Circuit realization, bifurcations, chaos and hyperchaos in a new 4D system

Applied Mathematics and Computation 239 (2014) 333–345

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Circuit realization, bifurcations, chaos and hyperchaos in a new 4D system A.M.A. El-Sayed a, H.M. Nour b, A. Elsaid b, A.E. Matouk c, A. Elsonbaty b,⇑ a

Faculty of Science, Alexandria University, Alexandria, Egypt Mathematics & Engineering Physics Department, Faculty of Engineering, Mansoura University, PO 35516, Mansoura, Egypt c Mathematics Department, Faculty of Science, Hail University, Hail 2440, Saudi Arabia b

a r t i c l e

i n f o

Keywords: Pitchfork bifurcation Hopf bifurcation Chaos Hyperchaos Circuit implementation

a b s t r a c t This paper is devoted to introduce a new four-dimensional hyperchaotic system. Existence and uniqueness of the solution of the proposed system are proved. Continuous dependence on initial conditions of the system’s solution and some stability conditions of system’s equilibrium points are studied. The existence of pitchfork bifurcation is demonstrated by using the center manifold theorem and the local bifurcation theory. The Hopf bifurcation is examined in double parameters bifurcation diagrams along with degenerate types of Hopf bifurcations. The rich dynamical behaviors of the system are explored, then circuit implementation of the system is proposed. Numerical simulations are carried out to verify theoretical analysis. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction In various disciplines of engineering, physics, biology, chemistry, and economy, we encounter systems that undergo spatial and temporal evolution [1,2]. To model, analyze, and understand these phenomena, the study of dynamical systems is a useful tool that helps in achieving these aims. One important and fascinating behavior that exists in some dynamical systems is chaos. Chaotic dynamical system is characterized by its sensitive dependence on initial conditions and by having positive Lyapunov exponents for its attractor. When the system’s attractor has more than one positive Lyapunov exponents, it is called a hyperchaotic system. A hyperchaotic system has the following properties: (i) an autonomous system with a phase space of dimension at least four, (ii) dissipative, and (iii) has at least two unstable directions. Thus, the hyperchaotic systems have higher unpredictability and more randomness than simple chaotic systems. So, hyperchaos is preferred in many applications including secure communications, chaos based image encryption, and cryptography. The applications of dynamical systems and chaos involve mathematical biology, financial systems, chaos control, synchronization, electronic circuits, secure communications, image encryption, cryptography, and neuroscience research [3–29]. In this paper, a new hyperchaotic system is proposed. Existence, uniqueness, and continuous dependence on initial conditions are studied for the solution of the system. Some stability conditions of the equilibrium points are examined. The pitchfork bifurcation of the new four-dimensional system is investigated by using center manifold theorem and the

⇑ Corresponding author. E-mail address: [email protected] (A. Elsonbaty). http://dx.doi.org/10.1016/j.amc.2014.04.109 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.

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bifurcation theory [30–34]. The rich dynamics of the new system are revealed via bifurcation diagrams and phase portraits of the system’s state variables. The rest of the paper is organized as follows: The existence, uniqueness, and continuous dependence on initial conditions of the proposed system’s solution are studied in Section 2. The pitchfork bifurcation of equilibrium points of the system is examined in Section 3. Numerical simulations are performed in Section 4. Circuit implementation of the model is presented in Section 5. Section 6 contains the conclusion and the general discussions of this work. 2. The new 4D system We propose following hyperchaotic system

dxðtÞ ¼ aðy  xÞ þ yw; dt dyðtÞ ¼ by  xz; dt dzðtÞ ¼ xy  dz; dt dwðtÞ ¼ cðx þ wÞ; dt

ð1Þ ð2Þ ð3Þ ð4Þ

where a; c; d are all positive parameters of the system and b 2 R are the parameters of system (1)–(4).  aa ab  The proposed system (1)–(4) has three equilibrium points given by E0 ¼ ð0; 0; 0; 0Þ; E1 ¼ a; a and a ; aa ; a   pffiffiffiffiffiffi aa ab E2 ¼ a; aþa ; aþa ; a where a ¼ bd . The equilibrium points E1 and E2 exist if a –  a and b > 0. Also, the system (1)–(4) is dissipative if it satisfies

r:V ¼

_ @ x_ @ y_ @ z_ @ w þ þ þ < 0; @x @y @z @w

ð5Þ

i.e.

 a þ b  d  c < 0;

or

ð6Þ

b < a þ c þ d:

ð7Þ ðaþcþdbÞt

This, means that a volume V 0 is contracted by the flow into a volume element V 0 e

at time t.

2.1. Existence and uniqueness of the solution System (1)–(4) can be written in the following form:

_ XðtÞ ¼ FðXðtÞÞ;

ð8Þ

t 2 ð0; T with initial conditions of the system given by

Xð0Þ ¼ X0 ;

ð9Þ

where

x

3

2

3

2

6y 6 X¼6 4z

7 7 7; 5

6y 7 6 07 X0 ¼ 6 7; 4 z0 5

6 6 and FðXÞ ¼ 6 4

2

w

x0

w0

aðy  xÞ þ yw by  xz xy  dz

3 7 7 7: 5

ð10Þ

cðx þ wÞ

The supremum norm utilized in the following analysis is defined for the class of continuous function C½0; T by

kWk ¼ sup jWðtÞj;

WðtÞ 2 C½0; T

ð11Þ

t2ð0;T

and for a matrix M ¼ ½mij ½t, it is defined by

kM k ¼

X

  sup mij ½t:

ð12Þ

i;j t2ð0;T

The existence and uniqueness of the solution are studied in the region X  J where J ¼ ð0; T and A > 0 is used to set the spatial boundary of the region of existence and uniqueness of the solution such that

X ¼ fðx; y; z; wÞ : maxfjxj; jyj; jzj; and jwjg 6 Ag:

ð13Þ

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The solution of (8) and (9) is given by:

X ¼ X0 þ

Z

t

FðXðsÞÞds:

ð14Þ

0

From the equivalence of the integral equation (14) and the system (8) and (9), denoting the right hand side of (14) by 2 3 2 3 x1 x2 6 y1 7 6 y2 7 7 6 7 GðXÞ; then for X1 ¼ 6 4 z1 5 and X2 ¼ 4 z2 5 we get w1 w2

GðX1 Þ  GðX2 Þ ¼

Z

t

ðFðX1 ðsÞÞ  FðX2 ðsÞÞÞds

ð15Þ

0

and therefore

2

jGðX1 Þ  GðX2 Þj 6

Z 0

t

6 6 jðFðX1 ðsÞÞ  FðX2 ðsÞÞÞjds 6 T max 6 4

aðjy1  y2 j þ jx1  x2 jÞ þ jy1 w1  y2 w2 j jbjjy1  y2 j þ jx1 z1  x2 z2 j jx1 y1  x2 y2 j þ djz1  z2 j cðjx1  x2 j þ jw1  w2 jÞ

3

2

!1

3

7 6! 7 7 6 27 7 6 T max 6 7: 5 4 !3 5

!4 ð16Þ

From (12), we know that kX1  X2 k ¼ supfjx1  x2 j þ jy1  y2 j þ jz1  z2 j þ jw1  w2 jg, also y1 w1  y2 w2 ¼ y1 ðw1  w2 Þþ w2 ðy1  y2 Þ 6 Aðjy1  y2 j þ jw1  w2 jÞ and the same procedure is applied to the other second order terms. Then after some calculations, the following results are obtained

kGðX1 Þ  GðX2 Þk ¼ T supf!1 þ !2 þ !3 þ !4 g 6 T maxf2A þ a þ c; 2A þ a þ jbj; A þ c; A þ dgkX1  X2 k; 6 K kX1  X2 k;

ð17Þ ð18Þ

where

K ¼ T maxf2A þ a þ c; 2A þ a þ jbj; A þ c; A þ dg > 0:

ð19Þ

Then if 0 < K < 1, the mapping X ¼ GðXÞ is a contraction mapping. The following theorem gives the sufficient condition for existence and uniqueness of the solution of system (1)–(4). Theorem 1. The sufficient condition for existence and uniqueness of the solution of system (1)–(4) with initial conditions Xð0Þ ¼ X0 in the region X  J is:0 < T maxf2A þ a þ c; 2A þ a þ jbj; A þ c; A þ dg < 1.

2.2. Continuous dependence on initial conditions The objective of this subsection is to determine the range of the parameters where the solution of system (1)–(4) shows continuous dependence on initial conditions. As continuous dependence on initial conditions opposes sensitive dependence on initial conditions that characterizes the chaotic behavior, the main benefit from knowing this range of parameters is that it theoretically enables researchers to determine the range of values of the system’s parameters and time T where the system do not exhibit chaotic behavior. Assume that there are two sets of initial conditions to system (8), X01 and X02 , which satisfy

kX01  X02 k 6 d:

ð20Þ

Also, assume that the condition of theorem (1) is satisfied. Then

X1 ¼ X01 þ X2 ¼ X02 þ

Z Z

t

FðX1 ðsÞÞds;

ð21Þ

FðX2 ðsÞÞds

ð22Þ

0 t 0

and the following inequality is obtained

kX1  X2 k 6 kX01  X02 k þ K kX1  X2 k;

ð23Þ

and therefore

ð1  KÞkX1  X2 k 6 kX01  X02 k;

ð24Þ

0
ð25Þ

where

is defined by (19).

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Denoting

d ð1KÞ

by , then the following relation holds

kX1  X2 k 6 :

ð26Þ

Theorem 2. For system (1)–(4) satisfying the condition of theorem (1) and K defined by ð19Þ. Then, 8 > 09dðÞ ¼ ð1  KÞ > 0 such that kX01  X02 k 6 d implies that kX1  X2 k 6  i.e. the solution exhibit continuous dependence on initial conditions. 3. Pitchfork bifurcation in the new 4D system In the following, we will choose the parameter b to be the dynamical parameter of the new system. 3.1. Some stability conditions of the equilibrium points The Jacobian matrix at the equilibrium point E0 is given as follows

2

a a 0 6 0 b 0 6 6 4 0 0 d c

0

0

0

3

0 7 7 7; 0 5

ð27Þ

c

whose characteristic equation is given by

PðkÞ ¼ k4 þ ða  b þ c þ dÞk3 þ ðab þ ad  bd þ ac  bc þ dcÞk2 þ ðabd  abc þ acd  bcdÞk  abcd ¼ 0:

ð28Þ

A necessary condition for any equilibrium point to be locally asymptotically stable is that the constant term of its characteristic equation n is positive. For b < 0; it is clear that the constant term of the characteristic equation (28) is positive which means that the unique equilibrium point E0 may achieve local stability below the value b ¼ 0. For b > 0; the equilibrium point E0 becomes unstable and two other equilibrium points E1 and E2 appear. The necessary conditions for equilibrium points E1 ðE2 Þ to be locally asymptotically stable are also satisfied because n ¼ 2abcdðn ¼ 2a2 bcd þ 2abcda), respectively, which means that E1 ðE2 Þ may achieve local stability above the value b ¼ 0. 3.2. Pitchfork bifurcation at the equilibrium point E0 Assume that the critical value of the dynamical parameter at which Pitchfork bifurcation occurs is bc . When b ¼ bc ¼ 0; the characteristic equation (28) of the equilibrium point E0 will have the eigenvalues:

k1 ¼ 0;

k2 ¼ a;

k3 ¼ d;

k4 ¼ c:

Therefore E0 is not hyperbolic and consequently we can use center manifold theorem [32] to discuss the dynamics near E0 : Now, we use the following transformation

2

x

3

2

1

6y7 6 1 6 7 6 6 7¼6 4z5 4 0 w 1

ac c

0 0 1

0

0

32

x1

3

6 7 07 76 x2 7 76 7; 5 1 0 4 x3 5 0 1 x4 0

l ¼ b  bc ;

ð29Þ

then system (1)–(4) is transformed into the standard form, and we get

3 2 32 3 2 3 0 0 0 0 x1 g1 x_ 1 6 x_ 7 6 0 a 0 6 7 6 7 0 7 76 x2 7 6 g 2 7 6 27 6 76 7 þ 6 7; 6 7¼6 4 x_ 3 5 4 0 0 d 0 54 x3 5 4 g 3 5 0 0 0 c x4 g4 x_ 4 2

ð30Þ

where

ða  cÞ x2 x3 ; c c c c c c lx1  x2 þ x1 x2 þ x1 x3 þ x1 x4 þ x2 x3 ; g2 ¼  ða  cÞ ða  cÞ 1 ða  cÞ ða  cÞ ða  cÞ ða  cÞ g 3 ¼ x21 þ x1 x2 ; c a c c a c a g4 ¼ lx1 þ x2  x1 x2  x1 x3  x1 x4  x2 x3 : ða  cÞ ða  cÞ 1 ða  cÞ ða  cÞ ða  cÞ c g 1 ¼ l x1  x1 x3 

Consider the parameter l to be the bifurcation parameter of system (30). Then, we reduce the system dimensions by applying the center manifold theorem

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 ; W c ð0Þ ¼ ðx1 ; x2 ; x3 ; x4 ; lÞ 2 R5 jx2 ¼ h1 ðx1 ; lÞ; x3 ¼ h2 ðx1 ; lÞ; x4 ¼ h3 ðx1 ; lÞ; jx1 j < q; jlj < q hi ð0; 0Þ ¼ 0; Dhi ð0; 0Þ ¼ 0; i ¼ 1; 2; 3g;

ð31Þ

c

 are sufficiently small. To compute the center manifold W ð0Þ and derive the vector field on the center manifold, where q; q we assume

x2 ¼ h1 ðx1 ; lÞ ¼ a1 x21 þ a2 x1 l þ a3 l2 þ    ; x3 ¼ h2 ðx1 ; lÞ ¼ b1 x21 þ b2 x1 l þ b3 l2 þ    ; 2 1 x1

x4 ¼ h3 ðx1 ; lÞ ¼ c

ð32Þ

2

þ c2 x1 l þ c3 l þ    :

According to the center manifold theorem [32], the center manifold must satisfy

@ðhðbx ; mÞÞ , Dbx hðbx ; mÞ½Kbx þ f ðbx ; hðbx ; mÞ; mÞ  Bhðbx ; mÞ  gðbx ; hðbx ; mÞ; mÞ ¼ 0;

ð33Þ

where

2 b x  x1 ;

0

a

m  l; K ¼ 0; f ¼ g 1 ; B ¼ 6 4 0

0

d

0

0

3

7 0 5; c

2

h1

3

6 7 h ¼ 4 h2 5;

2

g2

3

6 7 g ¼ 4 g 3 5: g4

h3

Substituting Eq. (32) into Eq. (33), and equating the like powers to zero, we get

c 1 1 ; b1 ¼ ; c1 ¼ ; aða  cÞ d ac c a x1 l : a2 ¼ ; b2 ¼ 0; c2 ¼ ; aða  cÞ cða  cÞ

x21 : a1 ¼

l2 : a3 ¼ 0; b3 ¼ 0; c3 ¼ 0: Thus, we have

x2 ¼ h1 ðx1 ; lÞ ¼ 

c c x2  x1 l þ    ; aða  cÞ 1 aða  cÞ

1 2 x þ ; d 1 1 2 a x4 ¼ h3 ðx1 ; lÞ ¼ x þ x1 l þ    : a  c 1 cða  cÞ x3 ¼ h2 ðx1 ; lÞ ¼

ð34Þ

Using Eqs. (30) and (34), we obtain the vector field reduced to the center manifold

x_ 1 ¼ lx1 

x31  l x1  1  þ ; d a a

ð35Þ

l_ ¼ 0: According to the bifurcation theory, the equilibrium point ðx1 ; lÞ ¼ ð0; 0Þ of system (35) undergoes a pitchfork bifurcation at

l ¼ 0, since the following conditions are satisfied Fð0; 0Þ ¼ 0;

 @F  ¼ 0; @x1 ð0;0Þ

 @F  ¼ 0; @ lð0;0Þ

 @ 2 F   @x21 

ð0;0Þ

¼ 0;

 @ 2 F   @x1 @ l

¼ 1 – 0; ð0;0Þ

 @ 3 F   @x31 

ð0;0Þ

¼

6 – 0; d

where

Fðx1 ; lÞ , lx1 

x31  l x1  1  : d a a

Consequently, system (1)–(4) undergoes a pitchfork bifurcation at E0 and the following theorem is now proved. Theorem 3. For b ¼ bc ¼ 0, system (1)–(4) undergoes a pitchfork bifurcation at E0 . Moreover, for b < bc , the unique equilibrium point E0 is locally asymptotically stable near bc ¼ 0. For b > bc , there are three equilibrium points

! pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi a bdða þ bdÞ abða þ bdÞ pffiffiffiffiffiffi ; ;  and bd; bd a2  bd a2  bd ! pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi a bdða  bdÞ abða  bdÞ pffiffiffiffiffiffi ; bd  bd; ; ; a2  bd a2  bd

E0 ¼ ð0; 0; 0; 0Þ; E2 ¼

E1 ¼

where E0 becomes unstable and both E1 ; E2 are locally asymptotically stable near bc ¼ 0.

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4. Phase portraits and bifurcation diagrams In this section, we choose the parameter b as a bifurcation parameter and change the values of other system’s parameters to obtain full view of change in the dynamical behavior of the system with b using several techniques as follows: 4.1. Regular and chaotic regions in double parameters planes We aim to get a global picture of the evolution of the proposed system (1)–(4) via locating the values of parameters that correspond to probable chaotic behavior of the new system. This goal is achieved by performing a systematic search in different planes of values of system parameters. As a large number of values of system parameters are used, we need a fast technique to distinguish regions of regular dynamics from chaotic behavior regions in system parameters planes. Using Lyapunov exponents as chaos indicator seems as a suitable choice but it requires a very long computational time. So, the Fast Lyapunov Indicator (FLI) method [35,36], a faster, computationally efficient and more sensitive tool [37,38], is applied to examine the ordered and chaotic behavior of the new system. In Fig.1, we take b as a bifurcation parameter and fix two of system’s parameters while changing the values of other parameter with b. Then using more than 105 values in each plane of system parameters, we plot in white the chaotic regions and in gray the non-chaotic ones. 4.2. Bifurcation diagrams The bifurcation diagrams of the proposed system are shown in Fig. 2 to investigate the dynamical behavior for different values of b and some chosen fixed values of the other parameters in the following cases: (a) (b) (c) (d) (e) (f) (g)

a ¼ 7; 2 6 b 6 5:5; c ¼ 14; d ¼ 1:6, a ¼ 5; 2 6 b 6 5:5; c ¼ 22; d ¼ 1:5, a ¼ 4:8; 2 6 b 6 5:5; c ¼ 22; d ¼ 1:5, a ¼ 4:8; 4 6 b 6 5:5; c ¼ 22; d ¼ 1:5, a ¼ 7:1; 2:5 6 b 6 5:5; c ¼ 18; d ¼ 2, a ¼ 5:3; 2:5 6 b 6 5:5; c ¼ 15; d ¼ 2, and a ¼ 5:3; 3:64 6 b 6 3:7; c ¼ 15; d ¼ 2.

Fig. 2 illustrates the rich dynamics of the proposed system which varies from stable equilibrium points to periodic orbits, period doubling cascades routs to chaos, boundary crises, interior crises, and chaotic behavior. 4.3. Hopf bifurcation As the stability of equilibrium points of the system (1)–(4) and pitchfork bifurcation are examined in previous section, we investigate the values of system’s parameters at which the system will undergo a Hopf bifurcation. Consider the system

_ XðtÞ ¼ FðX; gÞ;

X 2 Rn ; g 2 R;

ð36Þ

r

where F is a C ðr P 3Þ and the critical point X o exists for g ¼ g0 . Assume that the Jacobian matrix DFðX 0 ; g0 Þ has a simple pair kðg0 Þ ¼ ix0 and no other eigenvalues has zero real part. of pure imaginary eigenvalues kðg0 Þ ¼ aðg0 Þ þ ixðg0 Þ ¼ ix0 ;  Furthermore, assume that

  d ReðkðgÞÞ – 0; dg g¼g0

ð37Þ

then, an Andronov–Hopf bifurcation, corresponding to the birth of a limit cycle, occurs at g ¼ g0 [30–34]. The stability of generated limit cycle is determined by using first Lyapunov coefficient l1 which can be computed as follows: Suppose that the system (36) is written as

_ XðtÞ ¼ AðgÞ þ F1 ðX; gÞ;

ð38Þ

2

where F 1 ¼ OðkX k Þ. Then F1 ðX; g0 Þ can be represented by

F1 ðX; gÞ ¼

1 1 BðX; XÞ þ CðX; X; XÞ þ OðkX k4 Þ 2 6

ð39Þ

in which BðX; YÞ and CðX; Y; UÞ are bilinear and trilinear vector functions of X; Y, and U 2 Rn , respectively, and they can be obtained by

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339

Fig. 1. Chaotic regions in white and nonchaotic regions in gray in system’s parameters planes for (a) c ¼ 14; d ¼ 1:6, (b) c ¼ 17; d ¼ 2:7, (c) a ¼ 6:3; d ¼ 1:6, (d) a ¼ 5:3; d ¼ 1:6, (e) a ¼ 5:3; c ¼ 15, and (f) a ¼ 7:1; c ¼ 18.

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Fig. 2. Bifurcation diagram of system (1)–(4) illustrates dynamical behavior change of state variable x with b when (a) a ¼ 7; c ¼ 14; d ¼ 1:6, (b) a ¼ 5; c ¼ 22; d ¼ 1:5, (c, d) a ¼ 4:8; c ¼ 22; d ¼ 1:5, (e) a ¼ 7:1; c ¼ 18; d ¼ 2, and (f, g) a ¼ 5:3; c ¼ 15; d ¼ 2.

A.M.A. El-Sayed et al. / Applied Mathematics and Computation 239 (2014) 333–345

Bi ðX; YÞ ¼

 n X @ 2 F i ðf; g0 Þ  @fj @fk  j;k¼1

xj yk ;

i ¼ 1; 2; . . . ; n

341

ð40Þ

f¼f0

 n X @ 3 F i ðf; g0 Þ C i ðX; Y; UÞ ¼  @fj @fk @fm  j;k;m¼1

xj yk um ;

i ¼ 1; 2; . . . ; n:

ð41Þ

f¼f0

The first Lyapunov coefficient l1 is evaluated by

l1 ¼

1 Þi  2hp; Bðq; sÞi þ hp; Bðq; rÞi; Re½hp; Cðq; q; q 2x0

ð42Þ

where

Aq ¼ ix0 q; 1

AT p ¼ ix0 p;

Þ; s ¼ A Bðq; q

hp; qi ¼ 1; 1

r ¼ ð2ix0 I  AÞ Bðq; qÞ

ð43Þ ð44Þ

and I is the identity matrix. If l1 < 0ðl1 > 0Þ, then the Hopf bifurcation is nondegenerate and supercritical (subcritical). Generalized Hopf (GH) bifurcation or Bautin bifurcation occurs when the equilibrium point in two parameter family of autonomous system of differential equations has a pair of purely imaginary eigenvalues and the first Lyapunov coefficient for the Andronov–Hopf bifurcation vanishes. Zero-Hopf (ZH) bifurcation or saddle-node Hopf bifurcation occurs when the equilibrium point in two parameter family of autonomous system of differential equations has a zero eigenvalue and a pair of purely imaginary eigenvalues. Some results of numerical calculations of the values of parameters at which Hopf bifurcation exists are shown in Fig. 3 using Matcont continuation toolbox in the following cases:

Fig. 3. Double parameter Hopf bifurcation diagram at (a) a ¼ 5; d ¼ 1, (b) a ¼ 5:3; c ¼ 15, (c) a ¼ 7:1; d ¼ 1:6, and (d) c ¼ 18; d ¼ 2.

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Table 1 The values of Lyapunov exponents computed at different sets of the parameter values. The parameters set

LEs

a ¼ 11; b ¼ 8:5; c ¼ 0:02; d ¼ 1 a ¼ 7; b ¼ 6; c ¼ 14; d ¼ 1:6 a ¼ 4:8; b ¼ 5:4; c ¼ 22; d ¼ 1:5 a ¼ 5:3; b ¼ 2; c ¼ 15; d ¼ 0:9

k1 k1 k1 k1

¼ 0:54; ¼ 0:46; ¼ 0:26; ¼ 0; k2

Notes k2 ¼ 0:04; k3 ¼ 0; k4 ¼ 4:02 k2 ¼ 0; k3 ¼ 6; k4 ¼ 11:07 k2 ¼ 0; k3 ¼ 6:3; k4 ¼ 16:85 ¼ 0; k3 ¼ 8:46; k4 ¼ 10:72

Hyperchaos Chaos Chaos Quasi-periodicity

Fig. 4. Quasi-periodic phase portraits of system (1)–(4) with a ¼ 5:3; b ¼ 2; c ¼ 15; d ¼ 0:9: (a) x—y plane; (b) x—z plane; (c) x—w plane.

Fig. 5. Chaotic phase portraits of system (1)–(4) with a ¼ 7; b ¼ 6; c ¼ 14; d ¼ 1:6: (a) x—y plane; (b) x—z plane; (c) x—w plane.

Fig. 6. Hyperchaotic phase portraits of system (1)–(4) with a ¼ 11; b ¼ 8:5; c ¼ 0:02; d ¼ 1: (a) x—y plane; (b) x—z plane; (c) x—w plane.

(a) a ¼ 5; 1 6 b 6 6; 0 6 c 6 20; d ¼ 1, (b) a ¼ 5:3; 1 6 b 6 6; c ¼ 15; 0 6 d 6 3, (c) a ¼ 7:1; 1 6 b 6 6; 0 6 c 6 20; d ¼ 1:6, and (d) 0 6 a 6 11; 1 6 b 6 6; c ¼ 18; d ¼ 2.

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343

Fig. 7. (a) Circuit implementation of the new system (1)–(4), (b, c) Circuit output corresponding to x—y and x—z phase portrait of system for a ¼ 11; b ¼ 8:5; c ¼ 0:02; d ¼ 1; and (d, e) Circuit output corresponding to x—y and x—z phase portrait of system for a ¼ 7; b ¼ 6; c ¼ 14; d ¼ 1:6:

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4.4. Phase portraits of the system We calculate Lyapunov exponents (LEs) using the algorithm given in [39], for some selected values of systems’ parameters and illustrate them in Table 1. Then, the corresponding phase portraits of system (1)–(4) are shown in Figs. 4–6 to illustrate the cases of hyperchaos, chaos, and quasi-periodicity. 5. Circuit realization of the proposed system We design and implement an electronic circuit to realize the new system (1)–(4) by using TL074 operational amplifiers, analog multipliers such as AD633, and basic circuit elements of resistors and capacitors. Fig. 7 shows the proposed circuit schematics, corresponding to the values of parameters a ¼ 11; b ¼ 8:5; c ¼ 0:02; d ¼ 1, and some examples of the results of circuit simulations. The circuit simulations are carried out using Multisim 11 and the outputs of the proposed circuit are illustrated on the oscilloscope where it is clear that these results agree with the results of numerical simulations. 6. Conclusion This paper is an attempt to introduce and investigate the dynamical behavior of a new 4D system and realize these dynamics in a new circuit. The results of theoretical and numerical study illustrate the rich dynamics of the proposed system which includes periodic orbits, quasi-periodicity, crises, chaotic and hyperchaotic behaviors. Also, the pitchfork and Hopf bifurcations of equilibrium points are examined. The circuit implementation proposed for the new system makes it suitable for real applications such as secure communications and chaos cryptography. Acknowledgments The authors thank the referees for their valuable comments and suggestions. References [1] P.N.V. Tu, Dynamical Systems – An Introduction with Applications in Economics and Biology, Springer-Verlag, 1995. [2] S.H. Strogatz, Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry, and Engineering, Westview Press, 2001. [3] J. Zhaoa, Y. Wu, Q. 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