakamura ad A. Sekiguchi, The chaotic mobile robot, IEEE Trasactios o Robotics ad Automatio, 17, , [87] V.-T. Pham, C. K. Volos, S. Vaidyaatha ad V. Y. Vu, A memristor-based hyperchaotic system with hidde attractors: dyamics, sychroizatio ad circuital emulatig, Joural of Egieerig Sciece ad Techology Review, 8, , [88] C. K. Volos, I. M. Kypriaidis, I. N. Stouboulos, E. Tlelo-Cuautle ad S. Vaidyaatha, Memristor: A ew cocept i sychroizatio of coupled euromorphic circuits, Joural of Egieerig Sciece ad Techology Review, 8, , [89] V.-T. Pham, C. Volos, S. Jafari, X. Wag ad S. Vaidyaatha, Hidde hyperchaotic attractor i a ovel simple memristive eural etwork, Optoelectroics ad Advaced Materials, Rapid Commuicatios, 8, , [90] J.J. Buckley ad Y. Hayashi, Applicatios of fuzzy chaos to fuzzy simulatio, Fuzzy Sets ad Systems, 99, , [91] C.F. Hsu, Adaptive fuzzy wavelet eural cotroller desig for chaos sychroizatio, Expert Systems with Applicatios, 38, , [92] E. Ott, C. Grebogi ad J.A. Yorke, Cotrollig chaos, Physical Review Letters, 64, , [93] J. Wag, T. Zhag ad Y. Che, Chaos cotrol ad sychroizatio of two euros exposed to ELF exteral electric field, Chaos, Solitos ad Fractals, 34, , [94] V. Sudarapadia, Output regulatio of the Va der Pol oscillator, Joural of the Istitutio of Egieers (Idia): Electrical Egieerig Divisio, 88, 20-14, [95] V. Sudarapadia, Output regulatio of the Lorez attractor, Far East Joural of Mathematical Scieces, 42, , [96] S. Vaidyaatha, Output regulatio of the uified chaotic system, Commuicatios i Computer ad Iformatio Sciece, 198, 1-9, [97] S. Vaidyaatha, Output regulatio of Areodo-Coullet chaotic system, Commuicatios i Computer ad Iformatio Sciece, 133, , [98] S. Vaidyaatha, Output regulatio of the Liu chaotic system, Applied Mechaics ad Materials, 110, , [99] V. Sudarapadia, Adaptive cotrol ad sychroizatio of ucertai Liu-Che-Liu system, Iteratioal Joural of Computer Iformatio Systems, 3, 1-6, 2011.

16 276 S. Sivaperumal [100] V. Sudarapadia, Adaptive cotrol ad sychroizatio of the Shaw chaotic system, Iteratioal Joural i Foudatios of Computer Sciece ad Techology, 1, 1-11, [101] S. Vaidyaatha, Slidig mode cotrol based global chaos cotrol of Liu-Liu-Liu-Su chaotic system, Iteratioal Joural of Cotrol Theory ad Applicatios, 5, 15-20, [102] S. Vaidyaatha, Global chaos cotrol of hyperchaotic Liu system via slidig cotrol method, Iteratioal Joural of Cotrol Theory ad Applicatios, 5, , [103] S. Vaidyaatha, Global chaos sychroizatio of idetical Li-Wu chaotic systems via slidig mode cotrol, Iteratioal Joural of Web ad Grid Services, 22, , [104] M. Feki, A adaptive chaos sychroizatio scheme applied to secure commuicatio, Chaos, Solitos ad Fractals, 18, , [105] L. Kocarev ad U. Parlitz, Geeral approach for chaos sychroizatio with applicatios to commuicatios, Physical Review Letters, 74, , [106] K. Murali ad M. Lakshmaa, Secure commuicatio usig a compoud sigal usig sampled-data feedback, Applied Math. Mech., 11, , [107] J. Yag ad F. Zhu, Sychroizatio for chaotic systems ad chaos-based secure commuicatios via both reducedorder ad step-by-step slidig mode observers, Commuicatios i Noliear Sciece ad Numerical Simulatio, 18, , [108] L. Kocarev, Chaos-based cryptography: a brief overview, IEEE Circuits ad Systems, 1, 6-21, [109] H. Gao, Y. Zhag, S. Liag ad D. Li, A ew chaotic algorithm for image ecryptio, Chaos, Solitos ad Fractals, 29, , [110] Y. Wag, K.W. Wag, X. Liao ad G. Che, A ew chaos-based fast image ecryptio, Applied Soft Computig, 11, , [111] X. Zhag, Z. Zhao ad J. Wag, Chaotic image ecryptio based o circular substitutio box ad key stream buffer, Sigal Processig: Image Commuicatio, 29, , [112] L.M. Pecora ad T.I. Carroll, Sychroizatio i chaotic systems, Phys. Rev. Lett., 64, , [113] L.M. Pecora ad T.L. Carroll, Sychroizig i chaotic circuits, IEEE Tras. Circ. Sys., 38, , [114] L. Huag, R. Feg ad M. Wag, Sychroizatio of chaotic systems via oliear cotrol, Physics Letters A, 320, , [115] V. Sudarapadia ad R. Karthikeya, Global chaos sychroizatio of hyperchaotic Liu ad hyperchaotic Lorez systems by active oliear cotrol, Iteratioal Joural of Cotrol Theory ad Applicatios, 3, 79-91, [116] S. Vaidyaatha ad S. Rasappa, New results o the global chaos sychroizatio for Liu-Che-Liu ad Lü chaotic systems, Commuicatios i Computer ad Iformatio Sciece, 102, 20-27, [117] S. Vaidyaatha ad K. Rajagopal, Ati-sychroizatio of Li ad T chaotic systems by active oliear cotrol, Commuicatios i Computer ad Iformatio Sciece, 198, , [118] S. Vaidyaatha ad S. Rasappa, Global chaos sychroizatio of hyperchaotic Bao ad Xu systems by active oliear cotrol, Commuicatios i Computer ad Iformatio Sciece, 198, 10-17, [119] S. Vaidyaatha ad K. Rajagopal, Global chaos sychroizatio of hyperchaotic Pag ad Wag systems by active oliear cotrol, Commuicatios i Computer ad Iformatio Sciece, 204, 84-93, [120] P. Sarasu ad V. Sudarapadia, Active cotroller desig for geeralized projective sychroizatio of four-scroll chaotic systems, Iteratioal Joural of Systems Sigal Cotrol ad Egieerig Applicatio, 4, 26-33, [121] S. Vaidyaatha, Hybrid chaos sychroizatio of Liu ad Lü systems by active oliear cotrol, Commuicatios i Computer ad Iformatio Sciece, 204, 1-10, [122] P. Sarasu ad V. Sudarapadia, The geeralized projective sychroizatio of hyperchaotic Lorez ad hyperchaotic Qi systems via active cotrol, Iteratioal Joural of Soft Computig, 6, , [123] S. Vaidyaatha ad S. Rasappa, Hybrid sychroizatio of hyperchaotic Qi ad Lü systems by oliear cotrol, Commuicatios i Computer ad Iformatio Sciece, 131, , [124] S. Vaidyaatha ad S. Pakiriswamy, The desig of active feedback cotrollers for the geeralized projective sychroizatio of hyperchaotic Qi ad hyperchaotic Lorez systems, Commuicatios i Computer ad Iformatio Sciece, 245, , [125] S. Vaidyaatha ad K. Rajagopal, Hybrid sychroizatio of hyperchaotic Wag-Che ad hyperchaotic Lorez systems by active o-liear cotrol, Iteratioal Joural of Systems Sigal Cotrol ad Egieerig Applicatio, 4, 55-61, 2011.

17 Hybrid Sychroizatio of Idetical Hyperchaotic Systems via Novel Slidig Cotrol [126] V. Sudarapadia ad R. Karthikeya, Hybrid sychroizatio of hyperchaotic Lorez ad hyperchaotic Che systems via active cotrol, Joural of Egieerig ad Applied Scieces, 7, , [127] S. Pakiriswamy ad S. Vaidyaatha, Geeralized projective sychroizatio of three-scroll chaotic systems via active cotrol, Lecture Notes of the Istitute for Computer Scieces, Social-Iformatics ad Telecommuicatios Egieerig, 85, , [128] S. Vaidyaatha ad S. Pakiriswamy, Geeralized projective sychroizatio of double-scroll chaotic systems usig active feedback cotrol, Lecture Notes of the Istitute for Computer Scieces, Social-Iformatics ad Telecommuicatios Egieerig, 84, , [129] S. Pakiriswamy ad S. Vaidyaatha, Geeralized projective sychroizatio of hyperchaotic Lü ad hyperchaotic Cai systems via active cotrol, Lecture Notes of the Istitute for Computer Scieces, Social-Iformatics ad Telecommuicatios Egieerig, 84, 53-62, [130] S. Vaidyaatha, Complete chaos sychroizatio of six-term Sudarapadia chaotic systems with expoetial oliearity via active ad adaptive cotrol, Proceedigs of the 2013 Iteratioal Coferece o Gree Computig, Commuicatio ad Coservatio of Eergy, ICGCE 2013, , [131] R. Karthikeya ad V. Sudarapadia, Hybrid chaos sychroizatio of four-scroll systems via active cotrol, Joural of Electrical Egieerig, 65, , [132] S. Vaidyaatha, A. T. Azar, K. Rajagopal ad P. Alexader, Desig ad SPICE implemetatio of a 12-term ovel hyperchaotic system ad its sychroisatio via active cotrol, Iteratioal Joural of Modellig, Idetificatio ad Cotrol, 23 (3), , [133] B. Samuel, Adaptive sychroizatio betwee two differet chaotic dyamical systems, Adaptive Commu. Noliear Sci. Num. Simul., 12, , [134] J.H. Park, S.M. Lee ad O.M. Kwo, Adaptive sychroizatio of Geesio-Tesi system via a ovel feedback cotrol, Physics Letters A, 371, , [135] V. Sudarapadia ad R. Karthikeya, Ati-sychroizatio of hyperchaotic Lorez ad hyperchaotic Che systems by adaptive cotrol, Iteratioal Joural of System Sigal Cotrol ad Egieerig Applicatios, 4, 18-25, [136] S. Vaidyaatha ad K. Rajagopal, Global chaos sychroizatio of Lü ad Pa systems by adaptive oliear cotrol, Commuicatios i Computer ad Iformatio Sciece, 205, , [137] V. Sudarapadia ad R. Karthikeya, Ati-sychroizatio of Lü ad Pa chaotic systems by adaptive oliear cotrol, Europea Joural of Scietific Research, 64, , [138] V. Sudarapadia ad R. Karthikeya, Ati-sychroizatio of hyperchaotic Lorez ad hyperchaotic Che systems by adaptive cotrol, Iteratioal Joural of Systems Sigal Cotrol ad Egieerig Applicatio, 4, 18-25, [139] P. Sarasu ad V. Sudarapadia, Geeralized projective sychroizatio of three-scroll chaotic systems via adaptive cotrol, Europea Joural of Scietific Research, 72, , [140] V. Sudarapadia ad R. Karthikeya, Adaptive ati-sychroizatio of ucertai Tiga ad Li systems, Joural of Egieerig ad Applied Scieces, 7, 45-52, [141] S. Vaidyaatha ad K. Rajagopal, Global chaos sychroizatio of hyperchaotic Pag ad hyperchaotic Wag systems via adaptive cotrol, Iteratioal Joural of Soft Computig, 7, 28-37, [142] P. Sarasu ad V. Sudarapadia, Geeralized projective sychroizatio of two-scroll systems via adaptive cotrol, Iteratioal Joural of Soft Computig, 7, , [143] P. Sarasu ad V. Sudarapadia, Adaptive cotroller desig for the geeralized projective sychroizatio of 4-scroll systems, Iteratioal Joural of Systems Sigal Cotrol ad Egieerig Applicatio, 5, 21-30, [144] S. Vaidyaatha, Adaptive cotroller ad sychroizer desig for the Qi-Che chaotic system, Lecture Notes of the Istitute for Computer Scieces, Social-Iformatics ad Telecommuicatios Egieerig, 85, , [145] S. Vaidyaatha, Ati-sychroizatio of Sprott-L ad Sprott-M chaotic systems via adaptive cotrol, Iteratioal Joural of Cotrol Theory ad Applicatios, 5, 41-59, [146] V. Sudarapadia, Adaptive cotrol ad sychroizatio desig for the Lu-Xiao chaotic system, Spriger-Verlag Lecture Notes i Electrical Egieerig, 131, , [147] S. Vaidyaatha ad S. Pakiriswamy, Geeralized projective sychroizatio of six-term Sudarapadia chaotic systems by adaptive cotrol, Iteratioal Joural of Cotrol Theory ad Applicatios, 6, , [148] S. Vaidyaatha ad S. Pakiriswamy, Geeralized projective sychroizatio of Elhadj chaotic systems via adpative cotrol, Proceedigs of the 2013 Iteratioal Coferece o Gree Computig, Commuicatio ad Coservatio of Eergy, ICGCE 2013, , 2013.

18 278 S. Sivaperumal [149] S. Vaidyaatha, Aaysis, cotrol ad sychroizatio of hyperchaotic Zhou system via adaptive cotrol, Advaces i Itelliget Systems ad Computig, 177, 1-10, [150] T. Yag ad L.O. Chua, Cotrol of chaos usig sampled-data feedback cotrol, Iteratioal Joural of Bifurcatio ad Chaos, 9, , [151] N. Li, Y. Zhag, J. Hu ad Z. Nie, Sychroizatio for geeral complex dyamical etworks with sampled-data, Neurocomputig, 74, , [152] J.H. Park ad O.M. Kwo, A ovel criterio for delayed feedback cotrol of time-delay chaotic systems, Chaos, Solitos ad Fractals, 17, , [153] X. Wu ad J. Lü, Parameter idetificatio ad backsteppig cotrol of ucertai Lü system, Chaos, Solitos ad Fractals, 18, , [154] Y.G. Yu ad S.C. Zhag, Adaptive backsteppig sychroizatio of ucertai chaotic systems, Chaos, Solitos ad Fractals, 27, , [155] S. Vaidyaatha ad S. Rasappa, Global chaos sychroizatio of Che-Lee systems via backsteppig cotrol, IEEE- Iteratioal Coferece o Advaces i Egieerig, Sciece ad Maagemet, ICAESM-2012, 73-77, [156] R. Suresh ad V. Sudarapadia, Global chaos sychroizatio of WINDMI ad Coullet chaotic systems by backsteppig cotrol, Far East Joural of Mathematical Scieces, 67, , [157] S. Rasappa ad S. Vaidyaatha, Hybrid sychroizatio of -scroll Chua ad Lur e chaotic systems via backsteppig cotrol with ovel feedback, Archives of Cotrol Scieces, 22, , [158] S. Rasappa ad S. Vaidyaatha, Sychroizatio of hyperchaotic Liu via backsteppig cotrol with recursive feedback, Commuicatios i Computer ad Iformatio Sciece, 305, , [159] S. Vaidyaatha, Global chaos sychroizatio of Areodo chaotic system via backsteppig cotroller desig, ACM Iteratioal Coferece Proceedig Series, CCSEIT-12, 1-6, [160] R. Suresh ad V. Sudarapadia, Global chaos sychroizatio of a family of -scroll hyperchaotic Chua circuits usig backsteppig cotroller with recursive feedback, Far East Joural of Mathematical Scieces, 73, 73-95, [161] S. Rasappa ad S. Vaidyaatha, Hybrid sychroizatio of -scroll chaotic Chua circuits usig adaptive backsteppig cotrol desig with recursive feedback, Malaysia Joural of Mathematical Scieces, 7, , [162] S. Rasappa ad S. Vaidyaatha, Global chaos sychroizatio of WINDMI ad Coullet chaotic systems usig adaptive backsteppig cotrol desig, Kyugpook Mathematical Joural, 54, , [163] S. Vaidyaatha ad S. Rasappa, Global chaos sychroizatio of -scroll Chua circuit ad Lur e system usig backsteppig cotrol desig with recursive feedback, Arabia Joural for Sciece ad Egieerig, 39, , [164] S. Vaidyaatha, B.A. Idowu ad A.T. Azar, Backsteppig cotroller desig for the global chaos sychroizatio of Sprott s jerk systems, Studies i Computatioal Itelligece, 581, 39-58, [165] S. Vaidyaatha, Global chaos sychroizatio of Lorez-Steflo ad Qi chaotic systems by slidig mode cotrol, Iteratioal Joural of Cotrol Theory ad Applicatios, 4, , [166] S. Vaidyaatha ad S. Sampath, Global chaos sychroizatio of hyperchaotic Lorez systems by slidig mode cotrol, Commuicatios i Computer ad Iformatio Sciece, 205, , [167] V. Sudarapadia ad S. Sivaperumal, Slidig cotroller desig of hybrid sychroizatio of four-wig chaotic systems, Iteratioal Joural of Soft Computig, 6, , [168] S. Vaidyaatha ad S. Sampath, Ati-sychroizatio of four-wig chaotic systems via slidig mode cotrol, Iteratioal Joural of Automatio ad Computig, 9, , [169] S. Vaidyaatha ad S. Sampath, Slidig mode cotroller desig for the global chaos sychroizatio of Coullet systems, Lecture Notes of the Istitute for Computer Scieces, Social-Iformatics ad Telecommuicatios Egieerig, 84, , [170] S. Vaidyaatha, Global chaos sychroizatio of idetical Li-Wu chaotic systems via slidig mode cotrol, Iteratioal Joural of Modellig, Idetificatio ad Cotrol, 22, , [171] C.H. Lie, L. Zhag, S. Vaidyaatha ad H.R. Karimi, Switched dyamics with its applicatios, Abstract ad Applied Aalysis, 2014, art. o , [172] S. Vaidyaatha ad A.T. Azar, Ati-sychroizatio of idetical chaotic systems usig slidig mode cotrol ad a applicatio to Vaidyaatha-Madhava chaotic systems, Studies i Computatioal Itelligece, 576, , [173] S. Vaidyaatha ad A.T. Azar, Hybrid sychroizatio of idetical chaotic systems usig slidig mode cotrol ad a applicatio to Vaidyaatha chaotic systems, Studies i Computatioal Itelligece, 576, , [174] H.K. Khalil, Noliear Systems, Pretice Hall of Idia, New Jersey, USA, 2002.