electronics Article Multi-Scroll Chaotic Attractors in SC-CNN via Hyperbolic Tangent Function Enis Günay, * and Kenan Altun ID Department of Electrical and Electronics Engineering, Erciyes University, 89 Kayseri, Turkey Sivas Vocational College, Cumhuriyet University, 8 Sivas, Turkey; kaltun@cumhuriyetedutr * Correspondence: egunay@erciyesedutr; Tel:+9--7-6666 (ext ) Received: 9 April 8; Accepted: 7 May 8; Published: 9 May 8 Abstract: A State Controlled-Cellular Neural Network (SC-CNN) based chaotic model for generating multi-scroll attractors via hyperbolic tangent function series is proposed in this paper After presenting the double scroll generation, the presented SC-CNN system is used in multi-scroll chaotic attractor generation by adding hyperbolic tangent function series By using equilibrium analysis and their stability such as Lyapunov exponent analysis, bifurcation diagrams and Poincaré map, the dynamical behaviors of the proposed system are theoretically analyzed and numerically investigated Keywords: State Controlled-Cellular Neural Networks; multi-scroll chaotic attractors; hyperbolic tangent function Introduction It has been accepted that chaos can be quite useful in some engineering and technological applications Complex biological systems, information processing, secure communication, mechanism of memory, etc, can be listed among these applications Chaos, when under control, can provide useful properties and flexibility to the designer Some studies have been carried out to elucidate the effects of memory on the excitation dynamics of organic dynamical systems For example, cardiac action potential models demonstrate that memory can cause dynamical instabilities which result in complex excitation dynamics and chaos [] Moreover, chaos-based communication schemes and chaos-based cryptosystems using more complex attractors are preferable for enhancing the security of the systems Accordingly, simple dynamical systems like multi-scroll chaos generators that show more complex chaotic attractors are desirable [] Cellular Neural Network (CNN), introduced by Chua and Yang in 988, has been the subject of many theoretical and experimental studies as a subfield of nonlinear electrical systems [,] Various CNN based chaotic oscillators have facilitated the understanding of the chaotic phenomenon by complementing the research on chaos through analog simulations Designing more complex chaotic attractors using CNN among multi-scroll generation studies has yet to be well studied and it needs proper consideration The work on generating multi-scroll attractors has always grab researchers attention and has gradually become a new research field The functions such as piecewise linear (PWL), sawtooth, step wave, hysteresis series, switching, sine, saturated sequence and hyperbolic tangent have been proposed for generating a different type of multi-scroll chaotic attractors so far Suykens and Vandewalle proposed quasi-linear function approach to introduce a family of n-double scroll chaotic attractors [ 7] Alaoui et al presented multispiral chaotic attractors using PWL function approach for both autonomous and non-autonomous differential equations [8,9] Yalcin et al introduced a hyperchaotic attractor technique for generating a family of multi-scroll, and they also proposed a simple circuit model for generating multi-scroll chaotic attractors [,] Lü et al Electronics 8, 7, 67; doi:9/electronics767 wwwmdpicom/journal/electronics
Electronics 8, 7, 67 of 6 proposed multi-scroll chaotic attractors using hysteresis series method, saturated function series approach and thresholding approach [ ] Sine function method by Tang et al and nonlinear trans conductor method by Özoğuz et al and Salama et al are suggested for creating n-scroll chaotic attractors [ 7] For creating grid scroll hyperchaotic attractors from one-directional (-D) n-scroll to three-directional (-D) n-scroll attractors, Cafagna and Grassi developed a coupling Chua s circuit method [8,9] Yalcin et al presented a family of scroll-grid chaotic attractors using a stepping circuit [] Adjustable triangular, sawtooth and trans conductor wave functions were also utilized by Yuet al to generate n-scroll chaotic attractors from a general jerk circuit [,] Deng and Lü used a fractional order system such as stair function, saturated, and hysteresis series methods to generate n-scroll attractors [ ] Generation of multi-scroll chaotic attractors for the fractional-order system using the piecewise-linear, the stair, and the saturated function are utilized by Chen et al [6,7] While Xu and Yu presented hyperbolic tangent function in chaos control and chaos synchronization of multi-scroll chaotic attractors, Chen et al used hyperbolic tangent function series to present grid multi-scroll chaos generation [8,9] Wang et al showed multi-level pulses, multi-double-scroll attractors that are generated from the variable-boostable chaotic system [] Muñoz-Pacheco et al presented experimental verification of optimized multi-scroll chaotic attractors based on irregular saturated function [] Itis shown that multi-scrolls could be generated by adding additional breaking points in the output function of SC-CNN, which has a PWL characteristic [] Günay and Alçı revisited multi-scrolls in SC-CNN circuit via diode-based PWL function [] In addition to PWL approach, the trigonometric function was presented by Günay and Kılıç as an alternative way of generating multi-scroll attractors in SC-CNN [] In this paper, a novel methodfor multi-scroll chaotic attractor generation in SC-CNN is presented based on hyperbolic tangent function Firstly, the double scroll generation is analyzed via dynamical behaviors of the presented system, such as its equilibrium, stability, Lyapunov exponents, bifurcation diagrams, and Poincaré map Then, the dynamical mechanisms of multi-scroll chaos generations based on the corresponding model are theoretically analyzed and numerically investigated including one, two, and three directional multi-scroll chaotic attractors Finally, discussions, suggestions, and potential future work plans are presented in the Conclusions Section New Double Scroll Attractor As stated in the literature, besides being an image processing system, Cellular Neural Networks (CNNs) also offer an effective methodology and technology for the analysis and design of complex dynamics [] In 99, Arena et al showed complex dynamical systems can be imitated by CNN canonic model with an additional input that represents the feedbacks from the states of the cells [] The generalized dimensionless nonlinear state equations of SC-CNN can be given as follows: x j = x j + a j y j + G o + G s + i j, y j = [ xj + xj ] () where j, x j, and y j denote cell index, state variable and cell output, respectively a j and i j stand for constant parameter and threshold, respectively G o and G S represent the outputs and state variables of connected cells, respectively The dynamic model of three fully connected generalized CNN cells according to Equation () is defined in [] as follows: x = x + k= x = x + k= a k y k + s k x k + i k= a k y k + s k x k + i k= x = x + a k y k + s k x k + i k= k= ()
Electronics 8, 7, 67 of 6 where k is the cell index In this section, we discuss new double scroll generation and give its dynamical properties such as their equilibria and stability Consider the dynamical system obtained from Equation (): x = x + x x = x x + s x x = x tanh(ny ) + i y = [ x + x ] () where s = s = s = a = ; s = ; s = s = s = s = a = a = a = a = a = a = a = a = i = i = ; and s, i and n are constants The equilibrium points of Equation () exist in three subspaces defined as follows: D + = {(x, x, x ) x } : P + = ( ) k α, k β, k γ, D = {(x, x, x ) x } : P = ( ) l α, l β, l γ, D = {(x, x, x ) x } : P = ( () ) m α, m β, m γ where where k α = (s )(i ), k β = i, k γ = i m α = (s )(i + ), m β = i, m γ = i + l α = i (s ), l (n+) β = i (n+), l γ = i (n+) The equilibrium points P +, P *, and P can be found from the Jacobian matrices J + = J = (s ) The corresponding characteristic equation is: ( λ, = b q ; J = (s ) n () (6) aλ + bλ + cλ + d = (7) p = a c b, q = d a a bc + b a 7a λ = b + q + + q + + q ) ± i ( q = db 7 b c 8 bcd 6 + 7 c + d The equilibrium points P +, P *, and P are given by: + + q ) α ± βi, P ± (λ) = λ + ( s )λ + ( s )λ + P (λ) = λ + ( s )λ + ( s )λ + n + (8) Numerical computations show that proposed system in Equation () will produce chaotic behavior under the conditions of λ <, ( < s < 8), ( < i < ) and n > Thus, Equation () has a negative eigenvalue and one pair of complex conjugate eigenvalues with positive real parts Thus, the proposed
Electronics 8, 7, x FOR PEER REVIEW of Numerical computations show that proposed system in Equation () will produce chaotic behavior under the conditions of λ <, ( < s < 8), ( < i < ) and n > Thus, Equation () has a Electronics negative 8, eigenvalue 7, 67 and one pair of complex conjugate eigenvalues with positive real parts Thus, of 6 the proposed SC CNN system is unstable, and all equilibrium points P are saddle points of index SC-CNN [] Those system can be is given unstable, as follows: and all equilibrium points P are saddle points of index [] Those can be given as follows: Δ 7 8 6 7 Δ Δ = db 7 b c 8 bcd 6 + 7 c + d > ; λ = b + q + + q ( < α = b q + + q ) (9) (9) > Δ Δ The system can produce chaotic behaviors for most of the initial conditions and those are taken The system can produce chaotic behaviors for most of the initial conditions and those are taken as (,, ) in this paper The equilibrium points and eigenvalues of the Jacobian matrices are as (,, ) in this paper The equilibrium points and eigenvalues of the Jacobian matrices are calculated for the parameter values, namely: s = s = s a ; s ; s = ; i = ; and n calculated for the parameter values, namely: s = s = s = a = ; s = ; s = ; i = ; and n = = The eigenvalues for equilibrium point ( 6,, ) D* are calculated from the matrix J* The eigenvalues for equilibrium point ( 6,, ) D* are calculated from the matrix J* as λ 69 and λ, = 7 ± 96i as λ = 69 and λ, = 7 96i On the other hand, the eigenvalues for equilibrium points (,, ) D + and (8, 9, On the other hand, the eigenvalues for equilibrium points (,, ) D+ and (8, 9, 9) D are calculated from the matrix J ± as λ = and λ, = 6 ± 8i As seen from results, 9) D are calculated from the matrix J as λ = and λ, = 6 8i As seen from results, the the proposed system has one negative root and one pair of complex conjugate roots with positive real proposed system has one negative root and one pair of complex conjugate roots with positive real parts Thus, SC-CNN system is unstable, and all equilibrium P are saddle points of index parts Thus, SC CNN system is unstable, and all equilibrium P are saddle points of index To study the dynamics of the system in Equation (), firstly, phase portraits and time domain To study the dynamics of the system in Equation (), firstly, phase portraits and time domain responses are presented in the following figures In phase portraits, T-periodic solution for i = 8, responses are presented in the following figures In phase portraits, T periodic solution for i = 8, one band chaotic solution for i 9, double-scroll chaotic attractor for i, and similarly one band chaotic solution for i = 9, a double scroll chaotic attractor for i =, and similarly T-periodic solution for i = 8 can be seen in Figure a h with time domain representations T periodic solution for i = 8 can be seen in Figure a h with time domain representations x x - - - x - - - - (a) - - - x - - - 6 7 8 9 (b) x x - - - x - - - - (c) - - - x - - - 6 7 8 9 (d) Figure Cont
Electronics 8, 7, 67 of 6 Electronics 8, 7, x FOR PEER REVIEW of x - - x - x - - - (e) - - x - - - 6 7 8 9 (f) - x - - x - - - x - - - - - - x - - - 6 7 8 9 (g) (h) Figure Numerical results of the Switched SC CNN Based system: (a) T periodic solution for i = Figure Numerical results of the Switched-SC-CNN Based system: (a) T-periodic solution for i 8; (b) time response of x(t) dynamic for i = 8; (c) one band chaos for i = 9; (d) time = 8; (b) time response of x (t) dynamic for i = 8; (c) one band chaos for i = 9; (d) response of x(t) dynamic for i = 9; (e) double scroll for i = ; (f) time response of x(t) dynamic time response of x (t) dynamic for i = 9; (e) double scroll for i = ; (f) time response of x (t) for i = ; (g) T periodic solution for i = 8; and (h) time response of x(t) dynamic for i = 8 dynamic for i = ; (g) T-periodic solution for i = 8; and (h) time response of x (t) dynamic for i = 8 Bifurcation Diagrams, Lyapunov Exponents Spectra and Poincaré Map Bifurcation In this study, Diagrams, three bifurcation Lyapunovstudies Exponents are given Spectra In and the first Poincaré one, system Map parameters are fixed as s = s = s = a = ; s = ; s = ; and n =, with varying i The system is calculated numerically In this study, three bifurcation studies are given In the first one, system parameters are fixed fori [, ], and an increment of Δi = Period doubling route to chaos is clearly seen in the as s = s = s = a = ; s = ; s = ; and n =, with varying i The system is calculated bifurcation diagram, x versus i, as given in Figure a From the bifurcation diagram and the phase numerically for i [, ], and an increment of i = Period-doubling route to chaos is clearly domain figures given in Figure, the proposed system is symmetrical with respect to i seen in the bifurcation diagram, x versus i, as given in Figure a From the bifurcation diagram and In the second bifurcation study, system parameters are fixed as s = s = s = a = ; s = ; i = the phase domain figures given in Figure, the proposed system is symmetrical with respect to i ; and n =, with varying s The system is calculated numerically with s [, 8] for an In the second bifurcation study, system parameters are fixed as s = s = s = a = ; s = ; increment of Δs = Within [, 8], various curves seem to expand explosively and merge i = ; and n =, with varying s The system is calculated numerically with s [, 8] for together to produce an area of almost solid black, which are indicators of the onset of chaos an increment of s = Within [, 8], various curves seem to expand explosively and merge In the last bifurcation study, system parameters are fixed as s = s = s = a = ; s = ; i = ; together to produce an area of almost solid black, which are indicators of the onset of chaos and s =, with varying n The system is calculated numerically with n [, ] for an increment of In the last bifurcation study, system parameters are fixed as s = s = s = a = ; s = ; Δn = As shown in Figure c, when n exceeds, period orbit becomes, when n exceeds 8 i = ; and s =, with varying n The system is calculated numerically with n [, ] for an period orbit, becomes period orbit increment of n = As shown in Figure c, when n exceeds, period- orbit becomes, when n exceeds 8 period- orbit, becomes period- orbit
Electronics 8, 7, 67 6 of 6 Electronics 8, 7, x FOR PEER REVIEW 6 of (a) (b) (c) Figure (a) Bifurcation study with x versus i [, ] for an increment of Δi ; (b) bifurcation Figure (a) Bifurcation study with x versus i [, ] for an increment of i = ; (b) bifurcation study with x versus s [, ] for an increment of Δs ; and (c) bifurcation study with x study with x versus s [, ] for an increment of s = ; and (c) bifurcation study with x versus [, ] for an increment of Δn versus n [, ] for an increment of n = The alternative approach to determine whether a system is chaotic is to compute its Lyapunov exponents The alternative The system approach is accepted to determine as chaotic whether if it has a at system least one is chaotic positive is to Lyapunov compute exponent, its Lyapunov and exponents all the trajectories The system are ultimately is accepted bounded as chaotic [] if it has By using at least the one same positive parameter Lyapunov values exponent, in bifurcation and all the studies, trajectories three are Lyapunov ultimately exponent bounded investigations [] By using the are same presented parameter in Figure values in bifurcation As seen from studies, the three numerical Lyapunov results, exponent the Lyapunov investigations exponent are spectrums presented in with Figure respect As to seen i, from s and n, the are numerical consonant results, with the corresponding Lyapunov exponent phase and spectrums bifurcation with diagrams respect to In i, our s and study, n, are we consonant developed with an corresponding algorithm that phase estimates and the bifurcation dominant diagrams Lyapunov In our exponent study, we of time developed series an by algorithm monitoring that orbital estimates divergence the dominant of the Lyapunov proposed system exponent in of Equation time series () The by monitoring Lyapunov orbital exponent divergence spectrums of are the calculated proposed system using the in Equation numerical () methods The Lyapunov described exponent in [6] spectrums In addition, are a calculated Poincaré section using the of numerical the proposed methods system described in x x in domain [6] In can addition, be seen a in Poincaré Figure section of the proposed system in x -x domain can be seen in Figure
Electronics 8, 7, 67 7 of 6 Electronics 8, 7, x FOR PEER REVIEW 7 of L E - - - - - -6-8 LE LE LE - - -8-6 - - 6 8 i (a) - - -8-6 - - 6 8 i (b) L E - - -6-8 - - LE LE LE s s (c) (d) L E - - - - -6-8 LE LE LE - n (e) - n (f) Figure (a) Diagram of largest Lyapunov exponent for i [, ]; (b) Lyapunov exponents of the Figure (a) Diagram of largest Lyapunov exponent for i [, ]; (b) Lyapunov exponents of the system for i [, ]; (c) diagram of largest Lyapunov exponent for s [, ]; (d) Lyapunov system for i [, ]; (c) diagram of largest Lyapunov exponent for s [, ]; (d) Lyapunov exponents of the system for s [, ]; (e) diagram of largest Lyapunov exponent for [, ]; exponents of the system for s [, ]; (e) diagram of largest Lyapunov exponent for n [, ]; and (f) Lyapunov exponents of the system for n [, ] and (f) Lyapunov exponents of the system for n [, ]
Electronics 8, 7, 67 8 of 6 Electronics 8, 7, x FOR PEER REVIEW 8 of Poincare map x - - - - - - x Figure Poincaré mapping of the double scroll attractor (s = s = s = a ; s = ; s = ; n =, i Figure Poincaré mapping of the double-scroll attractor (s = s = s = a = ; s = ; s = ; = ) n =, i = ) Generating Multi Scroll Chaotic Attractors Generating Multi-Scroll Chaotic Attractors In this section, we introduce approaches in generating multi scroll chaotic attractors utilizing In this section, we introduce approaches in generating multi-scroll chaotic attractors utilizing hyperbolic tangent function series in the proposed system Firstly, hyperbolic tangent function series hyperbolic tangent function series in the proposed system Firstly, hyperbolic tangent function series are added along x to generate one direction ( D) multi scroll chaotic attractors Then, to generate are added along x two directions ( D) to generate one direction (-D) multi-scroll chaotic attractors Then, to generate multi scroll chaotic attractors, hyperbolic tangent function series are added to two directions (-D) multi-scroll chaotic attractors, hyperbolic tangent function series are added to the the system along x x directions and x x directions in two different ways Finally, three direction system along x ( D) multi scroll -x directions and x chaotic attractors -x are directions in two different ways Finally, three direction (-D) obtained by adding hyperbolic tangent function series to the multi-scroll chaotic attractors are obtained by adding hyperbolic tangent function series to the system system along x x x directions in three different ways along x -x -x directions in three different ways Generating One Direction ( D) Multi Scroll Chaotic Attractors Generating One Direction (-D) Multi-Scroll Chaotic Attractors Consider the following system: Consider the following system: x = x + x () x = x x + s x () x tanh tanh = x tanh(ny ) + i tanh(ny ) i where where s = s = s = a = ; s = ; s =, i =, n = ; and s = s = s = a = ; s = ; s =, i =, n = ; and s = s = s = s = a = a = a = a = a = a = a = a = i = i = s The equilibrium = s = s points = s of = a Equation = a = a () exist = a in = a these = a three = a subspaces, = a = i defined = i = as follows: The equilibrium points of Equation, () exist : in these three subspaces,,,, defined as follows:,, :,,, D () = {(x, x, x ) x } : P = (k, k, k ), D = {(x,, x,, x ) x : } : P = (l, l, l, ), () D where = {(x, x, x ) x } : P = (m, m, m ) where,, k = s, k =, k = m = s, m =, m =,, () l =, l =, l =,, The equilibrium points P, P, and P can be found from the Jacobian matrices The equilibrium points P, P, and P can be found from the Jacobian matrices J = J = (s ) ; ; J = (s ) () () n Thus, the corresponding characteristic equations and eigenvalues take the following form: ()
Electronics 8, 7, 67 9 of 6 Thus, the corresponding characteristic equations and eigenvalues take the following form: P (λ) = P (λ) = λ + ( s )λ + ( s )λ + P (λ) = λ ( s )λ + ( s )λ + n + Electronics 8, 7, x FOR PEER REVIEW } } 9 of λ + λ + λ = (s ) < λ f or D λ λ λ = <, D + λ + λ = (s ) < f or D () λ λ λ = ( n ) <, () The initial conditions are taken as (,, ) in this study The eigenvalues for equilibrium point (,, ) D are calculated from the matrix J as λ = 97 and λ, = 6 ± 7i, and the eigenvalues The initial conditions for equilibrium are taken points as (, (,,, ) ) in D this and study (,, The ) eigenvalues D are calculated for equilibrium from the point matrix(, J,, as ) λ D = are and calculated λ, = from ± 9987i the matrix J as λ = 97 and λ, = 6 7i, and the eigenvalues for equilibrium points (,, ) Dand (,, ) D are calculated from the matrix RemarkJ, as System λ = () and can λ, generate = -D 9987i multi-scroll chaotic attractors via two basic strategies that can be summarized as follows: Remark System () can generate D multi scroll chaotic attractors via two basic strategies that can be summarized adding as tanh follows: functions in x direction via y nonlinear function; and parameterss and n satisfy condition given in Equation () adding tanh functions in x direction via y nonlinear function; and parameterss and n satisfy condition given in Equation () System () has one negative root and one pair of complex conjugate roots with positive real parts for all System subspaces () has Then, one negative SC-CNNroot system and becomes one pair of unstable complex and conjugate all equilibrium roots with points positive P are real saddle parts for points all subspaces of index Then, Figure SC CNN a,b shows system x -xbecomes plane unstable projection and all of -double-scroll equilibrium points attractor P are saddle with variable points of x (t), index respectively Figure a,b Figure shows c,d presents x x x plane -double-scroll projection of attractor double scroll and timeattractor responsewith of x variable (t) for: x(t), respectively Figure c,d presents double scroll attractor and time response of x(t) for: x = x tanh(ny ) + i a tanh(ny ) i a tanh(ny ) + i b tanh(ny ) i b (6) tanh tanh tanh tanh (6) where i a = and i b = whereia = and ib = () x x - - - - x - - - - - - x - - 6 8 6 8 (a) (b) 6 Figure Cont x x - - - - -6 x (c) - - - - - x - - 6 8 6 8 (d) Figure Numerical results of the D multi scroll SC CNN Based system: (a) double scrollin
- - - - x - - - - x - - 6 8 6 8 Electronics 8, 7, 67 - - of 6 (a) (b) 6 x x - - - - -6 x (c) - - - - - x - - 6 8 6 8 (d) Figure Numerical results of the D multi scroll SC CNN Based system: (a) double scrollin Figure Numerical results of the -D multi-scroll SC-CNN Based system: (a) -double scrollin x -x -x x x x plane; (b) time response of x(t) dynamic for double scroll; (c) double scrollin x x x plane; (b) time response of x (t) dynamic for -double scroll; (c) -double scrollin x -x -x plane; and plane; and (d) time response of x(t) dynamic for double scroll (d) time response of x (t) dynamic for -double scroll Generating Two Direction (-D) Multi-Scroll Chaotic Attractors In this section, we introduce the ways of generating two-direction (-D) multi-scroll chaotic attractors in SC-CNN based system -D multi-scrolls can be generated by adding hyperbolic tangent function series to the SC-CNN system along x -x directions and x -x directions In the first model, hyperbolic tangent function series are added along x -x directions via y and y nonlinear functions, respectively In the second model, hyperbolic tangent function series are added along x -x directions by using y nonlinear function, respectively In the third model, hyperbolic tangent function series are injected to the system along x -x directions via y and y nonlinear functions, respectively In the fourth model, hyperbolic tangent function series are attached to the system along x -x directions by using y and y nonlinear functions, respectively x -x Direction -D Multi-Scroll Attractors-I: Consider -D multi-scroll chaotic attractor generator: x = x + x + tanh(ny ) + i x = x x + s x (7) x = x tanh(ny ) + i s = s = s = a = a = ; s = ; s =, i = i =, n = ; and s = s = s = s = a = a = a = a = a = a = a = i = The equilibrium points of Equation (7) exist in these three subspaces, defined as follows: D = {(x, x, x ) x, x } : P = (k, k, k 6 ), D = {(x, x, x ) x, x } : P = (l, l, l 6 ), D = {(x, x, x ) x, x } : P = (m, m, m 6 ) k = m 7 = (i + i )(s ), k = m 8 = i, k 6 = m 9 = i l = (i +i +i n i s ), l (n+)(n ) = i (n+), l 6 = i (n+) (8) (9)
Electronics 8, 7, 67 of 6 The equilibrium points P 6, P 7, and P 8 can be found from the Jacobian matrices J = J = (s ) The corresponding characteristic equation is: λ + λ + λ = (s ) < λ λ λ = <, J = a n + (s ) a n P (λ) = P (λ) = λ + ( s )λ + ( s )λ + P (λ) = λ + ( s )λ ( s )λ + a a n a n + } f or D, D λ + λ + λ = (s ) < λ λ λ = ( a a n + a n ) > () } () f or D () The system can produce chaotic behaviors for most of the initial conditions and those are taken as (,, ) in this model The eigenvalues for equilibrium point (,, 9) D are calculated from the matrix J as λ = 9 and λ, = 79 ± 97i, and the eigenvalues for equilibrium points (,, 9) D and (,, ) D are calculated from the matrix J, as λ = and λ, = ± 987i Remark The strategies which can generate -D multi-scroll chaotic attractors in System (7): adding tanh functions in x via y nonlinear function, and x direction via y nonlinear function; and parameterss, n, a, anda satisfy condition Equation () System (7) has one negative root and one pair of complex conjugate roots with positive real parts for all subspaces, and SC-CNN System (7) becomes unstable and equilibrium P is a saddle point index Figure 6a shows x -x -x plane projection of -double-scroll attractor Figure 6b presents time domain responses of x (t), Figure 6c,d presents -double-scroll attractor and time response of x (t) for: x = x + x + tanh(ny ) + i a + tanh(ny ) i a + tanh(ny ) + i b + tanh(ny ) i b + tanh(ny ) + i c + tanh(ny ) i c x = x x + s x x = x tanh(ny ) + i a tanh(ny ) i a tanh(ny ) + i b tanh(ny ) i b tanh(ny ) + i c tanh(ny ) i c () where i a = i a =, i b = i b =, and i c = i c =
tanh tanh tanh tanh () tanh tanh Electronics 8, 7, 67 of 6 where ia = ia =, ib = ib =, and ic = ic = x x - - - - - - - - x (a) - - - - 6 8 6 8 (b) x x - - - - -6 - - 6 x (c) - 6 7 8 9 (d) Figure 6 Numerical results of thex x direction D multi scroll SC CNN based system I: (a) Figure 6 Numerical results of the x -x direction-d multi-scroll SC-CNN based system-i: (a) -double double scrollin x x x plane; (b) variablex(t); (c) double scrollin x x plane; and (d) variable scrollin x -x -x plane; (b) variable x (t); (c) -double scrollin x -x plane; and (d) variable x (t) x(t) x x x -x Direction -D Multi-Scroll Attractor-II: Direction D Multi Scroll Attractor II: tanh x = x + x tanh(ny ) + i x = x x + s x () () x = x tanh tanh(ny ) + i s = s = s = a = a = ; s = ; s =, i = i =, n = ; and s = s = s = s = a = a = a = a = a = a = a = i = Three subspaces can be defined as follows: D 7 = {(x, x, x ) x } : P 7 = (k 7, k 8, k 9 ), D 6 = {(x, x, x ) x } : P 6 = (l 7, l 8, l 9 ), D 8 = {(x, x, x ) x } : P 8 = (m 7, m 8, m 9 ) k 7 = (s )(i + i ), k 8 = i i, k 9 = i m 7 = (s )(i + i + ), m 8 = i i, m 9 = i +, l 7 = [(s )(i +i a i n)], l (a n+)(a n ) 8 = (i +i a i n) (a n+)(a n ), l 9 = i (a n ) () (6)
Electronics 8, 7, 67 of 6 The equilibrium points P, P, and P can be found from the Jacobian matrices J 7 = J 8 = (s ) The corresponding characteristic equation is: ; J 6 = a n + (s ) a n P 7 (λ) = P 8 (λ) = λ + ( s )λ + ( s )λ + P 6 (λ) = λ (s + a n )λ (a n s a n + a ns + )λ a a n λ + λ + λ = (s ) < λ λ λ = < } +a n a n + f or D 7, D 8 λ + λ + λ = (s + a n ) > λ λ λ = (a n + )(a n ) > (7) } (8) f or D 6 (9) Remark System () can generate -D multi-scroll chaotic attractors: adding tanh functions in x and x direction via y nonlinear function; and parameterss, n, a, a satisfy condition as given in Equation (9) The system can produce chaotic behaviors for most of the initial conditions and those are taken as (,, ) in this paper The eigenvalues for equilibrium point (,, ) D 6 are calculated from the matrix J 6 as λ = 9 and λ, = ± 6i, and the eigenvalues for equilibrium points (8, 8, 9) D 7 and (,, ) D 8 are calculated from the matrix J 7,8 as λ = and λ, = ± 9987i System () has one positive root and one pair of complex conjugate roots with positive real parts for D 6 subspace, and SC-CNN System () becomes unstable and equilibria P 6 is afocus node On the other hand, System () has one negative root and one pair of complex conjugate roots with positive real parts for D 7,8 subspaces Thus, SC-CNN System () becomes unstable and equilibrium points P 7,8 are saddle point index Figure 7a shows x -x plane projection of -double-scroll attractor Figure 6b presents time domain responses of x (t), Figure 6c,d presents -double-scroll attractor and time response of x (t) for: x = x + x tanh(ny ) + i a tanh(ny ) i a tanh(ny ) + i b tanh(ny ) i b tanh(ny ) + i c tanh(ny ) i c x = x x + s x x = x tanh(ny ) + i a tanh(ny ) i a tanh(ny ) + i b tanh(ny ) i b tanh(ny ) + i c tanh(ny ) i c () where i a = i a =, i b = i b =, and i c = i c =
Electronics 8, 7, 67 of 6 Electronics 8, 7, x FOR PEER REVIEW of x x - - - - - - - - - - x (a) - 6 7 8 9 (b) x x - x - - - - x (c) - - - - - - 6 7 8 9 (d) Figure 7 Numerical results of the x x direction D multi scroll SC CNN based system II: (a) Figure 7 Numerical results of the x -x direction-d multi-scroll SC-CNN based system-ii: (a) -double double scrollin x x x plane; (b) variablex(t); (c) 7 double scrollin x x x plane; and (d) variable scrollin x -x -x plane; (b) variable x (t); (c) 7-double scrollin x -x -x plane; and (d) variable x (t) x(t) x -x Direction -D Multi-Scroll Attractors-I: x x Direction D Multi Scroll Attractors I: x = x + x tanh x = x x + s x tanh(ny ) + i () x = x + tanh tanh(ny ) + i s s = s s = = s s = a = a= a = a= ; = s ; = s ; = s ; =, s = i, = i = i, = i n =, ; n = ; ss = s s = s s = = s s = a = a= a = = a a = aa = a = a a = = i a= ; = i = ; The equilibrium points of Equation () exist in these three subspaces defined as follows:,,, :,, D = {(x, x, x ) x, x } : P = (k, k, k ) D,,, :,,, 9 = {(x, x, x ) x, x } : P 9 = (l, l, l ), D = {(x, x,, x, ) x,, x : } : P = (m,, m,, m ) k = i + i s i s, k = i, k = i + m = i + i + s i s,, m = i, m = i () () ()
Electronics 8, 7, 67 of 6 l = (i +i i s a i n) l = l = (a ns a n+a a n +), (i +a i n) (a ns a n+a a n +), (i +a i n) (a ns a n+a a n +) The equilibrium points P 9, P, and P can be found from the Jacobian matrices J = J = (s ) The corresponding characteristic equation is:, J 9 = (s + a n ) a n () P (λ) = P (λ) = λ + ( s )λ + ( s )λ + P 9 (λ) = λ (s + a n )λ (s + a n + a n )λ + a a n + a ns a n + λ + λ + λ = (s ) < λ λ λ = < } f or D, D } λ + λ + λ = (s + a n ) > λ λ λ = [ (a a s )n a a n ] < () f or D 9 (6) Remark System () can generate -D multi-scroll chaotic attractors via two basic strategies that can be summarized as follows: adding tanh functions in x direction via y nonlinear function and in x direction via y nonlinear function; and parameterss, n, a, and a satisfy condition as given in Equation (6) Initial conditions are taken as (,, ) in this paper The eigenvalues for equilibrium point ( 6,, ) D 9 are calculated from the matrix J 9 as λ = 69 and λ, = 7 ± 96i, and the eigenvalues for equilibrium points ( 9,, ) D and (, 9, 9) D are calculated from the matrix J, as λ = and λ, = ± 9997i System (9) has one negative root and one pair of complex conjugate roots with positive real parts for all subspaces Then, System () is unstable and all equilibrium points P are saddle points of index Figure 8a shows x -x -x plane projection of -double-scroll attractor Figure 8b presents time domain responses of x (t), Figure 8c,d presents -double-scroll attractor and time response of x (t) for: x = x + x x = x x + s x tanh(ny ) + i a tanh(ny ) i a tanh(ny ) + i b tanh(ny ) i b x = x + tanh(ny ) + i a + tanh(ny ) i a + tanh(ny ) + i b + tanh(ny ) i b (7) where i a = i a =, and i b = i b =
Electronics 8, 7, 67 6 of 6 Electronics 8, 7, x FOR PEER REVIEW of x - x - - - - - - x - (a) - - - - x - - - 6 8 6 8 (b) x - - - - x - - - x - - - (c) x - - - - - - 6 8 6 8 (d) Figure 8 Numerical results of thex x direction D multi scroll SC CNN based system I: (a) Figure 8 Numerical results of the x doublescroll in x x x plane; (b) variablex(t); -x direction -D multi-scroll SC-CNN based system-i: (a) (c) double scrollin x x x plane; and (d) variable -doublescroll x(t) in x -x -x plane; (b) variable x (t); (c) -double scrollin x -x -x plane; and (d) variable x (t) x x Direction D Multi Scroll Attractors II: x -x Consider Direction the -D dynamical Multi-Scroll system: Attractors-II: Consider the dynamical system: tanh (8) x = x + x tanh x = x x + s x + tanh(ny ) + i (8) s = s = s = a = a x = ; x s = ; tanh(ny s =, ) i + = ii =, n = ; s s = s = s = a a = a a = a a = a = i = ; s = s = s = a = a = ; s = ; s =, i = i =, n = ; Three subspaces can be defined as follows: s = s = s = s = a = a = a = a = a = a = a = i = ;,,, :,, Three subspaces can be defined as, follows:,, :,,, D,,, :,, = {(x, x, x ) x, x } : P = (k, k, k ) D = {(x, x, x ) x, x, } : P, = (l, l, l ), D = {(x, x, x ) x, x, } : P = (m,, m, m ),, k = i + i + s i s, k = i, k = i The equilibrium m points = i P, + i P, and s P i can s, be mfound = from the i, Jacobian m = imatrices + l = (i +i +i n i s ), l (n+)(n ) = i (n+), l = i (n+) (9) () (9) ()
Electronics 8, 7, 67 7 of 6 The equilibrium points P, P, and P can be found from the Jacobian matrices J = J = (s ) The corresponding characteristic equation is:, J = n (s ) n () Electronics 8, 7, x FOR PEER REVIEW 6 of P (λ) = P (λ) = λ + ( s )λ + ( s )λ + P (λ) = λ (s )λ (n + s )λ + n (), } () λ + λ + λ = (s ) < f or D λ λ λ = <, D The corresponding characteristic equation is: } () λ + λ + λ = (s ) < λ λ λ = ( n ) f or D > () The system can produce chaotic behaviors for most of the initial conditions and those are taken as (,, ) in this paper The eigenvalues for, equilibrium point (, 9, 9) () D are calculated from the matrix J as λ = 996 and λ, = 98 ± 777i, and the eigenvalues for equilibrium The points system (, can produce 9, 9) chaotic behaviors D and for ( 9, most of the, initial ) conditions D are and calculated those are taken from the matrix as J(,,, as λ) = in this andpaper λ, = The eigenvalues ± 9997i for System equilibrium (8) point has one (, positive 9, root 9) and D oneare pair of complex calculated conjugate from roots the matrix with negative J as λ real = 996 partsandλ, for D = 98 subspace, 777i, and SC-CNN and the System eigenvalues (8) becomes for unstable equilibrium and equilibria points (, P is9, a saddle 9) point D and index ( 9,, ) D are calculated from the matrix J, On the as λ other = andλ, hand, System = (8) 9997i has onesystem negative (8) root has and one one positive pairroot of complex and one conjugate pair of complex roots with positive conjugate real parts roots for with D negative, subspaces real parts Thus, for D SC-CNN subspace, System and (8) becomes SC CNN unstable System (8) and becomes equilibrium unstable and equilibria P is a saddle point index points P, are saddle point index Figure 9 shows x -x -x plane projection of -double-scroll On the other hand, System (8) has one negative root and one pair of complex conjugate roots attractor with x (t), x (t) and x (t) variables, respectively with positive real parts for D, subspaces Thus, SC CNN System (8) becomes unstable and equilibrium points P, are saddle point index Figure 9 shows x x x plane projection of Remark double scroll There are attractor two basic with mechanisms x(t), x(t) to and generate x(t) variables, -D multi-scroll respectively chaotic attractors in System (8): adding tanh functions in x direction via y nonlinear function and in x direction via y nonlinear Remark There are two basic mechanisms to generate D multi scroll chaotic attractors in System (8): function; and parameter adding sstanh and functions n satisfy in condition x direction asvia given y nonlinear Equation function () and in x direction via y nonlinear function; and parameter ss and n satisfy condition as given in Equation () x - - x x - - x - (a) - - - - - - - (b) Figure 9 Cont
Electronics 8, 7, 67 8 of 6 Electronics 8, 7, x FOR PEER REVIEW 7 of x x - - - - - (c) - (d) Figure 9 Numerical results of the x x direction D multi scroll attractors II: (a) double scrollin Figure 9 Numerical results of the x x x x plane; (b) variablex(t); (c) variable -x direction -D multi-scroll attractors-ii: (a) -double scrollin x(t); and (d) variable x(t) x -x -x plane; (b) variable x (t); (c) variable x (t); and (d) variable x (t) Generating Three Direction D Multi Scroll Chaotic Attractors Generating Three Direction -D Multi-Scroll Chaotic Attractors In this section, we introduce the ways of generating three directions ( D) multi scroll chaotic In attractors this section, in SC CNN we introduce based system the ways Hyperbolic of generating tangent three function directions series are (-D) added multi-scroll to SC CNN chaotic system along x x x directions, x x x directions and x x x attractors in SC-CNN based system Hyperbolic tangent functiondirections, series arerespectively, added to SC-CNN to generate system D multi scrolls Equilibrium analysis and their stability are given with phase and time domain along x -x -x directions, x -x -x directions and x -x -x directions, respectively, to generate -D responses of each system multi-scrolls Equilibrium analysis and their stability are given with phase and time domain responses of each system D multi Scroll Attractors alongx x x Directions: -D multi-scroll Attractors along x tanh -x -x Directions: tanh () tanh x = x + x + tanh(ny ) + i s = s = s = a = xa = a x = ; s x= ; + ss x=, + tanh(ny i = i = i ) = +, i n = ; and x s = s = s = s = x a = tanh(ny a = a = a = ) + i a = a = () sthe = equilibrium s = s = points a = aof Equation = a = ; () s exist = ; in sthese =, three isubspaces = i = i defined =, n as = follows: ; and s = s = s, = s, = a, =, a : = a = a = a, =, a, = The equilibrium points of Equation,, () exist,, in these : three subspaces,, defined, as follows: (),,,, :,, D 6 = {(x, x, x ) x, x, x } : P 6 = (k 6, k 7, k 8 ), D = {(x, x, x ) x, x,, x } : P =, (l 6, l 7, l 8 ), (6) () D 7 = {(x, x, x ) x, x, x, } : P 7 = (m 6,, m 7, m 8 ), k 6 = i + i + i i s i s +, k 7 = i i, k 8 = i m 6 = i + i + i i s i s, m 7 = i i, m, 8 = i + l 6 = (i +i +i i s i s a i n+a i n+a a i n, (a n+a n a s n+a a a n ) (a The equilibrium points P, l 7 P6, = a i n a i n+i +i ) and (a P7 n+a can n abe sfound n+a from a a nthe ) Jacobian matrices (6) l 8 = (i +a i n a i n+a i s n+a a i n ) (a n+a n a s, n+a a a n ) (7) The equilibrium points P, P 6, and P 7 can be found from the Jacobian matrices The corresponding characteristic equation is: J 6 = J 7 = (s ), J = a n (s ) a n a n (7)
Electronics 8, 7, 67 9 of 6 The corresponding characteristic equation is: P 6 (λ) = P 7 (λ) = λ + ( s )λ + ( s )λ + P (λ) = λ (s + a n )λ ( a ns a n s + a a n + ) λ a n a n + a s n a a a n + (8) The system can produce chaotic behaviors for most of the initial conditions and those are taken as (,, ) in this paper The eigenvalues for equilibrium point (,, ) D are calculated from the matrix J as λ = 999 and λ, = 76 ± 7i, and the eigenvalues for equilibrium points (9,, 9) D 6 and ( 9,, ) D 7 are calculated from the matrix J 6,7 as λ = and λ, = ± 9997i System () has one positive root and one pair of complex conjugate roots with negative real parts for D subspace, and SC-CNN System () becomes unstable and equilibria P is a saddle point index On the other hand, System () has one negative root and one pair of complex conjugate roots with positive real parts for D 6,7 subspaces Thus, SC-CNN System () becomes unstable and equilibrium points P 6,7 are saddle point index λ + λ + λ = (s ) < λ λ λ = < } f or D 6, D 7 λ + λ + λ = (s + a n ) > λ λ λ = [ (a + a a s )n + a a a n ] > Figure a shows x -x -x plane projection of -double-scroll attractor Remark 6 System () can generate -D multi-scroll chaotic attractors by: } f or D (9) adding tanh functions in x direction via y nonlinear function, and in x direction via y nonlinear function, and in x direction via y nonlinear function; and choosing parameters s, n, a, a, a to satisfy condition given in Equation (9) Figure b presents time domain responses of x (t), Figure c,d presents -double-scroll attractor and time response of x (t) for: x = x + x + tanh(ny ) + i a + tanh(ny ) i a + tanh(ny ) + i b + tanh(ny ) i b x = x x + s x + tanh(ny ) + i a + tanh(ny ) i a + tanh(ny ) + i b + tanh(ny ) i b x = x tanh(ny ) + i a tanh(ny ) i a tanh(ny ) + i b tanh(ny ) i b () where i a = i a = i a =, i b =, and i b = i b =
tanh tanh tanh tanh tanh tanh tanh tanh () tanh tanh Electronics 8, 7, 67 of 6 where ia = ia = ia =, ib =, and ib = ib = x - x - - - x - (a) - - - - x - - - 6 8 6 8 Electronics 8, 7, x FOR PEER REVIEW 9 of (b) 8 6 x x - - - - -6 - -8 - x (c) - - - - - x - - (d) Figure Numerical results of the D multi scroll attractors I: (a) double scrollin x x x plane; Figure Numerical results of the -D multi-scroll attractors-i: (a) -double scrollin x -x -x plane; (b) variablex(t); (c) double scrollin x x x plane; and (d) variable x(t) (b) variable x (t); (c) -double scrollin x -x -x plane; and (d) variable x (t) D Multi Scroll Attractors along x x x Directions: -D Multi-Scroll Attractors along x -x -x Directions: tanh tanh x = x + x + tanh(ny ) + i () x = x x + tanh s x + tanh(ny ) + i () x = x tanh(ny ) + i s = s = s = a = a = a = ; s = ; s =, i = i = i =, n = ; and s = s = s = a s = = a s = a s = ; s s = a = = ; s a = a =, i a = a = = i a = = i =, n = ; and s = s = s = s = a = a = a = a = a = a = The equilibrium points of Equation (8) exist in these three subspaces, defined as follows: The equilibrium points of Equation (8) exist in these three subspaces, defined as follows:,,,, :,,, D 9 = {(x,, x,, x ) x,, x, x, } : : P 9 = (k 9, k,, k, ),, () D 8 = {(x, x, x ) x, x, x } : P 8 = (l 9, l, l ), () D,,,, :,, = {(x, x, x ) x, x, x } : P = (m 9, m, m ),,,,, (), The equilibrium points P8, P9, Pcan be found through the Jacobian matrices
Electronics 8, 7, 67 of 6 k 9 = i + i + i i s i s +, k = i i, k = i m 9 = i + i + i i s i s, m = i i, m = i + l 9 = (i s i i i +i s +a i n+a i n a i n+a i s n+a a i n ), l = l = (a ns a n+a a n a a n +a a a n +a a s n +) (i +i +a i n+a i n+a a i n ) (a ns a n+a a n a a n +a a a n +a a s n +), (i +a i n+a i n a i s n a a i n ) (a ns a n+a a n a a n +a a a n +a a s n +) The equilibrium points P 8, P 9, P can be found through the Jacobian matrices J 9 = J = (s ) ; J 8 = The corresponding characteristic equation is: a n + (s + a n ) a n () () P 9 (λ) = P (λ) = λ + ( s )λ + ( s )λ + P 8 (λ) = λ (s + a n )λ ( s + a n + a n + a a n ) λ a n + a s n + a a n a a n + a a a n + a a s n + λ + λ + λ = (s ) < λ λ λ = < } f or D 9, D [ λ + λ + λ = (s + a n ) > ] (a λ λ λ = a s )n + (a a a a a s a )n (a a a )n > f or D 8 () (6) Initial conditions are taken as (,, ) and the equilibrium points and eigenvalues of the Jacobian matrices are calculated for the parameter values as s = s = s = a = a = a = ; s = ; s =, i = i = i =, and n = The eigenvalues for equilibrium point ( 8,, 8) D 8 are calculated from the matrix J 8 as λ = 977 and λ, = 87 ± 7i; the eigenvalues for equilibrium points (9,, 9) D 9 are calculated from the matrix J 9 as λ = and λ, = ± 9997i; and ( 9,, ) D are calculated from the matrix J as λ = and λ, = ± 9987i System () has one negative root and one pair of complex conjugate roots with positive real parts for all subspaces Remark 7 System () can generate -D multi-scroll chaotic attractors by: adding tanh functions in x direction via y nonlinear function, and in x direction via y nonlinear function, and in x direction via y nonlinear function; and choosing parameters s, n, a, a, a according to satisfy condition Equation (6) System () is unstable and all equilibrium points P are saddle points of index Figure a shows x -x -x plane projection of -double-scroll attractor Figure b presents time domain responses of x (t), Figure c,d presents -double-scroll attractor and time response of x (t) for: x = x + x + tanh(ny ) + i a + tanh(ny ) i a + tanh(ny ) + i b + tanh(ny ) i b x = x x + s x + tanh(ny ) + i a + tanh(ny ) i a + tanh(ny ) + i b + tanh(ny ) i b x = x tanh(ny ) + i a tanh(ny ) i a tanh(ny ) + i b tanh(ny ) i b (7) where i a = i a = i a =, and i b = i b = i b =
tanh tanh tanh tanh tanh tanh tanh tanh (7) tanh tanh Electronics 8, 7, 67 of 6 where ia = ia = ia =, andib = ib = ib = x x x - - - - - x (a) - - - - - 6 8 6 8 Electronics 8, 7, x FOR PEER REVIEW of (b) 8 6 x x - - - - -6 - - -8 x - (c) - - - - - - x - (d) Figure Numerical results of the D multi scroll attractors II: (a) double scrollin x x x plane; Figure Numerical results of the -D multi-scroll attractors-ii: (a) -double scrollin x -x -x plane; (b) variable x(t); (c) double scrollin x x x plane; and (d) variable x(t) (b) variable x (t); (c) -double scrollin x -x -x plane; and (d) variable x (t) D Multi Scroll Attractors along x x x Directions: -D Multi-Scroll Attractors along x -x -x Directions: tanh tanh x = x + x + tanh(ny ) + i (8) x = x x + tanh s x + tanh(ny ) + i (8) x = x tanh(ny ) + i s = s = s = a = a = a = ; s = ; s =, i = i = i =, n = ; and s = s = s = a s = = a s = = a s = ; s s a = = ; s a = a = =, i a a = = i = i a = =, n = ; and s = s = s = s = a = a = a = a = a = a = The equilibrium points of Equation () exist in these three subspaces, defined as follows: The equilibrium points of Equation () exist in these three subspaces, defined as follows:,,,, :,,, D = {(x,, x,, x ) x,, x, x, } : : P = (k, k,, k, ),, (9) D = {(x, x, x ) x, x, x } : P = (l, l, l ), (9),,,, :,,, D = {(x, x, x ) x, x, x } : P = (m, m, m ),,,,,, The equilibrium points P, P, and P can be found from the Jacobian matrices, (6)
Electronics 8, 7, 67 of 6 k = i + i + i i s i s +, k = i i, k = i m = i + i + i i s i s, m = i i, m = i + l = (i s i i i +i s +a i n a i n+a i s n+a a i n ), (a n )(a a n +a n ) l = (i +i +a i n) (a a n +a n ), l = (i +a i n) (a a n +a n ) The equilibrium points P, P, and P can be found from the Jacobian matrices J = J = (s ) J = The corresponding characteristic equation is: (a n + ) (a n ) (s ) a n (6) (6) P (λ) = P (λ) = λ + ( s )λ + ( s )λ + P (λ) = λ (s )λ (s + a n )λ a n a n + a a n a a n + a a a n + } (6) λ + λ + λ = (s ) < f or D, D λ λ λ = < λ + λ + λ = (s ) < λ λ λ = (a n ) ( a a n + a n ) < } f or D (6) The system can produce chaotic behaviors for most of the initial conditions and those are taken as (,, ) in this paper The equilibrium points and eigenvalues of the Jacobian matrices are calculated for the parameter values s = s = s = a = a = a = ; s = ; s = ; i = i = i = ; and n = The eigenvalues for equilibrium point ( 6,, ) D are calculated from the matrix J as λ = 69 and λ, = 7 ± 96i, and the eigenvalues for equilibrium points (,, ) D and (8, 9, 9) D are calculated from the matrix J, as λ = and λ, = 6 ± 8i System (8) has one negative root and one pair of complex conjugate roots with positive real parts for all subspaces Then, System (8) is unstable and all equilibrium points P are saddle points of index Remark 8 The two strategies that can generate -D multi-scroll chaotic attractors in System (8) are summarized as follows: adding tanh functions in x direction via y nonlinear function, and in x direction via y nonlinear function, and in x direction via y nonlinear function; and parameterss, n, a, a, a satisfy condition given in Equation (6) Figure a shows x -x -x plane projection of -double-scroll attractor Figure b presents time domain responses of x (t), Figure c,d presents 8-double-scroll attractor and time response of x (t) for: x = x + x + tanh(ny ) + i a + tanh(ny ) i a + tanh(ny ) + i b + tanh(ny ) i b x = x x + s x + tanh(ny ) + i a + tanh(ny ) i a + tanh(ny ) + i b + tanh(ny ) i b x = x tanh(ny ) + i a tanh(ny ) i a tanh(ny ) + i b tanh(ny ) i b (6) where i a = i a = i a =, and i b = i b = i b =
tanh tanh tanh tanh tanh tanh tanh tanh (6) tanh tanh Electronics 8, 7, 67 of 6 where ia = ia = ia =, andib = ib = ib = x x - - x - (a) - - - x - - - - 6 8 6 8 Electronics 8, 7, x FOR PEER REVIEW of (b) x - - x - - x - - - (c) x - - - - - - - (d) Figure Numerical results of the D multi scroll attractors III: (a) double scrollin x x x plane; Figure Numerical results of the -D multi-scroll attractors-iii: (a) -double scrollin x -x -x plane; (b) variable x(t); (c) 8 double scrollin x x x plane; and (d) variable x(t) (b) variable x (t); (c) 8-double scrollin x -x -x plane; and (d) variable x (t) Conclusions Conclusions In this paper, new SC CNN based chaotic system with multiple hyperbolic tangent functions In this paper, a new SC-CNN based chaotic system with multiple hyperbolic tangent functions is is proposed This paper presents a mathematical approach for generating multi scroll chaotic proposed This paper presents a mathematical approach for generating multi-scroll chaotic attractors, attractors, such as (one directional) D, (two directional) D scroll, and (three directional) D such as (one-directional) -D, (two-directional) -D scroll, and (three-directional) -D scroll attractors, scroll attractors, from a given D linear autonomous SC CNN system with a hyperbolic function from a given D linear autonomous SC-CNN system with a hyperbolic function series as the controller series as the controller The mechanism for generating multi scroll chaotic attractors is theoretically The mechanism for generating multi-scroll chaotic attractors is theoretically analyzed and numerically analyzed and numerically simulated Some dynamical behaviors of this system are investigated, simulated Some dynamical behaviors of this system are investigated, such as their equilibria, stability, such as their equilibria, stability, Lyapunov exponents and bifurcation diagrams A Poincaré map, Lyapunov exponents and bifurcation diagrams A Poincaré map, particularly is constructed for particularly is constructed for verifying the chaotic behaviors of the double scroll attractor verifying the chaotic behaviors of the double-scroll attractor The system under consideration displays complex nonlinear phenomena as period doubling The system under consideration displays complex nonlinear phenomena as period doubling bifurcation and multi scroll generation for different sets of system parameters The model bifurcation and multi-scroll generation for different sets of system parameters The model considered considered in this work represents an interesting tool for students and researchers to learn better in this work represents an interesting tool for students and researchers to learn better about nonlinear about nonlinear dynamics and chaos Furthermore, this system can be widely evaluated in data dynamics and chaos Furthermore, this system can be widely evaluated in data encryption and encryption and signal communication Finally, it can be predicted that various related bifurcation signal communication Finally, it can be predicted that various related bifurcation phenomena in the phenomena in the generated multi scroll chaotic systems need to be further investigated in the near generated multi-scroll chaotic systems need to be further investigated in the near future future Author Contributions: EG and KA performed theoretical and numerical analysis Author Contributions: EG and KA performed theoretical and numerical analysis Funding: This work is supported by Research Fund of Erciyes University (Project Code: FDK-6-677) Acknowledgments: This work is supported by Research Fund of Erciyes University (Project Code: FDK 6 677) Conflicts of Interest: The authors declare no conflict of interest References
Electronics 8, 7, 67 of 6 Conflicts of Interest: The authors declare no conflict of interest References Landaw, J; Garfinkel, A; Weiss, JN; Qu, Z Memory-induced chaos in cardiac excitation Phys Rev Lett 7, 8, 8 [CrossRef] [PubMed] Chen, G; Ueta, T Chaos in Circuits and Systems, World Scientific Series on Nonlinear Science; Series B (Book ); World Scientific Publishing: Singapore, Chua, LO; Yang, L Cellular neural networks: Theory IEEE Trans Circuits Syst 988,, 7 7 [CrossRef] Chua, LO CNN: A Paradigm for complexity In World Scientific Series on Nonlinear Science; Series A; World Scientific Publishing: Singapore, 998; Volume Suykens, JAK; Vandewalle, J Quasilinear approach to nonlinear systems and the design of n-double scroll (n =,,,,) IEE Proc G 99, 8, 9 6 6 Suykens, JAK; Vandewalle, J Generation of n-double scrolls (n =,,,,) IEEE Trans Circuits Syst I 99,, 86 867 [CrossRef] 7 Suykens, JAK; Vandewalle, J Between n-double sinks and n-double scrolls (n =,,,,) In Proceedings of the International Symposium on Nonlinear Theory and its Application (NOLTA 9), Honolulu, HI, USA, December 99; pp 89 8 8 Aziz-Alaoui, MA Differential equations with multispiral attractors Int J Bifurc Chaos 999, 9, 9 9 [CrossRef] 9 Aziz-Alaoui, MA Multispiral chaos In Proceedings of the nd International Conference Control Oscillations and Chaos, St Petersberg, Russsia, 7 July ; pp 88 9 Yalçın, ME; Suykens, JAK; Vandewalle, J Hyperchaotic n-scroll attractors In Proceedings of the IEEE Workshop on Nonlinear Dynamics of Electronic Systems (NDES ), Catania, Italy, 8 May ; pp 8 Yalçın, ME; Özoğuz, S; Suykens, JAK; Vandewalle, J n-scroll chaos generators: A simple circuit model Electron Lett, 7, 7 8 [CrossRef] Lü, J; Chen, G; Yu, X; Leung, H Generating multi-scroll chaotic attractors via switching control In Proceedings of the th Asian Control Conference, Melbourne, Australia, July ; pp 76 77 Lü, J; Chen, G; Yu, X; Leung, H Design and analysis of multi-scroll chaotic attractors from saturated function series IEEE Trans Circuits Syst I,, 76 9 [CrossRef] Lü, J; Chen, G Generating multi scroll chaotic attractors: Theories, methods and applications Int J Bifurc Chaos 6, 6, 77 88 [CrossRef] Tang, KS; Zhong, GQ; Chen, G; Man, KF Generation of n-scroll attractors via sine function IEEE Trans Circuits Syst I, 8, 69 7 [CrossRef] 6 Özoğuz, S; Elwakil, AS; Salama, KN n-scroll chaos generator using nonlinear transconductor Electron Lett, 8, 68 686 [CrossRef] 7 Salama, KN; Özoğuz, S; Elwakil, AS Generation of n-scroll chaos using nonlinear transconductors In Proceedings of the IEEE International Symposium Circuits and Systems (ISCAS ), Bangkok, Thailand, 8 May ; pp 76 79 8 Cafagna, D; Grassi, G Hyperchaotic coupled Chua circuits: An approach for generating new n m scroll attractors Int J Bifurc Chaos,, 7 [CrossRef] 9 Cafagna, D; Grassi, G New D-scroll attractors in hyperchaotic Chua s circuit forming a ring Int J Bifurc Chaos,, 889 9 [CrossRef] Yalçın, ME; Suykens, JAK; Vandewalle, J; Özoğuz, S Families of scroll grid attractors Int J Bifurc Chaos,, [CrossRef] Yu, SM; Lü, J; Leung, H; Chen, G Design and circuit implementation of n-scroll chaotic attractor from a general jerk system IEEE Trans Circuits Syst I,, 9 76 Yu, SM; Lü, J; Leung, H; Chen, G N-scroll chaotic attractors from a general jerk circuit In Proceedings of the IEEE International Symposium Circuits and Systems (ISCAS ), Kobe, Japan, 6 May ; pp 7 76 Deng, W Generating -D scroll grid attractors of fractional differential systems via stair function Int J Bifurc Chaos 7, 7, 96 98 [CrossRef]
Electronics 8, 7, 67 6 of 6 Deng, W; Lü, J Design of multidirectional multiscroll chaotic attractors based on fractional differential systems via switching control Chaos Interdiscip J Nonlinear Sci 6, 6, [CrossRef] [PubMed] Deng, W; Lü, J Generating multi-directional multi-scroll chaotic attractors via a fractional differential hysteresis system Phys Lett A 7, 69, 8 [CrossRef] 6 Chen, L; Pan, W; Wu, R; Wang, K; He, Y Generation and circuit implementation of fractional-order multi-scroll attractors Chaos Solitons Fractals 6, 8, [CrossRef] 7 Chen, L; Pan, W; Wu, R; Machado, JAT; Lopes, AM Design and implementation of grid multi-scroll fractional-order chaotic attractors Chaos Interdiscip J Nonlinear Sci 6, 6, 8 [CrossRef] [PubMed] 8 Xu, F; Yu, P Chaos control and chaos synchronization for multi-scroll chaotic attractors generated using hyperbolic functions J Math Anal Appl, 6, 7 [CrossRef] 9 Chen, Z; Wen, G; Zhou, H; Chen, J Generation of grid multi-scroll chaotic attractors via hyperbolic tangent function series Opt Int J Light Elect Opt 7,, 9 6 [CrossRef] Wang, N; Bao, B-C; Xu, Q; Chen, M; Wu, PY Emerging multi-double-scroll attractor from variable-boostable chaotic system excited by multi-level pulse J Eng 8, 8, [CrossRef] Muñoz-Pacheco, JM; Guevara-Flores, DK; Félix-Beltrán, OG; Tlelo-Cuautle, E; Barradas-Guevara, JE; Volos, CK Experimental Verification of Optimized Multiscroll Chaotic Oscillators Based on Irregular Saturated Functions Complexity 8, 8 [CrossRef] Arena, P; Baglio, S; Fortuna, L; Manganaro, G Generation of n-double scrolls via cellular neural networks Int J Circuit Theory Appl 996,, [CrossRef] Günay, E; Alçı, M n-double Scrolls in SC-CNN Circuit via Diode-Based PWL Function Int J Bifurc Chaos 6, 6, [CrossRef] Günay, E; Kılıç, R A New Way of Generating n-scroll Attractors via Trigonometric Function Int J Bifurc Chaos,, 897 9 [CrossRef] Arena, P; Baglio, S; Fortuna, L; Manganaro, G Chua s circuit can be generated by CNN cells IEEE Trans Circuits Syst I 99,, [CrossRef] 6 Wolf, A; Swift, JB; Swinney, HL; Vasano, A Determining Lyapunov Exponents from A Time Series Physical D 98, 6, 8 7 [CrossRef] 8 by the authors Licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommonsorg/licenses/by//)