# Self-Excited and Hidden Chaotic Attractors in Matouk’s Hyperchaotic Systems

Authors: A. Othman Almatroud; A. E. Matouk; Wael W. Mohammed; Naveed Iqbal; Saleh Alshammari
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

Self-excited and hidden chaotic attractors are interesting complex dynamical phenomena. Here, Matouk’s hyperchaotic systems are shown to have self-excited and hidden chaotic attractors, respectively. Two case studies of hidden chaotic attractors are provided which are examined with orders 3.08 and 3.992, respectively. Moreover, self-excited chaotic attractors are found in the fractional-order system and its integer-order counterpart. The existence of one-eyed face self-excited chaotic attractors is also reported in this work. Our results show that the fractional derivative affects the appearances of hidden chaotic attractors in this system.


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## Body

## 1. Introduction

Recently, fractional-order calculus (FC) has been utilized to describe complex dynamics arising from nonlinear dynamical systems (NDS) [1–8]. FC also has some interdisciplinary applications in several scientific fields. For example, it is widely used in engineering [9], heat conduction, and thermoelectricity models [10, 11].In fact,self-excited attractors and hidden attractors are very interesting complex dynamical phenomena that can be found in some NDS involving fractional- or integer-order derivatives [12, 13]. The basin of attraction of a hidden attractor is not connected with small neighborhoods of any stationary state (equilibrium point) unlike the case of self-excited attractors. Therefore, hidden attractors occur in systems with unique stable stationary (equilibrium) states or with no equilibria. One of the famous problems dealing with the hidden attractors is studying mutual disposition and the number of limit cycles in 2D polynomial differential systems [14, 15]. The former problem had firstly been arisen by Hilbert in 1900, in association with Hilbert’s sixteenth problem (2nd part), to investigate hidden periodic oscillations [16]. Hidden attractors had also been arisen in the study of Aizerman and Kalman conjectures. It is found that hidden attractors have potential applications in many fields like ecosystems, climate science, physics, engineering, financial market, aircraft control systems, and phase-locked loop-based circuits. Recently, some NDS involving hidden attractors with stable equilibria and without equilibrium points have appeared [17–22]. For example, Yang et al. observed hidden chaotic attractors in a system with two stable node foci [17]; Dudkowski et al. found hidden chaotic attractors in Chua’s circuit [18]; Munmuangsaen and Srisuchinwong reported hidden chaotic attractors in the well-known Lorenz model for atmospheric convection [19]; and Kuznetsov and Leonov studied hidden attractors in some engineering and fundamental problems [20, 21]. Singh and Roy obtained hidden attractors in a new 4D chaotic system involving a line of equilibria [22]. Furthermore, the interesting phenomena of coexisting self-excited attractors (CS-EAs) occur in a system when several attractors with independent basin sets of attraction take place with different initial conditions for a given parameter set. The CS-EA is usually determined via the standard computational procedures. In [23], Singh and Roy showed the coexistence of self-excited chaotic attractors in a new 3D chaotic system.In this work, we mainly focus on exploring hidden chaotic attractors and self-excited coexisting attractors in a novel hyperchaotic system given by Matouk [24]. The proposed system is described by a set of four coupled nonlinear integer or fractional-order differential equations [25] and is known here as Matouk’s hyperchaotic system. Matouk’s system was applied to solve some potential engineering problems [3, 24]. Here, the novel dynamical behaviors of Matouk’s hyperchaotic systems have the following different types: (i) hidden chaotic attractors exist in the integer-order and the fractional-order Matouk’s hyperchaotic systems with one saddle equilibrium point and two locally asymptotically stable equilibria; (ii) variety of self-excited chaotic attractors exist in this system including the newly found one-eyed face attractors; (iii) coexistence of multihidden attractors, with different topological shapes, are also found in this system.The organization of this paper is described as follows: Section2 is devoted to represent the basic notations related to the fractional analysis used in this work. In Section 3, we present some dynamical behaviors of Matouk’s hyperchaotic system including some stability conditions of its equilibrium solutions and some conditions for its chaoticity based on Shil’nikov’s Theorem. In Section 4, we investigate the hidden chaotic attractors and self-excited chaotic attractors that existed in Matouk’s systems. Finally, this work is ended by a conclusion in Section 5.


## 2. Preliminaries

The Caputo fractional differential operator [26] can be expressed by(1)DCap0ηωt=∫0tt−γp−η−1ωpγdγΓp−η,where η∈R+ and ωp represents the pth-order derivative of ωγ such that p−1<η<p∈Z+. Obviously, this operator is nonlocal with a singular kernel. Therefore, it is convenient to describe the complex dynamics arising from the NDS.Consider the autonomous system(2)dYηdtη=fY,where 0<η≤1 and f is a nonlinear vector function. If J represents the Jacobian of the linearized counterpart of autonomous system (2), then the local stability of an equilibrium point Υ¯ of system (2) is governed by the following Matignon’s inequalities:(3)η<2πargςj,;j=1,…,n,where ςj is an arbitrary eigenvalue of JY¯.Throughout this work, we will refer by LAS an equilibrium pointY¯ that is locally asymptotically stable; i.e., all the eigenvalues of Y¯ satisfies conditions (3).


## 3. The Systems’ Description

Matouk’s system with caputo fractional operators [25] is modeled by the following set of equations:(4)DCap0αy1=ay4−y2+hy1−y1y4,DCap0αy2=by1+y4−y1y3,DCap0αy3=y12−cy3,DCap0αy4=dy4,where the parameter set Α=a,b,c,d,h⊂R and α∈0,1. System (4) is simple in comparison with similar types of 4D systems and reports studies of important problems [3, 24, 25]. It has the equilibria(5)Eq1=0,0,0,0,Eq2=σ,σha,b,0,Eq3=−σ,−σha,b,0,σ=bc.Assume thatςi is an eigenvalue of the Jacobian matrix J computed after the linearization of Matouk’s system (4) and J is evaluated at any of the equilibrium points described in equations (5). For α=1,Eq1 has the following eigenvalues:(6)ς1=−c,ς2=d,ς3,4=h±h2−4ab2.Thus,Eq1 is LAS if and only if(7)ab∈R+,c∈R+,d∈R−,h∈R−.The other equilibrium pointsEq2 and Eq3 have the same characteristic equation:(8)ς−dς3+ρ1ς2+ρ2ς+ρ3=0,where(9)ρ1=c−h,ρ2=−hc,ρ3=−2abc.According to the classic Routh–Hurwitz criterion and the stability theory of integer-order differential systems, the equilibrium statesEq2 and Eq3 are LAS if and only if(10)d<0,ρ1>0,ρ3>0,ρ1ρ2>ρ3.Forα∈0,1 and according to Theorem 2 in Ref. [25], the original equilibrium point Eq1 satisfies the following cases:(1)
Eq1 is LAS inside the interval α∈0,1 when c∈R+andd∈R− provided that(i)
h∈R−,ab∈R+(ii)
ab>h2/4;ab∈R+;α<2/πtan−14ab−h2/h(2)
Eq1 is saddle if dc∈R+ or ab∈R− and the parameter h has not vanishedAlso, the local stability of the other nonorigin equilibrium points is determined by Matignon’s conditions (3) or by testing the conditions of Theorem 3 in Ref. [25].To discuss the chaoticity in system (4), we firstly assume that this system, with α=1, has a saddle foci equilibria type in which the eigenvalues satisfy one of the following three cases:(A)ς1,2=−υ±iϖ¯,ς3∈R+,andς4≤0, where ϖ¯ is not vanished and υ∈R+ such that ς3−υ∈R+((B)ς1,2=υ±iϖ¯,ς3∈R−,andς4≤0, where ϖ¯ is not vanished and υ∈R+ such that ς3+υ∈R−(C)ς1,2=υ1±iϖ¯1andς3,4=−υ2±iϖ¯2, where ϖ¯1andϖ¯2 are not vanished and υ1,υ2∈R+ such that υ1≠υ2.Then, we introduce the following theorems [27, 28].Theorem 1.
Assume that the original equilibrium stateEq1 satisfies one of conditions (A–C), and assume that Eq1 is connected to itself by a homoclinic orbit. Then, the integer-order form of system (4) has Smale horseshoe chaos if eigenvalues (6) of Eq1 satisfy any of the following conditions:(I)ς3−2υ∈R− and conditions (A) hold(II)ς3+2υ∈R− and conditions (B) hold(III) Conditions (C) holdTheorem 2.
Assume that the nonorigin equilibrium statesEq2 and Eq3 satisfy one of conditions (A–C), and also assume that Eq2 and Eq3 are connected with heteroclinic orbits. Then, the integer-order form of system (4) has Smale horseshoe chaos if the eigenvalues of (8) satisfy any of the following conditions:(I)ς3−2υ∈R− and conditions (A) hold(II)ς3+2υ∈R− and conditions (B) hold(III) Conditions (C) holdFor example, the parameter selectiona=12,b=15,c=7.35,d=−0.85,h=−0.95, and α=1 fulfills condition (II) of Theorem 1 since the eigenvalues of the saddle-focus origin equilibrium state Eq1 satisfy this condition. Figure 1 shows the existence of homoclinic orbits shrinking to the origin state Eq1 for the aforementioned selection of the parameter values.Figure 1
3D view of the integer-order Matouk’s system (4) with a=12,b=15,c=7.35,d=−0.85,h=−0.95.


## 4. Hidden and Self-Excited Chaotic Attractors in Matouk’s System

In this part, we will discuss the occurrence of hidden chaotic attractors and self-excited chaotic attractors in Matouk’s systems (4). Indeed, the computational classification of an attractor is divided into two main types: hidden and self-excited attractors. These types can clearly be defined as follows.Definition 1.
(see [12]). A self-excited attractor is an attractor whose basin of attraction intersects with any open neighborhood of an equilibrium (stationary) state; otherwise, the attractor is said to be a hidden attractor.
Thus, the basin of attraction of a hidden attractor is not connected with stationary states (equilibria).
In [29], Prakash et al. pointed out that hyperchaotic systems can also be classified into two main types: (i) hidden attractors and (ii) self-excited attractors.
Matouk’s equations (4) are coded based on the PECE (Predict, Evaluate, Correct, Evaluate) numerical scheme [30, 31]. These codes are executed to obtain attractors with different topological shapes involving a variety of dynamical behaviors such as hidden and self-excited chaotic attractors and coexisting multihidden attractors.
Indeed, there exist different techniques to locate hidden attractors such as the method of perpetual points [32, 33]. To illustrate the hidden chaotic attractors, the PECE scheme is employed to obtain sets of 100 trajectories initiating from the unstable steady state. However, to visualize the hidden chaotic attractor, only representative trajectories have been depicted.
In the following, we will show the existence of different types of hidden and self-excited chaotic attractors in Matouk’s equations (4) with fractional orders.


### 4.1. The First Case Study

In this section, the PECE method is used to obtain the numerical solution of the fractional Matouk’s system with step size 0.005. Using the parameter setΑ1=−3,15,0.6,−0.0001,−1.5, the system’s nonorigin equilibrium points are(11)Eq2=3,1.5,15,0,Eq3=−3,−1.5,15,0,with eigen-spectrum Λ=−0.0001,−4.531316699,1.21565835±3.230981389i,i=−1. So, they are LAS when α=0.77 since they satisfy Matignon’s stability conditions (3). Meanwhile, the original equilibrium point Eq1 is a saddle for any α∈0,1, since it satisfies stability condition (2) of Section 3. Moreover, it has the eigen-spectrum Λ=6,−0.0001,−0.6,−7.5 which means that it has an index one type of saddleness.To explain the occurrence of the hidden chaotic attractors in the fractional case of system (4), we will show that trajectories originate from small neighborhoods of the unstable origin state Eq1 and are all attracted by the nonorigin LAS points Eq2 and Eq3. The results are depicted in Figure 2 in which the hidden chaotic attractor has the green color that surrounds the emanating trajectories from the close vicinity of the original state Eq1 which either approach the nonorigin state Eq3=−3,−1.5,15,0 (blue plot) or tend to the nonorigin state Eq2=3,1.5,15,0 (red plot). Thus, the selection of the initial conditions y10,y20,y30,y40=0.1,0,1,0.01 leads to a chaotic attractor whereas, the initial conditions y10,y20,y30,y40=3,1,15,0.0199 and y10,y20,y30,y40=−3,−1,15,0.0199 lead to the red nonchaotic (asymptotic) attractor of Eq2 and the blue nonchaotic (asymptotic) attractor of Eq3, respectively. Thus, according to the abovementioned definition, a hidden chaotic attractor (the green domain in Figure 2) emerges if there exists another region from which this motion is chaotic. Obviously, the hidden chaotic attractor occurs when the order is 4×0.77=3.08.Figure 2
Hidden chaotic attractor (green plot) of system (4), with parameter selection Α1 and α=0.77, showing the one-point attractor corresponding to Eq2=3,1.5,15,0 (red plot) and one-point attractor corresponding to Eq3=−3,−1.5,15,0 (blue plot): (a)y1y2-view; (b)y1y3-view.
(a)(b)Obviously, the fractional derivative affects the stability of the fractional form of system (4) according to Matignon’s inequalities (3) and the abovementioned discussion. However, when α=1, the nonorigin states Eq2 and Eq3 lose their stability and the corresponding two LAS attractors are replaced by two unstable focus-node points with an index 2 (saddle foci). However, Eq1 remains a saddle point of an index 1. Clearly, conditions (II) of Theorem 2 hold in this case. Then, if there exists a heteroclinic orbit that connects Eq2 and Eq3, system (4) exhibits a Smale horse-type chaos. Accordingly, the occurrence of chaotic attractors is expected. Figure 3 shows the existence of a self-excited chaotic attractor using the abovementioned selection of initial conditions along with the parameter set Α1 and α=1. The computation of the corresponding basin set of attraction is illustrated in Figure 4.Figure 3
3D plot of a self-excited chaotic attractor in the integer-order Matouk’s system using the parameter setΑ1 and initial conditions 0.1,0,1,0.01 for green trajectory, 3,1,15,0.0199 for red trajectory, and −3,−1,15,0.0199 for blue trajectory.Figure 4
A cross section fory3=0andy4=0 of basins of attraction for the self-excited attractor.The bifurcation diagrams corresponding to the abovementioned sets of initial conditions are illustrated in Figure5. Moreover, Figure 5(c) shows that hidden chaotic attractors in system (4) exist for the selection of the parameter set Α1 where c∈0.36,0.65 and α=0.77. Also, according to the selection of parameters in Figure 5(c), a new shape of self-excited chaotic attractors, one-eyed face, is found when c=0.30,0.32,0.33,0.34,0.35 and using different selections of initial conditions. The calculations of the corresponding spectrum of Lyapunov exponents λi,s (or LEs) are depicted in Figure 6. Also, a variety of self-excited chaotic attractors in the fractional system are depicted in Figure 7 where trajectories with initial conditions in neighborhoods of the nonorigin points Eq2 and Eq3 (red domain) and (blue domain), respectively, tend to the chaotic attractor (green domain) surrounding the saddle origin point Eq1. Appearance of the newly found one-eyed face self-excited chaotic attractors is depicted in Figure 8. Hence, according to [34], system (4) fulfills the standard criteria for a new chaotic system. Furthermore, the coexistence of multihidden attractors is given in Figure 9 where their basins of attraction do not intersect with a certain neighborhood of the equilibrium points. Obviously, Figure 9 shows hidden attractors with different topological shapes such as coexistence of unstable saddle-focus attractors, coexistence of homoclinic orbits, coexistence of invariant closed curves, coexistence of higher periodic orbits, and existence of strange hidden chaotic attractors.Figure 5
The bifurcation diagrams of the fractional-order Matouk’s system with varying system’s parameters, using the parameter setΑ1 and α=0.77 and initial conditions 0.1,0,1,0.01 (green plot); 3,1,15,0.0199 (red plot), and −3,−1,15,0.0199 (blue plot).
(a)(b)(c)(d)(e)Figure 6
Lyapunov spectrum of system (4) with fixing the parameter values according to the set Α1Α1, (a) except varying the parameter a and setting α=0.77α=0.77, (b) except varying the parameter b and setting α=0.77, (c) except varying the parameter c and setting α=0.77, (d) except varying the parameter h and setting α=0.77, and (e) varyingα.
(a)(b)(c)(d)(e)Figure 7
The projection of the self-excited attractor of the fractional-order Matouk’s system in they1y2−plane with initial conditions 0.1,0,1,0.01 for green trajectory; 3,1,15,0.0199 for red trajectory; and −3,−1,15,0.0199 for blue trajectory and using the following parameter values: (a)a=−3.2,b=15,c=0.6,d=−0.0001,h=−1.5,α=0.77; (b)a=−3,b=15.75,c=0.6,d=−0.0001,h=−1.5,α=0.77; (c) a=−3,b=15,c=0.6,d=−0.0001,h=−1.46,α=0.77; (d)a=−3,b=15,c=0.6,d=−0.0001,h=−1.5,α=0.775.
(a)(b)(c)(d)Figure 8
The one-eyed face self-excited chaotic attractor of the fractional-order Matouk’s system in they1y3−plane with initial conditions 0.01,0.01,0.01,0.01, a=−3,b=15,d=−0.0001,h=−1.5,α=0.77, and (a)c=0.34 and (b)c=0.35.
(a)(b)Figure 9
3D view of multihidden attractors in the fractional-order Matouk’s system, with the fractional parameterα=0.77 and initial conditions 0.1,0,1,0.01 for green trajectory, 3,1,15,0.0199 for red trajectory, and −3,−1,15,0.0199 for blue trajectory, showing the following: (a) coexisting unstable saddle-focus attractors for a=−3,b=15,c=0.8,d=−0.0001,h=−1.5; (b) coexisting homoclinic orbits for a=−3,b=15,c=0.6,d=−0.0001,h=0.0.18; (c) coexisting invariant closed curves for a=−3,b=15,c=0.6,d=−0.0001,h=0.3; (d) coexisting higher periodic orbits and homoclinic orbits for a=−3,b=15,c=0.6,d=−0.0001,h=0.25; and (e) existence of strange hidden chaotic attractor in the integer-order Matouk’s system, with a=−3,b=15,c=0.1,d=−1,h=−1.5 using initial conditions 0.1,0,1,0.01 for green trajectory.
(a)(b)(c)(d)(e)Obviously, Figure5(e) shows that Matouk’s system still exhibits chaotic attractors for a wide range as the derivative order is varied (0.77≤α≤1). This interesting result has also never been shown in an existing fractional system.


### 4.2. The Second Case Study

In this case, the PECE method is also used with step size 0.005 along with the set of parametersA2=−2,15,0.45,−0.15,−7.775; the system’s nonorigin equilibrium points are(12)Eq2=2.5981,10.1,15,0,Eq3=−2.5981,−10.1,15,0.Forα∈0,1, the nonorigin points are LAS since they satisfy stability conditions (3) and (9). However, the origin point Eq1 is the saddle of an index 1. This case also fulfills conditions (II) of Theorem 2. Hence, if there exists a heteroclinic orbit that connects Eq2 and Eq3, the integer-order case of system (4) exhibits a Smale horse-type chaos.Also, in this case, a hidden chaotic attractor for system (4), α∈0,1, is created by trajectories starting from the close vicinity of the origin point Eq1 that either approaches Eq2=2.5981,10.1,15,0 or Eq3=−2.5981,−10.1,15,0. Hence, the choice of initial data y10,y20,y30,y40=0.01,0.01,0.01,0.01 leads to the chaotic attractor (green domain) whereas, the other selections y10,y20,y30,y40=2.7,10.6,15,0.0001 and y10,y20,y30,y40=−2.7,−10.6,15,0.0001 lead to the asymptotic attractor of Eq2 (red domain) and the asymptotic attractor of Eq3 (blue domain), respectively. These hidden chaotic attractors are depicted in Figures 10 and 11, for the integer-order (α=1) and the fractional-order (α=0.998) cases, respectively. Consequently, in this case, the hidden chaotic attractor exists as the order is 4×0.998=3.992. The computation of a basin set of attraction corresponding to Figure 10 is illustrated in Figure 12. Moreover, computations of the corresponding bifurcation diagrams are illustrated in Figures 13 and 14 for α=1 and α=0.998, respectively.Figure 10
View of the integer-order Matouk’s system (4) in the y1y3−plane with parameter selection Α2, showing the hidden chaotic attractor (green plot), one-point attractor corresponding to Eq2=2.5981,10.1,15,0 (red plot), and one-point attractor corresponding to Eq3=−2.5981,−10.1,15,0 (blue plot).Figure 11
View of Matouk’s system (4) in y1y3−plane with parameter selection Α2 and α=0.998, showing the hidden chaotic attractor (green plot), one-point attractor corresponding to Eq2=2.5981,10.1,15,0 (red plot), and one-point attractor corresponding to Eq3=−2.5981,−10.1,15,0 (blue plot).Figure 12
A cross section fory3=0andy4=0 of three colored basins of attraction for the hidden chaotic attractor (green), the equilibrium Eq2 (orange), and the equilibrium Eq3 (turquoise).Figure 13
The bifurcation diagrams of the integer-order Matouk’s system with varying one parameter and fixing the others according to the setΑ2 and using initial conditions 0.01,0.01,0.01,0.01 (green domain), 2.7,10.6,15,0.0001 (red domain), and −2.7,−10.6,15,0.0001 (blue domain).
(a)(b)(c)(d)Figure 14
The bifurcation diagrams of the fractional-order Matouk’s system withα=0.998, varying one parameter and fixing the others according to the set Α2 and using initial conditions 0.01,0.01,0.01,0.01 (green domain), 2.7,10.6,15,0.0001 (red domain), and −2.7,−10.6,15,0.0001 (blue domain).
(a)(b)(c)(d)(e)


## 4.1. The First Case Study

In this section, the PECE method is used to obtain the numerical solution of the fractional Matouk’s system with step size 0.005. Using the parameter setΑ1=−3,15,0.6,−0.0001,−1.5, the system’s nonorigin equilibrium points are(11)Eq2=3,1.5,15,0,Eq3=−3,−1.5,15,0,with eigen-spectrum Λ=−0.0001,−4.531316699,1.21565835±3.230981389i,i=−1. So, they are LAS when α=0.77 since they satisfy Matignon’s stability conditions (3). Meanwhile, the original equilibrium point Eq1 is a saddle for any α∈0,1, since it satisfies stability condition (2) of Section 3. Moreover, it has the eigen-spectrum Λ=6,−0.0001,−0.6,−7.5 which means that it has an index one type of saddleness.To explain the occurrence of the hidden chaotic attractors in the fractional case of system (4), we will show that trajectories originate from small neighborhoods of the unstable origin state Eq1 and are all attracted by the nonorigin LAS points Eq2 and Eq3. The results are depicted in Figure 2 in which the hidden chaotic attractor has the green color that surrounds the emanating trajectories from the close vicinity of the original state Eq1 which either approach the nonorigin state Eq3=−3,−1.5,15,0 (blue plot) or tend to the nonorigin state Eq2=3,1.5,15,0 (red plot). Thus, the selection of the initial conditions y10,y20,y30,y40=0.1,0,1,0.01 leads to a chaotic attractor whereas, the initial conditions y10,y20,y30,y40=3,1,15,0.0199 and y10,y20,y30,y40=−3,−1,15,0.0199 lead to the red nonchaotic (asymptotic) attractor of Eq2 and the blue nonchaotic (asymptotic) attractor of Eq3, respectively. Thus, according to the abovementioned definition, a hidden chaotic attractor (the green domain in Figure 2) emerges if there exists another region from which this motion is chaotic. Obviously, the hidden chaotic attractor occurs when the order is 4×0.77=3.08.Figure 2
Hidden chaotic attractor (green plot) of system (4), with parameter selection Α1 and α=0.77, showing the one-point attractor corresponding to Eq2=3,1.5,15,0 (red plot) and one-point attractor corresponding to Eq3=−3,−1.5,15,0 (blue plot): (a)y1y2-view; (b)y1y3-view.
(a)(b)Obviously, the fractional derivative affects the stability of the fractional form of system (4) according to Matignon’s inequalities (3) and the abovementioned discussion. However, when α=1, the nonorigin states Eq2 and Eq3 lose their stability and the corresponding two LAS attractors are replaced by two unstable focus-node points with an index 2 (saddle foci). However, Eq1 remains a saddle point of an index 1. Clearly, conditions (II) of Theorem 2 hold in this case. Then, if there exists a heteroclinic orbit that connects Eq2 and Eq3, system (4) exhibits a Smale horse-type chaos. Accordingly, the occurrence of chaotic attractors is expected. Figure 3 shows the existence of a self-excited chaotic attractor using the abovementioned selection of initial conditions along with the parameter set Α1 and α=1. The computation of the corresponding basin set of attraction is illustrated in Figure 4.Figure 3
3D plot of a self-excited chaotic attractor in the integer-order Matouk’s system using the parameter setΑ1 and initial conditions 0.1,0,1,0.01 for green trajectory, 3,1,15,0.0199 for red trajectory, and −3,−1,15,0.0199 for blue trajectory.Figure 4
A cross section fory3=0andy4=0 of basins of attraction for the self-excited attractor.The bifurcation diagrams corresponding to the abovementioned sets of initial conditions are illustrated in Figure5. Moreover, Figure 5(c) shows that hidden chaotic attractors in system (4) exist for the selection of the parameter set Α1 where c∈0.36,0.65 and α=0.77. Also, according to the selection of parameters in Figure 5(c), a new shape of self-excited chaotic attractors, one-eyed face, is found when c=0.30,0.32,0.33,0.34,0.35 and using different selections of initial conditions. The calculations of the corresponding spectrum of Lyapunov exponents λi,s (or LEs) are depicted in Figure 6. Also, a variety of self-excited chaotic attractors in the fractional system are depicted in Figure 7 where trajectories with initial conditions in neighborhoods of the nonorigin points Eq2 and Eq3 (red domain) and (blue domain), respectively, tend to the chaotic attractor (green domain) surrounding the saddle origin point Eq1. Appearance of the newly found one-eyed face self-excited chaotic attractors is depicted in Figure 8. Hence, according to [34], system (4) fulfills the standard criteria for a new chaotic system. Furthermore, the coexistence of multihidden attractors is given in Figure 9 where their basins of attraction do not intersect with a certain neighborhood of the equilibrium points. Obviously, Figure 9 shows hidden attractors with different topological shapes such as coexistence of unstable saddle-focus attractors, coexistence of homoclinic orbits, coexistence of invariant closed curves, coexistence of higher periodic orbits, and existence of strange hidden chaotic attractors.Figure 5
The bifurcation diagrams of the fractional-order Matouk’s system with varying system’s parameters, using the parameter setΑ1 and α=0.77 and initial conditions 0.1,0,1,0.01 (green plot); 3,1,15,0.0199 (red plot), and −3,−1,15,0.0199 (blue plot).
(a)(b)(c)(d)(e)Figure 6
Lyapunov spectrum of system (4) with fixing the parameter values according to the set Α1Α1, (a) except varying the parameter a and setting α=0.77α=0.77, (b) except varying the parameter b and setting α=0.77, (c) except varying the parameter c and setting α=0.77, (d) except varying the parameter h and setting α=0.77, and (e) varyingα.
(a)(b)(c)(d)(e)Figure 7
The projection of the self-excited attractor of the fractional-order Matouk’s system in they1y2−plane with initial conditions 0.1,0,1,0.01 for green trajectory; 3,1,15,0.0199 for red trajectory; and −3,−1,15,0.0199 for blue trajectory and using the following parameter values: (a)a=−3.2,b=15,c=0.6,d=−0.0001,h=−1.5,α=0.77; (b)a=−3,b=15.75,c=0.6,d=−0.0001,h=−1.5,α=0.77; (c) a=−3,b=15,c=0.6,d=−0.0001,h=−1.46,α=0.77; (d)a=−3,b=15,c=0.6,d=−0.0001,h=−1.5,α=0.775.
(a)(b)(c)(d)Figure 8
The one-eyed face self-excited chaotic attractor of the fractional-order Matouk’s system in they1y3−plane with initial conditions 0.01,0.01,0.01,0.01, a=−3,b=15,d=−0.0001,h=−1.5,α=0.77, and (a)c=0.34 and (b)c=0.35.
(a)(b)Figure 9
3D view of multihidden attractors in the fractional-order Matouk’s system, with the fractional parameterα=0.77 and initial conditions 0.1,0,1,0.01 for green trajectory, 3,1,15,0.0199 for red trajectory, and −3,−1,15,0.0199 for blue trajectory, showing the following: (a) coexisting unstable saddle-focus attractors for a=−3,b=15,c=0.8,d=−0.0001,h=−1.5; (b) coexisting homoclinic orbits for a=−3,b=15,c=0.6,d=−0.0001,h=0.0.18; (c) coexisting invariant closed curves for a=−3,b=15,c=0.6,d=−0.0001,h=0.3; (d) coexisting higher periodic orbits and homoclinic orbits for a=−3,b=15,c=0.6,d=−0.0001,h=0.25; and (e) existence of strange hidden chaotic attractor in the integer-order Matouk’s system, with a=−3,b=15,c=0.1,d=−1,h=−1.5 using initial conditions 0.1,0,1,0.01 for green trajectory.
(a)(b)(c)(d)(e)Obviously, Figure5(e) shows that Matouk’s system still exhibits chaotic attractors for a wide range as the derivative order is varied (0.77≤α≤1). This interesting result has also never been shown in an existing fractional system.


## 4.2. The Second Case Study

In this case, the PECE method is also used with step size 0.005 along with the set of parametersA2=−2,15,0.45,−0.15,−7.775; the system’s nonorigin equilibrium points are(12)Eq2=2.5981,10.1,15,0,Eq3=−2.5981,−10.1,15,0.Forα∈0,1, the nonorigin points are LAS since they satisfy stability conditions (3) and (9). However, the origin point Eq1 is the saddle of an index 1. This case also fulfills conditions (II) of Theorem 2. Hence, if there exists a heteroclinic orbit that connects Eq2 and Eq3, the integer-order case of system (4) exhibits a Smale horse-type chaos.Also, in this case, a hidden chaotic attractor for system (4), α∈0,1, is created by trajectories starting from the close vicinity of the origin point Eq1 that either approaches Eq2=2.5981,10.1,15,0 or Eq3=−2.5981,−10.1,15,0. Hence, the choice of initial data y10,y20,y30,y40=0.01,0.01,0.01,0.01 leads to the chaotic attractor (green domain) whereas, the other selections y10,y20,y30,y40=2.7,10.6,15,0.0001 and y10,y20,y30,y40=−2.7,−10.6,15,0.0001 lead to the asymptotic attractor of Eq2 (red domain) and the asymptotic attractor of Eq3 (blue domain), respectively. These hidden chaotic attractors are depicted in Figures 10 and 11, for the integer-order (α=1) and the fractional-order (α=0.998) cases, respectively. Consequently, in this case, the hidden chaotic attractor exists as the order is 4×0.998=3.992. The computation of a basin set of attraction corresponding to Figure 10 is illustrated in Figure 12. Moreover, computations of the corresponding bifurcation diagrams are illustrated in Figures 13 and 14 for α=1 and α=0.998, respectively.Figure 10
View of the integer-order Matouk’s system (4) in the y1y3−plane with parameter selection Α2, showing the hidden chaotic attractor (green plot), one-point attractor corresponding to Eq2=2.5981,10.1,15,0 (red plot), and one-point attractor corresponding to Eq3=−2.5981,−10.1,15,0 (blue plot).Figure 11
View of Matouk’s system (4) in y1y3−plane with parameter selection Α2 and α=0.998, showing the hidden chaotic attractor (green plot), one-point attractor corresponding to Eq2=2.5981,10.1,15,0 (red plot), and one-point attractor corresponding to Eq3=−2.5981,−10.1,15,0 (blue plot).Figure 12
A cross section fory3=0andy4=0 of three colored basins of attraction for the hidden chaotic attractor (green), the equilibrium Eq2 (orange), and the equilibrium Eq3 (turquoise).Figure 13
The bifurcation diagrams of the integer-order Matouk’s system with varying one parameter and fixing the others according to the setΑ2 and using initial conditions 0.01,0.01,0.01,0.01 (green domain), 2.7,10.6,15,0.0001 (red domain), and −2.7,−10.6,15,0.0001 (blue domain).
(a)(b)(c)(d)Figure 14
The bifurcation diagrams of the fractional-order Matouk’s system withα=0.998, varying one parameter and fixing the others according to the set Α2 and using initial conditions 0.01,0.01,0.01,0.01 (green domain), 2.7,10.6,15,0.0001 (red domain), and −2.7,−10.6,15,0.0001 (blue domain).
(a)(b)(c)(d)(e)


## 5. Conclusion

We have obtained a rich variety of complex dynamics including the occurrence of hidden chaotic attractors in a new 4D system, named Matouk’s fractional-order hyperchaotic system. We have introduced two case studies, in which hidden chaotic attractors are found in the fractional-order system, with orders 3.08 and 3.992, respectively. In the second case, we have shown that hidden chaotic attractors are obtained in both the integer and fractional-order Matouk’s systems. In addition, self-excited chaotic attractors have been obtained in the fractional-order system and its integer-order form. Moreover, the coexistence of multihidden attractors has been shown in the integer-order Matouk’s system and its fractional version. The obtained results verify that the fractional form is a better candidate to discuss the complex dynamics of Matouk’s hyperchaotic system since it exhibits a variety of complex dynamics including self-excited and hidden chaotic attractors because of its higher degrees of freedom.Numerical simulations, based on the PECE algorithm, have been used to confirm these important foundations and successful results through the reported strange attractors, computations of bifurcation diagrams, Lyapunov exponents, and basins of attractions. Thus, the system shows a variety of rich dynamics especially chaotic attractors that still exist in this system for a wide dynamical range as the derivative order is varied (0.77≤α≤1) in some cases. Also, the existence of new shaped attractors in this simple 4D system has been depicted such as the existence of the one-eyed face self-excited chaotic attractors. Consequently, the system has been shown to be a new chaotic system according to the basic criteria in the literature.Future work may investigate the analytical conditions for the existence of Shil’nikov’s chaos in the fractional-order case.



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