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Article

Multi-Scroll Chaotic Attractors in SC-CNN via Hyperbolic Tangent Function

1
Department of Electrical and Electronics Engineering, Erciyes University, 38039 Kayseri, Turkey
2
Sivas Vocational College, Cumhuriyet University, 58140 Sivas, Turkey
*
Author to whom correspondence should be addressed.
Electronics 2018, 7(5), 67; https://doi.org/10.3390/electronics7050067
Submission received: 9 April 2018 / Revised: 6 May 2018 / Accepted: 7 May 2018 / Published: 9 May 2018

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.

1. 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 [1].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 [2].
Cellular Neural Network (CNN), introduced by Chua and Yang in 1988, has been the subject of many theoretical and experimental studies as a subfield of nonlinear electrical systems [3,4]. 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 [5,6,7]. 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 [10,11]. Lü et al. proposed multi-scroll chaotic attractors using hysteresis series method, saturated function series approach and thresholding approach [12,13,14].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 [15,16,17]. For creating grid scroll hyperchaotic attractors from one-directional (1-D) n-scroll to three-directional (3-D) n-scroll attractors, Cafagna and Grassi developed a coupling Chua’s circuit method [18,19]. Yalcin et al. presented a family of scroll-grid chaotic attractors using a stepping circuit [20]. 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 [21,22]. Deng and Lü used a fractional order system such as stair function, saturated, and hysteresis series methods to generate n-scroll attractors [23,24,25]. 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. [26,27]. 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 [28,29]. Wang et al. showed multi-level pulses, multi-double-scroll attractors that are generated from the variable-boostable chaotic system [30].Muñoz-Pacheco et al. presented experimental verification of optimized multi-scroll chaotic attractors based on irregular saturated function [31].
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 [32]. Günay and Alçı revisited multi-scrolls in SC-CNN circuit via diode-based PWL function [33]. 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 [34].
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.

2. 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 [2]. In 1995, 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 [34]. 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 = 1 2 [ | x j + 1 | | x j 1 | ]
where j, xj, and yj denote cell index, state variable and cell output, respectively. aj and ij stand for constant parameter and threshold, respectively. Go and GS represent the outputs and state variables of connected cells, respectively. The dynamic model of three fully connected generalized CNN cells according to Equation (1) is defined in [34] as follows:
x ˙ 1 = x 1 + k = 1 3 a 1 k y k + k = 1 3 s 1 k x k + i 1 x ˙ 2 = x 2 + k = 1 3 a 2 k y k + k = 1 3 s 2 k x k + i 2 x ˙ 3 = x 3 + k = 1 3 a 3 k y k + k = 1 3 s 3 k x k + i 3
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 (2):
x ˙ 1 = x 2 + x 3 x ˙ 2 = x 1 x 2 + s 22 x 2 x ˙ 3 = x 3 tanh ( n y 2 ) + i 3 y 2 = 1 2 [ | x 2 + 1 | | x 2 1 | ]
where
s11 = s12 = s13 = a32 = 1; s21 = −1;
s23 = s31 = s32 = s33 = a11 = a12 = a13 = a21 = a22 = a23 = a31 = a33 = i1 = i2 = 0; and
s22, i3 and n are constants. The equilibrium points of Equation (3) exist in three subspaces defined as follows:
D + = { ( x 1 , x 2 , x 3 ) | x 2 1 } : P + = ( k α , k β , k γ ) , D * = { ( x 1 , x 2 , x 3 ) | | x 2 | 1 } : P * = ( l α , l β , l γ ) , D = { ( x 1 , x 2 , x 3 ) | x 2 1 } : P = ( m α , m β , m γ )
where
k α = ( s 22 1 ) ( i 3 1 ) , k β = 1 i 3 , k γ = i 3 1 m α = ( s 22 1 ) ( i 3 + 1 ) , m β = 1 i 3 , m γ = i 3 + 1 l α = i 3 ( s 22 1 ) ( n + 1 ) ,   l β = i 3 ( n + 1 ) ,   l γ = i 3 ( n + 1 )
The equilibrium points P+, P*, and P can be found from the Jacobian matrices.
J + = J = [ 0 1 1 1 ( s 22 1 ) 0 0 0 1 ] ; J * = [ 0 1 1 1 ( s 22 1 ) 0 0 n 1 ]
The corresponding characteristic equation is:
a λ 3 + b λ 2 + c λ + d = 0
where
p = c a b 2 3 a 2 ,   q = d a b c 3 a 2 + 2 b 3 27 a 3 λ 1 = b 3 + q 2 2 + Δ 3 + q 2 2 Δ 3 λ 2 , 3 = b 3 1 2 ( q 2 2 + Δ 3 + q 2 2 Δ 3 ) ± i 3 2 ( q 2 2 + Δ 3 + q 2 2 Δ 3 ) α ± β i , Δ = d b 3 27 b 2 c 2 108 b c d 6 + c 3 27 + d 2 4
The equilibrium points P+, P*, and P are given by:
P ± ( λ ) = λ 3 + ( 1 s 22 ) λ 2 + ( 1 s 22 ) λ + 1 P * ( λ ) = λ 3 + ( 1 s 22 ) λ 2 + ( 1 s 22 ) λ + n + 1
Numerical computations show that proposed system in Equation (3) will produce chaotic behavior under the conditions of λ1 < 0, (1 < s22 < 1.28), (−1 < i3 < 1) and n > 0. Thus, Equation (3) has a negative eigenvalue and one pair of complex conjugate eigenvalues with positive real parts. Thus, the proposed SC-CNN system is unstable, and all equilibrium points P are saddle points of index 2 [35]. Those can be given as follows:
Δ = d b 3 27 b 2 c 2 108 b c d 6 + c 3 27 + d 2 4 > 0 ;   λ 1 = b 3 + q 2 2 + Δ 3 + q 2 2 Δ 3 < 0 α = b 3 1 2 ( q 2 2 + Δ 3 + q 2 2 Δ 3 ) > 0
The system can produce chaotic behaviors for most of the initial conditions and those are taken as (0.1, 0.1, 0.1) in this paper. The equilibrium points and eigenvalues of the Jacobian matrices are calculated for the parameter values, namely: s11 = s12 = s13 = a32 = 1; s21 = −1; s22 = 1.2; i33 = 0.1; and n = 10. The eigenvalues for equilibrium point (−0.06, −0.05, 0.05) ∈ D* are calculated from the matrix J* as λ1 = −1.2469 and λ2,3 = 0.7234 ± 1.0396i.
On the other hand, the eigenvalues for equilibrium points (−1.32, −1.1, 1.1) ∈ D+ and (1.08, 0.9, −0.9) ∈ D are calculated from the matrix J± as λ1 = −1 and λ2,3 = 0.6 ± 0.8i. As seen from results, the 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 2.
To study the dynamics of the system in Equation (3), firstly, phase portraits and time domain responses are presented in the following figures. In phase portraits, 3T-periodic solution for i33 = −0.83, one band chaotic solution for i33 = −0.59, a double-scroll chaotic attractor for i33 = 0.1, and similarly 3T-periodic solution for i33 = 0.83 can be seen in Figure 1a–h with time domain representations.

3. Bifurcation Diagrams, Lyapunov Exponents Spectra and Poincaré Map

In this study, three bifurcation studies are given. In the first one, system parameters are fixed as s11 = s12 = s13 = a32 = 1; s21 =−1; s22 = 1.2; and n = 10, with varying i3. The system is calculated numerically for i3 ∈ [−1, 1], and an increment of Δi3 = 0.001. Period-doubling route to chaos is clearly seen in the bifurcation diagram, x1 versus i3, as given in Figure 2a. From the bifurcation diagram and the phase domain figures given in Figure 1, the proposed system is symmetrical with respect to i3.
In the second bifurcation study, system parameters are fixed as s11 = s12 = s13 = a32 = 1; s21 = −1; i3 = 0.1; and n = 10, with varying s22. The system is calculated numerically with s22 ∈ [1, 1.28] for an increment of Δs22 = 0.001. Within [1, 1.28], various curves seem to expand explosively and merge together to produce an area of almost solid black, which are indicators of the onset of chaos.
In the last bifurcation study, system parameters are fixed as s11 = s12 = s13 = a32 = 1; s21 = −1; i3 = 0.1; and s22 = 1.2, with varying n. The system is calculated numerically with n ∈ [1, 3.5] for an increment of Δn = 0.001. As shown in Figure 2c, when n exceeds 1.21, period-1 orbit becomes, when n exceeds 1.85 period-2 orbit, becomes period-4 orbit.
The alternative approach to determine whether a system is chaotic is to compute its Lyapunov exponents. The system is accepted as chaotic if it has at least one positive Lyapunov exponent, and all the trajectories are ultimately bounded [13]. By using the same parameter values in bifurcation studies, three Lyapunov exponent investigations are presented in Figure 3. As seen from the numerical results, the Lyapunov exponent spectrums with respect to i3, s22 and n, are consonant with corresponding phase and bifurcation diagrams. In our study, we developed an algorithm that estimates the dominant Lyapunov exponent of time series by monitoring orbital divergence of the proposed system in Equation (3). The Lyapunov exponent spectrums are calculated using the numerical methods described in [36]. In addition, a Poincaré section of the proposed system in x1-x3 domain can be seen in Figure 4.

4. Generating Multi-Scroll Chaotic Attractors

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 are added along x3 to generate one direction (1-D) multi-scroll chaotic attractors. Then, to generate two directions (2-D) multi-scroll chaotic attractors, hyperbolic tangent function series are added to the system along x1-x3 directions and x2-x3 directions in two different ways. Finally, three direction (3-D) multi-scroll chaotic attractors are obtained by adding hyperbolic tangent function series to the system along x1-x2-x3 directions in three different ways.

4.1. Generating One Direction (1-D) Multi-Scroll Chaotic Attractors

Consider the following system:
x ˙ 1 = x 2 + x 3 x ˙ 2 = x 1 x 2 + s 22 x 2 x ˙ 3 = x 3 tanh ( n y 2 ) + i 3 tanh ( n y 2 ) i 3
where
s11 = s12 = s13 = a32 = 1; s21 = −1; s22 = 1.1, i3 = 0.1, n = 10; and
s23 = s31 = s32 = s33 = a11 = a12 = a13 = a21 = a22 = a23 = a31 = a33 = i1 = i2 = 0.
The equilibrium points of Equation (10) exist in these three subspaces, defined as follows:
D 1 = { ( x 1 , x 2 , x 3 ) | x 2 1 } : P 1 = ( k 1 , k 2 , k 3 ) , D 0 = { ( x 1 , x 2 , x 3 ) | | x 2 | 1 } : P 0 = ( l 1 , l 2 , l 3 ) , D 2 = { ( x 1 , x 2 , x 3 ) | x 2 1 } : P 2 = ( m 1 , m 2 , m 3 )
where
k 1 = 2 s 22 2 ,   k 2 = 2 ,   k 3 = 2 m 1 = 2 2 s 22 ,   m 2 = 2 ,   m 3 = 2 l 1 = 0 ,   l 2 = 0 ,   l 3 = 0
The equilibrium points P1, P0, and P2 can be found from the Jacobian matrices.
J 1 = J 2 = [ 0 1 1 1 ( s 22 1 ) 0 0 0 1 ] ;   J 0 = [ 0 1 1 1 ( s 22 1 ) 0 0 2 n 1 ]
Thus, the corresponding characteristic equations and eigenvalues take the following form:
P 1 ( λ ) = P 2 ( λ ) = λ 3 + ( 2 s 22 ) λ 2 + ( 2 s 22 ) λ + 1 P 0 ( λ ) = λ 3 ( 2 s 22 ) λ 2 + ( 2 s 22 ) λ + 2 n + 1
λ 1 + λ 2 + λ 3 = ( s 22 2 ) < 0 λ 1 λ 2 λ 3 = 1 < 0 } f o r   D 1 , D 2 λ 1 + λ 2 + λ 3 = ( s 22 2 ) < 0 λ 1 λ 2 λ 3 = ( 2 n 1 ) < 0 } f o r   D 0
The initial conditions are taken as (0.1, 0.1, 0.1) in this study. The eigenvalues for equilibrium point (0, 0, 0) ∈ D0 are calculated from the matrix J0 as λ1 = −2.9731 and λ2,3 = 1.0365 ± 2.4472i, and the eigenvalues for equilibrium points (−0.2, −2, 2) ∈ D1 and (0.2, 2, −2) ∈ D2 are calculated from the matrix J1,2 as λ1 = −1 and λ2,3 = 0.05 ± 0.9987i.
Remark 1.
System (10) can generate 1-D multi-scroll chaotic attractors via two basic strategies that can be summarized as follows:
  • adding tanh functions in x3 direction via y2 nonlinear function; and
  • parameterss22 and n satisfy condition given in Equation (15).
System (10) has one negative root and one pair of complex conjugate roots with positive real parts for all subspaces. Then, SC-CNN system becomes unstable and all equilibrium points P are saddle points of index 2. Figure 5a,b shows x1-x2-x3 plane projection of 2-double-scroll attractor with variable x1(t), respectively. Figure 5c,d presents 4-double-scroll attractor and time response of x1(t) for:
x ˙ 3 = x 3 tanh ( n y 2 ) + i 3 a tanh ( n y 2 ) i 3 a tanh ( n y 2 ) + i 3 b tanh ( n y 2 ) i 3 b
where i3a = 0.5 and i3b = 1.

4.2. Generating Two Direction (2-D) Multi-Scroll Chaotic Attractors

In this section, we introduce the ways of generating two-direction (2-D) multi-scroll chaotic attractors in SC-CNN based system. 2-D multi-scrolls can be generated by adding hyperbolic tangent function series to the SC-CNN system along x1-x3 directions and x2-x3 directions.
  • In the first model, hyperbolic tangent function series are added along x1-x3 directions via y3 and y2 nonlinear functions, respectively.
  • In the second model, hyperbolic tangent function series are added along x1-x3 directions by using y2 nonlinear function, respectively.
  • In the third model, hyperbolic tangent function series are injected to the system along x2-x3 directions via y2 and y1 nonlinear functions, respectively.
  • In the fourth model, hyperbolic tangent function series are attached to the system along x2-x3 directions by using y1 and y2 nonlinear functions, respectively.

4.2.1. x1-x3 Direction 2-D Multi-Scroll Attractors-I:

Consider 2-D multi-scroll chaotic attractor generator:
x ˙ 1 = x 2 + x 3 + tanh ( n y 3 ) + i 1 x ˙ 2 = x 1 x 2 + s 22 x 2 x ˙ 3 = x 3 tanh ( n y 2 ) + i 3
s11 = s12 = s13 = a11 = a32 = 1; s21 = −1; s22 = 1.1, i1 = i3 = 0.1, n = 10; and
s23 = s31 = s32 = s33 = a12 = a13 = a21 = a22 = a23 = a31 = a33 = i2 = 0.
The equilibrium points of Equation (17) exist in these three subspaces, defined as follows:
D 4 = { ( x 1 , x 2 , x 3 ) | x 2 , x 3 1 } : P 4 = ( k 4 , k 5 , k 6 ) , D 3 = { ( x 1 , x 2 , x 3 ) | | x 2 | , | x 3 | 1 } : P 3 = ( l 4 , l 5 , l 6 ) , D 5 = { ( x 1 , x 2 , x 3 ) | x 2 , x 3 1 } : P 5 = ( m 4 , m 5 , m 6 )
k 4 = m 7 = ( i 1 + i 3 ) ( s 22 1 ) ,    k 5 = m 8 = 1 i 3 ,    k 6 = m 9 = i 3 1 l 4 = ( i 2 + i 3 + i 2 n i 3 s 22 ) ( n + 1 ) ( n 1 ) ,   l 5 = i 3 ( n + 1 ) ,   l 6 = i 3 ( n + 1 )
The equilibrium points P6, P7, and P8 can be found from the Jacobian matrices.
J 4 = J 5 = [ 0 1 1 1 ( s 22 1 ) 0 0 0 1 ] , J 3 = [ 0 1 a 32 n + 1 1 ( s 22 1 ) 0 0 a 32 n 1 ]
The corresponding characteristic equation is:
P 4 ( λ ) = P 5 ( λ ) = λ 3 + ( 2 s 22 ) λ 2 + ( 2 s 22 ) λ + 1 P 3 ( λ ) = λ 3 + ( 2 s 22 ) λ 2 ( 2 s 22 ) λ + a 32 a 13 n 2 a 32 n + 1
λ 1 + λ 2 + λ 3 = ( s 22 2 ) < 0 λ 1 λ 2 λ 3 = 1 < 0 } f o r   D 4 , D 5 λ 1 + λ 2 + λ 3 = ( s 22 2 ) < 0 λ 1 λ 2 λ 3 = ( a 32 a 13 n 2 + a 32 n 1 ) > 0 } f o r   D 3
The system can produce chaotic behaviors for most of the initial conditions and those are taken as (0.1, 0.1, 0.1) in this model. The eigenvalues for equilibrium point (−0.02, −0.2, −0.9) ∈ D3 are calculated from the matrix J3 as λ1 = −5.0591 and λ2,3 = 2.0795 ± 4.1972i, and the eigenvalues for equilibrium points (−0.02, −0.2, −0.9) ∈ D4 and (−0.02, −0.2, 1.1) ∈ D5 are calculated from the matrix J4,5 as λ1 = −1 and λ2,3 = 0.05 ± 0.987i.
Remark 2.
The strategies which can generate 2-D multi-scroll chaotic attractors in System (17):
  • adding tanh functions in x1 via y3 nonlinear function, and x3 direction via y2 nonlinear function; and
  • parameterss22, n, a13, anda32 satisfy condition Equation (22).
System (17) has one negative root and one pair of complex conjugate roots with positive real parts for all subspaces, and SC-CNN System (17) becomes unstable and equilibrium P is a saddle point index 2. Figure 6a shows x1-x2-x3 plane projection of 2-double-scroll attractor. Figure 6b presents time domain responses of x2(t), Figure 6c,d presents 12-double-scroll attractor and time response of x1(t) for:
x ˙ 1 = x 2 + x 3 + tanh ( n y 3 ) + i 1 a + tanh ( n y 3 ) i 1 a + tanh ( n y 3 ) + i 1 b + tanh ( n y 3 ) i 1 b + tanh ( n y 3 ) + i 1 c + tanh ( n y 3 ) i 1 c x ˙ 2 = x 1 x 2 + s 22 x 2 x ˙ 3 = x 3 tanh ( n y 2 ) + i 3 a tanh ( n y 2 ) i 3 a tanh ( n y 2 ) + i 3 b tanh ( n y 2 ) i 3 b tanh ( n y 2 ) + i 3 c tanh ( n y 2 ) i 3 c
where i1a = i3a = 0.1, i1b = i3b = 0.2, and i1c = i3c = 0.3.

4.2.2. x1-x3 Direction 2-D Multi-Scroll Attractor-II:

x ˙ 1 = x 2 + x 3 tanh ( n y 2 ) + i 1 x ˙ 2 = x 1 x 2 + s 22 x 2 x ˙ 3 = x 3 tanh ( n y 2 ) + i 3
s11 = s12 = s13 = a12 = a32 = 1; s21 = −1; s22 = 1.1, i1 = i3 = 0.1, n = 10; and
s23 = s31 = s32 = s33 = a11 = a13 = a21 = a22 = a23 = a31 = a33 = i2 = 0.
Three subspaces can be defined as follows:
D 7 = { ( x 1 , x 2 , x 3 ) | x 2 1 } : P 7 = ( k 7 , k 8 , k 9 ) , D 6 = { ( x 1 , x 2 , x 3 ) | | x 2 | 1 } : P 6 = ( l 7 , l 8 , l 9 ) , D 8 = { ( x 1 , x 2 , x 3 ) | x 2 1 } : P 8 = ( m 7 , m 8 , m 9 )
k 7 = ( s 22 1 ) ( i 1 + i 3 2 ) ,   k 8 = 2 i 3 i 1 ,   k 9 = i 3 1 m 7 = ( s 22 1 ) ( i 1 + i 3 + 2 ) ,   m 8 = 2 i 3 i 1 ,   m 9 = i 3 + 1 , l 7 = [ ( s 22 1 ) ( i 1 + i 3 a 32 i 1 n ) ] ( a 12 n + 1 ) ( a 32 n 1 ) ,   l 8 = ( i 1 + i 3 a 32 i 1 n ) ( a 12 n + 1 ) ( a 32 n 1 ) ,   l 9 = i 3 ( a 32 n 1 )
The equilibrium points P12, P13, and P14 can be found from the Jacobian matrices.
J 7 = J 8 = [ 0 1 1 1 ( s 22 1 ) 0 0 0 1 ] ;   J 6 = [ 0 a 12 n + 1 1 1 ( s 22 1 ) 0 0 0 a 32 n 1 ]
The corresponding characteristic equation is:
P 7 ( λ ) = P 8 ( λ ) = λ 3 + ( 2 s 22 ) λ 2 + ( 2 s 22 ) λ + 1 P 6 ( λ ) = λ 3 ( s 22 + a 32 n 2 ) λ 2 ( a 12 n s 22 a 32 n + a 32 n s 22 + 2 ) λ a 12 a 32 n 2 + a 12 n a 32 n + 1
λ 1 + λ 2 + λ 3 = ( s 22 2 ) < 0 λ 1 λ 2 λ 3 = 1 < 0 } f o r   D 7 , D 8 λ 1 + λ 2 + λ 3 = ( s 22 + a 32 n 2 ) > 0 λ 1 λ 2 λ 3 = ( a 12 n + 1 ) ( a 32 n 1 ) > 0 } f o r   D 6
Remark 3.
System (24) can generate 2-D multi-scroll chaotic attractors:
  • adding tanh functions in x1 and x3 direction via y2 nonlinear function; and
  • parameterss22, n, a12, a32 satisfy condition as given in Equation (29).
The system can produce chaotic behaviors for most of the initial conditions and those are taken as (0.1, 0.1, 0.1) in this paper. The eigenvalues for equilibrium point (0, −0.01, −0.01) ∈ D6 are calculated from the matrix J6 as λ1 = 9 and λ2,3 = 0.05 ± 3.3162i, and the eigenvalues for equilibrium points (0.18, 1.8, −0.9) ∈ D7 and (−0.22, −2.2, 1.1) ∈ D8 are calculated from the matrix J7,8 as λ1 = −1 and λ2,3 = 0.05 ± 0.9987i. System (24) has one positive root and one pair of complex conjugate roots with positive real parts for D6 subspace, and SC-CNN System (24) becomes unstable and equilibria P6 is afocus node. On the other hand, System (24) has one negative root and one pair of complex conjugate roots with positive real parts for D7,8 subspaces. Thus, SC-CNN System (24) becomes unstable and equilibrium points P7,8 are saddle point index 2. Figure 7a shows x1-x3 plane projection of 2-double-scroll attractor. Figure 6b presents time domain responses of x2(t), Figure 6c,d presents 12-double-scroll attractor and time response of x1(t) for:
x ˙ 1 = x 2 + x 3 tanh ( n y 2 ) + i 1 a tanh ( n y 2 ) i 1 a tanh ( n y 2 ) + i 1 b tanh ( n y 2 ) i 1 b tanh ( n y 2 ) + i 1 c tanh ( n y 2 ) i 1 c x ˙ 2 = x 1 x 2 + s 22 x 2 x ˙ 3 = x 3 tanh ( n y 2 ) + i 3 a tanh ( n y 2 ) i 3 a tanh ( n y 2 ) + i 3 b tanh ( n y 2 ) i 3 b tanh ( n y 2 ) + i 3 c tanh ( n y 2 ) i 3 c
where i1a = i3a = 0.1, i1b = i3b = 0.2, and i1c = i3c = 0.3.

4.2.3. x2-x3 Direction 2-D Multi-Scroll Attractors-I:

x ˙ 1 = x 2 + x 3 x ˙ 2 = x 1 x 2 + s 22 x 2 tanh ( n y 2 ) + i 2 x ˙ 3 = x 3 + tanh ( n y 1 ) + i 3
s11 = s12 = s13 = a31 = a22 = 1; s21 = −1; s22 = 1.05, i2 = i3 = 0.1, n = 10;
s23 = s31 = s32 = s33 = a11 = a12 = a13 = a23 = a32 = a33 = i1 = 0;
The equilibrium points of Equation (31) exist in these three subspaces defined as follows:
D 10 = { ( x 1 , x 2 , x 3 ) | x 1 , x 2 1 } : P 10 = ( k 10 , k 11 , k 12 ) D 9 = { ( x 1 , x 2 , x 3 ) | | x 1 | , | x 2 | 1 } : P 9 = ( l 10 , l 11 , l 12 ) , D 11 = { ( x 1 , x 2 , x 3 ) | x 1 , x 2 1 } : P 11 = ( m 10 , m 11 , m 12 )
k 10 = i 2 + i 3 s 22 i 3 s 22 ,    k 11 = 1 i 3 ,    k 12 = i 3 + 1 m 10 = i 2 + i 3 + s 22 i 3 s 22 ,    m 11 = 1 i 3 ,    m 12 = i 3 1
l 10 = ( i 2 + i 3 i 3 s 22 a 22 i 3 n ) ( a 31 n s 22 a 31 n + a 31 a 22 n 2 + 1 ) , l 11 = ( i 3 + a 31 i 2 n ) ( a 31 n s 22 a 31 n + a 31 a 22 n 2 + 1 ) ,   l 12 = ( i 3 + a 31 i 2 n ) ( a 31 n s 22 a 31 n + a 31 a 22 n 2 + 1 )
The equilibrium points P9, P10, and P11 can be found from the Jacobian matrices.
J 10 = J 11 = [ 0 1 1 1 ( s 22 1 ) 0 0 0 1 ] ,   J 9 = [ 0 1 1 1 ( s 22 + a 22 n 1 ) 0 a 31 n 0 1 ]
The corresponding characteristic equation is:
P 10 ( λ ) = P 11 ( λ ) = λ 3 + ( 2 s 22 ) λ 2 + ( 2 s 22 ) λ + 1 P 9 ( λ ) = λ 3 ( s 22 + a 22 n 2 ) λ 2 ( s 22 + a 31 n + a 22 n 2 ) λ + a 31 a 22 n 2 + a 31 n s 22 a 31 n + 1
λ 1 + λ 2 + λ 3 = ( s 22 2 ) < 0 λ 1 λ 2 λ 3 = 1 < 0 } f o r   D 10 , D 11 λ 1 + λ 2 + λ 3 = ( s 22 + a 22 n 2 ) > 0 λ 1 λ 2 λ 3 = [ ( a 31 a 31 s 22 ) n a 31 a 22 n 2 1 ] < 0 } f o r   D 9
Remark 4.
System (31) can generate 2-D multi-scroll chaotic attractors via two basic strategies that can be summarized as follows:
  • adding tanh functions in x2 direction via y2 nonlinear function and in x3 direction via y1 nonlinear function; and
  • parameterss22, n, a22, and a31 satisfy condition as given in Equation (36).
Initial conditions are taken as (0.1, 0.1, 0.1) in this paper. The eigenvalues for equilibrium point (−0.06, −0.05, 0.05) ∈ D9 are calculated from the matrix J9 as λ1 = −1.2469 and λ2,3 = 0.7234 ± 1.0396i, and the eigenvalues for equilibrium points (−0.9550, −1.1, 1.1) ∈ D10 and (1.1450, 0.9, −0.9) ∈ D11 are calculated from the matrix J10,11 as λ1 = −1 and λ2,3 = 0.0250 ± 0.9997i. System (29) has one negative root and one pair of complex conjugate roots with positive real parts for all subspaces. Then, System (31) is unstable and all equilibrium points P are saddle points of index 2. Figure 8a shows x1-x2-x3 plane projection of 2-double-scroll attractor. Figure 8b presents time domain responses of x1(t), Figure 8c,d presents 4-double-scroll attractor and time response of x2(t) for:
x ˙ 1 = x 2 + x 3 x ˙ 2 = x 1 x 2 + s 22 x 2 tanh ( n y 2 ) + i 2 a tanh ( n y 2 ) i 2 a tanh ( n y 2 ) + i 2 b tanh ( n y 2 ) i 2 b x ˙ 3 = x 3 + tanh ( n y 1 ) + i 3 a + tanh ( n y 1 ) i 3 a + tanh ( n y 1 ) + i 3 b + tanh ( n y 1 ) i 3 b
where i2a = i3a = 0.1, and i2b = i3b = 0.2.

4.2.4. x2-x3 Direction 2-D Multi-Scroll Attractors-II:

Consider the dynamical system:
x ˙ 1 = x 2 + x 3 x ˙ 2 = x 1 x 2 + s 22 x 2 + tanh ( n y 1 ) + i 2 x ˙ 3 = x 3 tanh ( n y 2 ) + i 3
s11 = s12 = s13 = a21 = a32 = 1; s21 = −1; s22 = 1.05, i2 = i3 = 0.1, n = 10;
s23 = s31 = s32 = s33 = a11 = a12 = a13 = a22 = a23 = a31 = a33 = i1 = 0;
Three subspaces can be defined as follows:
D 13 = { ( x 1 , x 2 , x 3 ) | x 1 , x 2 1 } : P 13 = ( k 13 , k 14 , k 15 ) D 12 = { ( x 1 , x 2 , x 3 ) | | x 1 | , | x 2 | 1 } : P 12 = ( l 13 , l 14 , l 15 ) , D 14 = { ( x 1 , x 2 , x 3 ) | x 1 , x 2 1 } : P 14 = ( m 13 , m 14 , m 15 )
k 13 = i 2 + i 3 + s 22 i 3 s 22 ,    k 14 = 1 i 3 ,    k 15 = i 3 1 m 13 = i 2 + i 3 s 22 i 3 s 22 ,    m 14 = 1 i 3 ,    m 15 = i 3 + 1 l 13 = ( i 2 + i 3 + i 2 n i 3 s 22 ) ( n + 1 ) ( n 1 ) ,    l 14 = i 3 ( n + 1 ) ,    l 15 = i 3 ( n + 1 )
The equilibrium points P12, P13, and P14 can be found from the Jacobian matrices.
J 13 = J 14 = [ 0 1 1 1 ( s 22 1 ) 0 0 0 1 ] ,   J 12 = [ 0 1 1 n 1 ( s 22 1 ) 0 0 n 1 ]
The corresponding characteristic equation is:
P 13 ( λ ) = P 14 ( λ ) = λ 3 + ( 2 s 22 ) λ 2 + ( 2 s 22 ) λ + 1 P 12 ( λ ) = λ 3 ( s 22 2 ) λ 2 ( n + s 22 2 ) λ + 1 n 2
λ 1 + λ 2 + λ 3 = ( s 22 2 ) < 0 λ 1 λ 2 λ 3 = 1 < 0 } f o r   D 13 , D 14 λ 1 + λ 2 + λ 3 = ( s 22 2 ) < 0 λ 1 λ 2 λ 3 = ( n 2 1 ) > 0 } f o r   D 12
The system can produce chaotic behaviors for most of the initial conditions and those are taken as (0.1, 0.1, 0.1) in this paper. The eigenvalues for equilibrium point (−0.011, −0.09, 0.09) ∈ D12 are calculated from the matrix J12 as λ1 = 4.9396 and λ2,3 = −2.9448 ± 3.3777i, and the eigenvalues for equilibrium points (1.145, 0.9, −0.9) ∈ D13 and (−0.9550, −1.1, 1.1) ∈ D14 are calculated from the matrix J13,14 as λ1 = −1 and λ2,3 = 0.0250 ± 0.9997i. System (38) has one positive root and one pair of complex conjugate roots with negative real parts for D12 subspace, and SC-CNN System (38) becomes unstable and equilibria P12 is a saddle point index 1.
On the other hand, System (38) has one negative root and one pair of complex conjugate roots with positive real parts for D13,14 subspaces. Thus, SC-CNN System (38) becomes unstable and equilibrium points P13,14 are saddle point index 2. Figure 9 shows x1-x2-x3 plane projection of 4-double-scroll attractor with x1(t), x2(t) and x3(t) variables, respectively.
Remark 5.
There are two basic mechanisms to generate 2-D multi-scroll chaotic attractors in System (38):
  • adding tanh functions in x1 direction via y1 nonlinear function and in x3 direction via y2 nonlinear function; and
  • parameter ss22 and n satisfy condition as given in Equation (43).

4.3. Generating Three Direction 3-D Multi-Scroll Chaotic Attractors

In this section, we introduce the ways of generating three directions (3-D) multi-scroll chaotic attractors in SC-CNN based system. Hyperbolic tangent function series are added to SC-CNN system along x1-x3-x2 directions, x3-x2-x1 directions and x3-x1-x2 directions, respectively, to generate 3-D multi-scrolls. Equilibrium analysis and their stability are given with phase and time domain responses of each system.

4.3.1. 3-D multi-Scroll Attractors along x1-x3-x2 Directions:

x ˙ 1 = x 2 + x 3 + tanh ( n y 1 ) + i 1 x ˙ 2 = x 1 x 2 + s 22 x 2 + tanh ( n y 3 ) + i 2 x ˙ 3 = x 3 tanh ( n y 2 ) + i 3
s11 = s12 = s13 = a11 = a23 = a32 = 1; s21 = −1; s22 = 1.05, i1 = i2 = i3 = 0.1, n = 10; and
s23 = s31 = s32 = s33 = a12 = a13 = a21 = a22 = a31 = a33 = 0.
The equilibrium points of Equation (44) exist in these three subspaces defined as follows:
D 16 = { ( x 1 , x 2 , x 3 ) | x 1 , x 2 , x 3 1 } : P 16 = ( k 16 , k 17 , k 18 ) , D 15 = { ( x 1 , x 2 , x 3 ) | | x 1 | , | x 2 | , | x 3 | 1 } : P 15 = ( l 16 , l 17 , l 18 ) , D 17 = { ( x 1 , x 2 , x 3 ) | x 1 , x 2 , x 3 1 } : P 17 = ( m 16 , m 17 , m 18 )
k 16 = i 1 + i 2 + i 3 i 1 s 22 i 3 s 22 + 1 ,   k 17 = i 3 i 1 ,   k 18 = i 3 1 m 16 = i 1 + i 2 + i 3 i 1 s 22 i 3 s 22 1 ,   m 17 = i 3 i 1 ,   m 18 = i 3 + 1
l 16 = ( i 1 + i 2 + i 3 i 1 s 22 i 3 s 22 a 23 i 2 n + a 32 i 3 n + a 23 a 32 i 1 n 2 ( a 11 n + a 23 n a 11 s 22 n + a 11 a 23 a 32 n 3 1 ) , l 17 = ( a 11 a 32 i 3 n 2 a 1 i 23 n + i 1 + i 3 ) ( a 11 n + a 23 n a 11 s 22 n + a 11 a 23 a 32 n 3 1 ) ,   l 18 = ( i 3 + a 23 i 1 n a 11 i 3 n + a 1 i 3 s 22 n + a 11 a 23 i 2 n 2 ) ( a 11 n + a 23 n a 11 s 22 n + a 11 a 23 a 32 n 3 1 )
The equilibrium points P15, P16, and P17 can be found from the Jacobian matrices.
J 16 = J 17 = [ 0 1 1 1 ( s 22 1 ) 0 0 0 1 ] ,   J 15 = [ a 11 n 1 1 1 ( s 22 1 ) a 32 n 0 a 23 n 1 ]
The corresponding characteristic equation is:
P 16 ( λ ) = P 17 ( λ ) = λ 3 + ( 2 s 22 ) λ 2 + ( 2 s 22 ) λ + 1 P 15 ( λ ) = λ 3 ( s 22 + a 11 n 2 ) λ 2 ( a 11 n s 22 2 a 11 n s 22 + a 23 a 32 n 2 + 2 ) λ a 23 n a 11 n + a 11 s 22 n a 11 a 23 a 32 n 3 + 1
The system can produce chaotic behaviors for most of the initial conditions and those are taken as (0.1, 0.1, 0.1) in this paper. The eigenvalues for equilibrium point (−0.01, 0.0111, −0.0111) ∈ D15 are calculated from the matrix J15 as λ1 = 9.9952 and λ2,3 = −0.4726 ± 10.0337i, and the eigenvalues for equilibrium points (1.09, −0.2, −0.9) ∈ D16 and (−0.91, −0.2, 1.1) ∈ D17 are calculated from the matrix J16,17 as λ1 = −1 and λ2,3 = 0.0250 ± 0.9997i. System (44) has one positive root and one pair of complex conjugate roots with negative real parts for D15 subspace, and SC-CNN System (44) becomes unstable and equilibria P15 is a saddle point index 1. On the other hand, System (44) has one negative root and one pair of complex conjugate roots with positive real parts for D16,17 subspaces. Thus, SC-CNN System (44) becomes unstable and equilibrium points P16,17 are saddle point index 2.
λ 1 + λ 2 + λ 3 = ( s 22 2 ) < 0 λ 1 λ 2 λ 3 = 1 < 0 } f o r   D 16 , D 17 λ 1 + λ 2 + λ 3 = ( s 22 + a 11 n 2 ) > 0 λ 1 λ 2 λ 3 = [ ( a 11 + a 23 a 11 s 22 ) n + a 11 a 23 a 32 n 3 1 ] > 0 } f o r   D 15
Figure 10a shows x1-x2-x3 plane projection of 2-double-scroll attractor.
Remark 6.
System (44) can generate 3-D multi-scroll chaotic attractors by:
  • adding tanh functions in x1 direction via y1 nonlinear function, and in x2 direction via y3 nonlinear function, and in x3 direction via y2 nonlinear function; and
  • choosing parameters s22, n, a11, a23, a32 to satisfy condition given in Equation (49).
Figure 10b presents time domain responses of x1(t), Figure 10c,d presents 4-double-scroll attractor and time response of x3(t) for:
x ˙ 1 = x 2 + x 3 + tanh ( n y 1 ) + i 1 a + tanh ( n y 1 ) i 1 a + tanh ( n y 1 ) + i 1 b + tanh ( n y 1 ) i 1 b x ˙ 2 = x 1 x 2 + s 22 x 2 + tanh ( n y 3 ) + i 2 a + tanh ( n y 3 ) i 2 a + tanh ( n y 3 ) + i 2 b + tanh ( n y 3 ) i 2 b x ˙ 3 = x 3 tanh ( n y 2 ) + i 3 a tanh ( n y 2 ) i 3 a tanh ( n y 2 ) + i 3 b tanh ( n y 2 ) i 3 b
where i1a = i2a = i3a = 0.1, i1b = 0.3, and i2b = i3b = 0.2.

4.3.2. 3-D Multi-Scroll Attractors along x3-x2-x1 Directions:

x ˙ 1 = x 2 + x 3 + tanh ( n y 3 ) + i 1 x ˙ 2 = x 1 x 2 + s 22 x 2 + tanh ( n y 2 ) + i 2 x ˙ 3 = x 3 tanh ( n y 1 ) + i 3
s11 = s12 = s13 = a13 = a22 = a31 = 1; s21 = −1; s22 = 1.1, i1 = i2 = i3 = 0.1, n = 10; and
s23 = s31 = s32 = s33 = a11 = a12 = a21 = a23 = a32 = a33 = 0.
The equilibrium points of Equation (48) exist in these three subspaces, defined as follows:
D 19 = { ( x 1 , x 2 , x 3 ) | x 1 , x 2 , x 3 1 } : P 19 = ( k 19 , k 20 , k 21 ) , D 18 = { ( x 1 , x 2 , x 3 ) | | x 1 | , | x 2 | , | x 3 | 1 } : P 18 = ( l 19 , l 20 , l 21 ) , D 20 = { ( x 1 , x 2 , x 3 ) | x 1 , x 2 , x 3 1 } : P 20 = ( m 19 , m 20 , m 21 )
k 19 = i 1 + i 2 + i 3 i 1 s 22 i 3 s 22 + 1 ,    k 20 = i 3 i 1 ,    k 21 = i 3 1 m 19 = i 1 + i 2 + i 3 i 1 s 22 i 3 s 22 1 ,   m 20 = i 3 i 1 ,   m 21 = i 3 + 1 l 19 = ( i 1 s 22 i 2 i 3 i 1 + i 3 s 22 + a 22 i 1 n + a 22 i 3 n a 31 i 3 n + a 31 i 3 s 22 n + a 22 a 31 i 3 n 2 ) ( a 13 n s 22 a 13 n + a 13 a 22 n 2 a 13 a 31 n 2 + a 13 a 22 a 31 n 3 + a 13 a 31 s 22 n 2 + 1 ) , l 20 = ( i 1 + i 3 + a 13 i 2 n + a 31 i 3 n + a 13 a 31 i 2 n 2 ) ( a 13 n s 22 a 13 n + a 13 a 22 n 2 a 13 a 31 n 2 + a 13 a 22 a 31 n 3 + a 13 a 31 s 22 n 2 + 1 ) ,   l 21 = ( i 3 + a 13 i 1 n + a 13 i 2 n a 13 i 1 s 22 n a 13 a 22 i 1 n 2 ) ( a 13 n s 22 a 13 n + a 13 a 22 n 2 a 13 a 31 n 2 + a 13 a 22 a 31 n 3 + a 13 a 31 s 22 n 2 + 1 )
The equilibrium points P18, P19, P20 can be found through the Jacobian matrices.
J 19 = J 20 = [ 0 1 1 1 ( s 22 1 ) 0 0 0 1 ] ;   J 18 = [ 0 1 a 31 n + 1 1 ( s 22 + a 2 n 1 ) 0 a 13 n 0 1 ]
The corresponding characteristic equation is:
P 19 ( λ ) = P 20 ( λ ) = λ 3 + ( 2 s 22 ) λ 2 + ( 2 s 22 ) λ + 1 P 18 ( λ ) = λ 3 ( s 22 + a 22 n 2 ) λ 2 ( s 22 + a 13 n + a 22 n + a 11 a 31 n 2 2 ) λ a 13 n + a 13 s 22 n + a 13 a 22 n 2 a 13 a 31 n 2 + a 13 a 22 a 31 n 3 + a 13 a 31 s 22 n 2 + 1
λ 1 + λ 2 + λ 3 = ( s 22 2 ) < 0 λ 1 λ 2 λ 3 = 1 < 0 } f o r   D 19 , D 20 λ 1 + λ 2 + λ 3 = ( s 22 + a 22 n 2 ) > 0 λ 1 λ 2 λ 3 = [ ( a 13 a 13 s 22 ) n + ( a 13 a 31 a 13 a 22 a 13 s 22 a 31 ) n 2 ( a 13 a 22 a 31 ) n 3 1 ] > 0 } f o r   D 18
Initial conditions are taken as (0.1, 0.1, 0.1) and the equilibrium points and eigenvalues of the Jacobian matrices are calculated for the parameter values as s11 = s12 = s13 = a13 = a22 = a31 = 1; s21 = −1; s22 = 1.1, i1 = i2 = i3 = 0.1, and n = 10. The eigenvalues for equilibrium point (−0.0108, −0.0110, 0.0081) ∈ D18 are calculated from the matrix J18 as λ1 = −10.9774 and λ2,3 = 10.0387 ± 0.7231i; the eigenvalues for equilibrium points (1.09, −0.2, −0.9) ∈ D19 are calculated from the matrix J19 as λ1 = −1 and λ2,3 = 0.0250 ± 0.9997i; and (−0.92, −0.2, 1.1) ∈ D20 are calculated from the matrix J20 as λ1 = −1 and λ2,3 = 0.0500 ± 0.9987i. System (51) has one negative root and one pair of complex conjugate roots with positive real parts for all subspaces.
Remark 7.
System (51) can generate 3-D multi-scroll chaotic attractors by:
  • adding tanh functions in x1 direction via y3 nonlinear function, and in x2 direction via y2 nonlinear function, and in x3 direction via y1 nonlinear function; and
  • choosing parameters s22, n, a13, a22, a31 according to satisfy condition Equation (56)
System (51) is unstable and all equilibrium points P are saddle points of index 2. Figure 11a shows x1-x2-x3 plane projection of 2-double-scroll attractor. Figure 11b presents time domain responses of x1(t), Figure 11c,d presents 4-double-scroll attractor and time response of x3(t) for:
x ˙ 1 = x 2 + x 3 + tanh ( n y 3 ) + i 1 a + tanh ( n y 3 ) i 1 a + tanh ( n y 3 ) + i 1 b + tanh ( n y 3 ) i 1 b x ˙ 2 = x 1 x 2 + s 22 x 2 + tanh ( n y 2 ) + i 2 a + tanh ( n y 2 ) i 2 a + tanh ( n y 2 ) + i 2 b + tanh ( n y 2 ) i 2 b x ˙ 3 = x 3 tanh ( n y 1 ) + i 3 a tanh ( n y 1 ) i 3 a tanh ( n y 1 ) + i 3 b tanh ( n y 1 ) i 3 b
where i1a = i2a = i3a = 0.2, and i1b = i2b = i3b = 0.3.

4.3.3. 3-D Multi-Scroll Attractors along x3-x1-x2 Directions:

x ˙ 1 = x 2 + x 3 + tanh ( n y 3 ) + i 1 x ˙ 2 = x 1 x 2 + s 22 x 2 + tanh ( n y 1 ) + i 2 x ˙ 3 = x 3 tanh ( n y 2 ) + i 3
s11 = s12 = s13 = a13 = a21 = a32 = 1; s21 = −1; s22 = 1.1, i1 = i2 = i3 = 0.1, n = 10; and
s23 = s31 = s32 = s33 = a11 = a12 = a22 = a23 = a31 = a33 = 0.
The equilibrium points of Equation (53) exist in these three subspaces, defined as follows:
D 22 = { ( x 1 , x 2 , x 3 ) | x 1 , x 2 , x 3 1 } : P 22 = ( k 22 , k 23 , k 24 ) , D 21 = { ( x 1 , x 2 , x 3 ) | | x 1 | , | x 2 | , | x 3 | 1 } : P 21 = ( l 22 , l 23 , l 24 ) , D 23 = { ( x 1 , x 2 , x 3 ) | x 1 , x 2 , x 3 1 } : P 23 = ( m 22 , m 23 , m 24 ) ,
k 22 = i 1 + i 2 + i 3 i 1 s 22 i 3 s 22 + 1 ,    k 23 = i 3 i 1 ,    k 24 = i 3 1 m 22 = i 1 + i 2 + i 3 i 1 s 22 i 3 s 22 1 ,    m 23 = i 3 i 1 ,    m 24 = i 3 + 1 l 22 = ( i 1 s 22 i 2 i 3 i 1 + i 3 s 22 + a 21 i 2 n a 32 i 3 n + a 32 i 3 s 22 n + a 21 a 32 i 2 n 2 ) ( a 13 n 1 ) ( a 21 a 32 n 2 + a 21 n 1 ) , l 23 = ( i 1 + i 3 + a 32 i 3 n ) ( a 21 a 32 n 2 + a 21 n 1 ) , l 21 = ( i 3 + a 21 i 1 n ) ( a 21 a 32 n 2 + a 21 n 1 )
The equilibrium points P21, P22, and P23 can be found from the Jacobian matrices.
J 22 = J 23 = [ 0 1 1 1 ( s 22 1 ) 0 0 0 1 ] J 21 = [ 0 1 ( a 32 n + 1 ) ( a 13 n 1 ) ( s 22 1 ) 0 0 a 21 n 1 ]
The corresponding characteristic equation is:
P 22 ( λ ) = P 23 ( λ ) = λ 3 + ( 2 s 22 ) λ 2 + ( 2 s 22 ) λ + 1 P 21 ( λ ) = λ 3 ( s 22 2 ) λ 2 ( s 22 + a 13 n 2 ) λ a 13 n a 21 n + a 13 a 21 n 2 a 21 a 32 n 2 + a 13 a 21 a 32 n 3 + 1
λ 1 + λ 2 + λ 3 = ( s 22 2 ) < 0 λ 1 λ 2 λ 3 = 1 < 0 } f o r   D 22 , D 23 λ 1 + λ 2 + λ 3 = ( s 22 2 ) < 0 λ 1 λ 2 λ 3 = ( a 13 n 1 ) ( a 21 a 32 n 2 + a 21 n 1 ) < 0 } f o r   D 21
The system can produce chaotic behaviors for most of the initial conditions and those are taken as (0.1, 0.1, 0.1) in this paper. The equilibrium points and eigenvalues of the Jacobian matrices are calculated for the parameter values s11 = s12 = s13 = a13 = a21 = a32 = 1; s21 = −1; s22 = 1.1; i1 = i2 = i3 = 0.1; and n = 10. The eigenvalues for equilibrium point (−0.06, −0.05, 0.05) ∈ D21 are calculated from the matrix J21 as λ1 = −1.2469 and λ2,3 = 0.7234 ± 1.0396i, and the eigenvalues for equilibrium points (−1.32, −1.1, 1.1) ∈ D22 and (1.08, 0.9, −0.9) ∈ D23 are calculated from the matrix J1,2 as λ1 = −1 and λ2,3 = 0.6 ± 0.8i. System (58) has one negative root and one pair of complex conjugate roots with positive real parts for all subspaces. Then, System (58) is unstable and all equilibrium points P are saddle points of index 2.
Remark 8.
The two strategies that can generate 3-D multi-scroll chaotic attractors in System (58) are summarized as follows:
  • adding tanh functions in x1 direction via y3 nonlinear function, and in x2 direction via y1 nonlinear function, and in x3 direction via y2 nonlinear function; and
  • parameterss22, n, a13, a21, a32 satisfy condition given in Equation (63).
Figure 12a shows x1-x2-x3 plane projection of 4-double-scroll attractor. Figure 12b presents time domain responses of x1(t), Figure 12c,d presents 8-double-scroll attractor and time response of x3(t) for:
x ˙ 1 = x 2 + x 3 + tanh ( n y 3 ) + i 1 a + tanh ( n y 3 ) i 1 a + tanh ( n y 3 ) + i 1 b + tanh ( n y 3 ) i 1 b x ˙ 2 = x 1 x 2 + s 22 x 2 + tanh ( n y 1 ) + i 2 a + tanh ( n y 1 ) i 2 a + tanh ( n y 1 ) + i 2 b + tanh ( n y 1 ) i 2 b x ˙ 3 = x 3 tanh ( n y 2 ) + i 3 a tanh ( n y 2 ) i 3 a tanh ( n y 2 ) + i 3 b tanh ( n y 2 ) i 3 b
where i1a = i2a = i3a = 0.1, and i1b = i2b = i3b = 0.2.

5. Conclusions

In this paper, a new SC-CNN based chaotic system with multiple hyperbolic tangent functions is proposed. This paper presents a mathematical approach for generating multi-scroll chaotic attractors, such as (one-directional) 1-D, (two-directional) 2-D scroll, and (three-directional) 3-D scroll attractors, from a given 3D linear autonomous SC-CNN system with a hyperbolic function series as the controller. The mechanism for generating multi-scroll chaotic attractors is theoretically analyzed and numerically simulated. Some dynamical behaviors of this system are investigated, such as their equilibria, stability, Lyapunov exponents and bifurcation diagrams. A Poincaré map, particularly is constructed for verifying the chaotic behaviors of the double-scroll attractor.
The system under consideration displays complex nonlinear phenomena as period doubling bifurcation and multi-scroll generation for different sets of system parameters. The model considered in this work represents an interesting tool for students and researchers to learn better about nonlinear dynamics and chaos. Furthermore, this system can be widely evaluated in data encryption and signal communication. Finally, it can be predicted that various related bifurcation phenomena in the generated multi-scroll chaotic systems need to be further investigated in the near future.

Author Contributions

E.G. and K.A. performed theoretical and numerical analysis.

Funding

This work is supported by Research Fund of Erciyes University (Project Code: FDK-2016-6757).

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Numerical results of the Switched-SC-CNN Based system: (a) 3T-periodic solution for i33 = −0.83; (b) time response of x1(t) dynamic for i33 = −0.83; (c) one band chaos for i33 = −0.59; (d) time response of x1(t) dynamic for i33 = −0.59; (e) double scroll for i33 = 0.1; (f) time response of x2(t) dynamic for i33 = 0.1; (g) 3T-periodic solution for i33 = 0.83; and (h) time response of x1(t) dynamic for i33 = 0.83.
Figure 1. Numerical results of the Switched-SC-CNN Based system: (a) 3T-periodic solution for i33 = −0.83; (b) time response of x1(t) dynamic for i33 = −0.83; (c) one band chaos for i33 = −0.59; (d) time response of x1(t) dynamic for i33 = −0.59; (e) double scroll for i33 = 0.1; (f) time response of x2(t) dynamic for i33 = 0.1; (g) 3T-periodic solution for i33 = 0.83; and (h) time response of x1(t) dynamic for i33 = 0.83.
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Figure 2. (a) Bifurcation study with x1 versus i3 ∈ [−1, 1] for an increment of Δi3 = 0.001; (b) bifurcation study with x1 versus s22 ∈ [1, 1.3] for an increment of Δs22 = 0.001; and (c) bifurcation study with x1 versus n ∈ [1, 3.5] for an increment of Δn = 0.001.
Figure 2. (a) Bifurcation study with x1 versus i3 ∈ [−1, 1] for an increment of Δi3 = 0.001; (b) bifurcation study with x1 versus s22 ∈ [1, 1.3] for an increment of Δs22 = 0.001; and (c) bifurcation study with x1 versus n ∈ [1, 3.5] for an increment of Δn = 0.001.
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Figure 3. (a) Diagram of largest Lyapunov exponent for i3 ∈ [−1, 1]; (b) Lyapunov exponents of the system for i3 ∈ [−1, 1]; (c) diagram of largest Lyapunov exponent for s22 ∈ [1, 1.3]; (d) Lyapunov exponents of the system for s22 ∈ [1, 1.3]; (e) diagram of largest Lyapunov exponent for n ∈ [1, 3.5]; and (f) Lyapunov exponents of the system for n ∈ [1, 3.5].
Figure 3. (a) Diagram of largest Lyapunov exponent for i3 ∈ [−1, 1]; (b) Lyapunov exponents of the system for i3 ∈ [−1, 1]; (c) diagram of largest Lyapunov exponent for s22 ∈ [1, 1.3]; (d) Lyapunov exponents of the system for s22 ∈ [1, 1.3]; (e) diagram of largest Lyapunov exponent for n ∈ [1, 3.5]; and (f) Lyapunov exponents of the system for n ∈ [1, 3.5].
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Figure 4. Poincaré mapping of the double-scroll attractor (s11 = s12 = s13 = a32 = 1; s21 =−1; s22 = 1.2; n = 10, i3 = 0.1).
Figure 4. Poincaré mapping of the double-scroll attractor (s11 = s12 = s13 = a32 = 1; s21 =−1; s22 = 1.2; n = 10, i3 = 0.1).
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Figure 5. Numerical results of the 1-D multi-scroll SC-CNN Based system: (a) 2-double scrollin x1-x2-x3 plane; (b) time response of x1(t) dynamic for 2-double scroll; (c) 4-double scrollin x1-x2-x3 plane; and (d) time response of x1(t) dynamic for 4-double scroll.
Figure 5. Numerical results of the 1-D multi-scroll SC-CNN Based system: (a) 2-double scrollin x1-x2-x3 plane; (b) time response of x1(t) dynamic for 2-double scroll; (c) 4-double scrollin x1-x2-x3 plane; and (d) time response of x1(t) dynamic for 4-double scroll.
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Figure 6. Numerical results of the x1-x3 direction2-D multi-scroll SC-CNN based system-I: (a) 2-double scrollin x1-x2-x3 plane; (b) variable x1(t); (c) 12-double scrollin x1-x3 plane; and (d) variable x1(t).
Figure 6. Numerical results of the x1-x3 direction2-D multi-scroll SC-CNN based system-I: (a) 2-double scrollin x1-x2-x3 plane; (b) variable x1(t); (c) 12-double scrollin x1-x3 plane; and (d) variable x1(t).
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Figure 7. Numerical results of the x1-x3 direction2-D multi-scroll SC-CNN based system-II: (a) 2-double scrollin x1-x2-x3 plane; (b) variable x1(t); (c) 7-double scrollin x1-x2-x3 plane; and (d) variable x2(t).
Figure 7. Numerical results of the x1-x3 direction2-D multi-scroll SC-CNN based system-II: (a) 2-double scrollin x1-x2-x3 plane; (b) variable x1(t); (c) 7-double scrollin x1-x2-x3 plane; and (d) variable x2(t).
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Figure 8. Numerical results of the x2-x3 direction 2-D multi-scroll SC-CNN based system-I: (a) 2-doublescroll in x1-x2-x3 plane; (b) variable x1(t); (c) 4-double scrollin x1-x2-x3 plane; and (d) variable x2(t).
Figure 8. Numerical results of the x2-x3 direction 2-D multi-scroll SC-CNN based system-I: (a) 2-doublescroll in x1-x2-x3 plane; (b) variable x1(t); (c) 4-double scrollin x1-x2-x3 plane; and (d) variable x2(t).
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Figure 9. Numerical results of the x2-x3 direction 2-D multi-scroll attractors-II: (a) 4-double scrollin x1-x2-x3 plane; (b) variable x1(t); (c) variable x2(t); and (d) variable x3(t).
Figure 9. Numerical results of the x2-x3 direction 2-D multi-scroll attractors-II: (a) 4-double scrollin x1-x2-x3 plane; (b) variable x1(t); (c) variable x2(t); and (d) variable x3(t).
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Figure 10. Numerical results of the 3-D multi-scroll attractors-I: (a) 2-double scrollin x1-x2-x3 plane; (b) variable x1(t); (c) 4-double scrollin x1-x2-x3 plane; and (d) variable x3(t).
Figure 10. Numerical results of the 3-D multi-scroll attractors-I: (a) 2-double scrollin x1-x2-x3 plane; (b) variable x1(t); (c) 4-double scrollin x1-x2-x3 plane; and (d) variable x3(t).
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Figure 11. Numerical results of the 3-D multi-scroll attractors-II: (a) 2-double scrollin x1-x2-x3 plane; (b) variable x1(t); (c) 4-double scrollin x1-x2-x3 plane; and (d) variable x3(t).
Figure 11. Numerical results of the 3-D multi-scroll attractors-II: (a) 2-double scrollin x1-x2-x3 plane; (b) variable x1(t); (c) 4-double scrollin x1-x2-x3 plane; and (d) variable x3(t).
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Figure 12. Numerical results of the 3-D multi-scroll attractors-III: (a) 2-double scrollin x1-x2-x3 plane; (b) variable x1(t); (c) 8-double scrollin x1-x2-x3 plane; and (d) variable x3(t).
Figure 12. Numerical results of the 3-D multi-scroll attractors-III: (a) 2-double scrollin x1-x2-x3 plane; (b) variable x1(t); (c) 8-double scrollin x1-x2-x3 plane; and (d) variable x3(t).
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Günay, E.; Altun, K. Multi-Scroll Chaotic Attractors in SC-CNN via Hyperbolic Tangent Function. Electronics 2018, 7, 67. https://doi.org/10.3390/electronics7050067

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Günay E, Altun K. Multi-Scroll Chaotic Attractors in SC-CNN via Hyperbolic Tangent Function. Electronics. 2018; 7(5):67. https://doi.org/10.3390/electronics7050067

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Günay, Enis, and Kenan Altun. 2018. "Multi-Scroll Chaotic Attractors in SC-CNN via Hyperbolic Tangent Function" Electronics 7, no. 5: 67. https://doi.org/10.3390/electronics7050067

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