MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi 10.1155/2019/6908607 6908607 Research Article Hidden Coexisting Attractors in a Fractional-Order System without Equilibrium: Analysis, Circuit Implementation, and Finite-Time Synchronization https://orcid.org/0000-0002-9497-1830 Zheng Guangchao 1 2 3 Liu Ling 1 2 Liu Chongxin 1 2 Petras Ivo 1 State Key Laboratory of Electrical Insulation and Power Equipment Xi’an Jiaotong University Xi’an 710049 China xjtu.edu.cn 2 School of Electrical Engineering Xi’an Jiaotong University Xi’an 710049 China xjtu.edu.cn 3 Baoding Power Supply Branch of State Grid Hebei Electric Power Supply Co. Ltd. Baoding 071000 China 2019 2272019 2019 24 02 2019 05 06 2019 03 07 2019 2272019 2019 Copyright © 2019 Guangchao Zheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, a novel three-dimensional fractional-order chaotic system without equilibrium, which can present symmetric hidden coexisting chaotic attractors, is proposed. Dynamical characteristics of the fractional-order system are analyzed fully through numerical simulations, mainly including finite-time local Lyapunov exponents, bifurcation diagram, and the basins of attraction. In particular, the system can generate diverse coexisting attractors varying with different orders, which presents ample and complex dynamic characteristics. And there is great potential for secure communication. Then electronic circuit of the fractional-order system is designed to help verify its effectiveness. What is more, taking the disturbances into account, a finite-time synchronization of the fractional-order chaotic system without equilibrium is achieved and the improved controller is proven strictly by applying finite-time stable theorem. Eventually, simulation results verify the validity and rapidness of the proposed method. Therefore, the fractional-order chaotic system with hidden attractors can present better performance for practical applications, such as secure communication and image encryption, which deserve further investigation.

National Natural Science Foundation of China 51877162
1. Introduction

Since Lorenz system  was put forward in the 1960s, chaos theory has made a great leap forward and researchers have never ceased to delve deeply into it. Among them, three-dimensional autonomous chaotic systems are investigated most extensively at home and abroad, such as Rössler system , Chen system , Lü system , and Liu system . And these chaotic systems have achieved comprehensive application in real life, including secure communication, image encryption, medicine, and aerospace.

In recent years, however, increasing attention has been paid to hidden attractors, which were discovered first in the Chua’circuit . For those conventional chaotic systems above, their attractors are called self-excited attractors , whose basins of attraction can intersect unstable equilibrium points. In contrast, the attractors are defined as hidden attractors if its basins of attraction do not touch any unstable equilibrium point. The current literatures indicate that hidden attractors originate from several types of particular dynamical systems. And to be specific, the dynamical systems with one stable equilibrium [8, 9], a line equilibrium [10, 11], or no equilibrium  can all fall into this category. Even there are hidden attractors found in some special chaotic systems  which have both unstable equilibrium and stable equilibria. It is the presence of hidden attractors that pose a new challenge to Shilnikov criteria which illustrate that no less than one unstable equilibrium is needed to generate chaos. Meanwhile, with the lack of correlation between hidden attractors and equilibria, precise localization of these unpredictable attractors can be of great intractability. In addition, it also has been a focus of concern about coexistence of hidden attractors , which presents more complex and diverse dynamic behavior, and enhances the security ability in the field of cryptography application.

However, the existing research findings about hidden attractors have been mostly focused on integer-order dynamical systems and relatively few efforts have been done for fractional-order systems with hidden attractors [16, 2124]. The latter are worthy of further investigation owing to excellent characteristics of fractional calculus. To begin with, fractional-order systems, a further generalization, are more applicable by contrast with integer-order systems. Indeed, it can be said integer-order systems are the particular case of fractional-order systems and the idealization of real matter. Accordingly, fractional-order systems can describe the physical characteristics of the system engineering more clearly and reflect the essence and law of things in application domain. Then, fractional-order systems have special historical memory function, which are suitable for describing systems that rely on historical information. Furthermore, they can exhibit more complex and richer dynamic characteristics, with parameter selection more flexible. Thus, fractional calculus shows strong potential in practical engineering applications .

Based on the above issues, this paper discusses the fractional-order form of a new 3D symmetric system without equilibrium and gives emphasis to the influence of the order on dynamic behavior of the system. The detailed dynamic analysis was carried out, mainly including existence of multiple attractors. And a finite-time synchronization of the fractional-order chaotic system with hidden attractors is realized. It has enormous value for the application of chaos technology in practical engineering field, especially in secure communication.

This paper is organized as follows. In Section 2, several definitions of fractional-order derivative and the fractional-order chaotic system model without equilibrium are introduced. In Section 3, dynamic analysis of the system is carried out at length. Section 4 gives the electronic circuit implementation results of the fractional-order system with hidden coexisting attractors. In Section 5, based on the finite-time stable theorem, a finite-time synchronization of the proposed system is implemented. Section 6 summarizes the conclusions.

2. Fractional-Order Form of the System without Equilibrium 2.1. Definitions of Fractional-Order Derivative

So far, a few different definitions [29, 30] of fractional-order derivative have been proposed, such as Grunwald–Letnikov (G-L), Riemann–Liouville (R-L), Caputo, and Conformable definitions. In general, the first three definitions are extensively employed and the Caputo definition is more suitable for engineering applications. In this paper, we conduct the study on the basis of the Caputo definition, which is given by(1)aCDtqft=1Γn-q×atfnτt-τq+1-ndτ

In definition (1), n is the first integer which is not less than q and n-1<q<n; Γ(·) is the Gamma function.

2.2. The Fractional-Order Chaotic System Model without Equilibrium

Sprott A system  is a three-dimensional conservative system and it has no equilibrium which depends on its third equation (1-y2). Based on Sprott A system, we construct a new 3D fractional-order chaotic system without equilibrium by imposing nonlinear functions associated with variables x and y on the third equation of Sprott A system, which can generate diverse coexisting attractors, described as(2)dqxdtq=ydqydtq=-x-yzdqzdtq=ax2+by2+cxy+dwhere the parameters a, b, c, d are real constants, q is the fractional-order and 0<q<1. When setting the system parameters and the fractional-order as a=1.2, b=1, c=1, d=-1.3 and q=0.992, the fractional-order system (2) with initial conditions (x(0),y(0),z(0))=(1,1,0) presents hidden attractors as shown in Figure 1 and system (2) shows four-wing structure on the x-z plane. The Poincaré map on the x-z plane with y=0 shown in the three-dimensional space in Figure 2 also can verify it.

The chaotic attractor of the fractional-order system (2): (a) x-y-z space; (b) x-y plane; (c) x-z plane; (d) y-z plane.

3D Poincaré map of the fractional-order system (2) on the x-z plane with y=0 (both directions through the cross-section in red and blue, respectively).

To further confirm the chaotic motion, by the Benettin–Wolf algorithm, the finite-time local Lyapunov exponents on the time interval t[0,1500] with the same initial conditions and parameters as above are shown in Figure 3. And it can be seen that the largest one of the finite-time local LEs on the time interval t[0,1500] always maintains to be positive.

Dynamics of the finite-time local LEs of the fractional-order system (2) with initial conditions (x(0),y(0),z(0))=(1,1,0) on the time interval t[0,1500].

Remark 1.

In numerical simulations, the Lyapunov exponents may differ significantly from different trajectories and only a finite-time interval can be considered for their computation. Therefore, we follow the concept of the finite-time local Lyapunov exponents in the article.

3. Dynamic Analysis of the Fractional-Order System without Equilibrium 3.1. Basic Dynamic Properties

For system (2), it is symmetric about the z-axis on account of the coordinate transformation (x,y,z)(-x,-y,z). And if let the right hand side of system (2) to zero, then it can be written as(3)y=0-x-yz=0ax2+by2+cxy+d=0

While x=y=0, it is obvious that there is no equilibrium due to d0. Even so, through the above phase portraits, finite-time local Lyapunov exponents, and Poincaré map, the fractional-order system can show obvious chaotic characteristics and be regarded as the sort of hidden attractors.

3.2. Bifurcation Analysis of System Parameters

Bifurcation diagram can help us observe the further dynamic behaviors of the fractional-order system (2). While setting b=1, c=1, d=-1.3 and q=0.99, the bifurcation diagram and the largest finite-time local LEs varying with the parameter a are shown in Figures 4 and 5. The performance via period-doubling bifurcations and hidden coexisting attractors are observed in Figure 6. The fractional-order system presents several different symmetric pair of coexisting limit cycles for a1.11. As shown on Figures 6(a) and 6(b), there coexist a symmetric pair of limit cycles with period-1 orbits and another symmetric limit cycle with period-2 orbits, respectively. For a>1.11, the fractional system begins to exhibit obvious chaotic oscillations and hidden coexisting chaotic attractors while the parameter is not over 1.21 as shown in Figure 6(c).

Bifurcation diagram with the variable parameter a, initial conditions (x(0),y(0),z(0))=(1,1,0) (blue) and (x(0),y(0),z(0))=(-1,-1,0) (red).

Largest finite-time local LEs on the time interval t[0,1500] with the variable parameter a, initial conditions (x(0),y(0),z(0))=(1,1,0) (blue) and (x(0),y(0),z(0))=(-1,-1,0) (red).

Coexisting attractors with initial conditions (x(0),y(0),z(0))=(1,1,0) (blue) and (x(0),y(0),z(0))=(-1,-1,0) (red) when the parameter a is kept as (a) a=0.9; (b) a=1.0; (c) a=1.2.

When the parameters and fractional-order are chosen as a=1.2, b=1, c=1, and q=0.99, the bifurcation diagram of state variable x varying with d is provided in Figure 7. Similarly, the states of the fractional-order system evolve from periodic state to multiperiodic state and then to chaos, which also exists periodic window in the neighborhood of d=-1.2. For purpose of further study in system parameter analysis, when keeping b=1, c=1 and q=0.99, Figure 8 plots the largest finite-time local LEs diagram with the mutual variation of parameters a and d, and the right color bar denotes different levels of largest Lyapunov exponent. In Figure 8, different colors intuitively represent influences of parameters on dynamical behavior where the warmer colors mean the higher values of largest finite-time local LEs.

Bifurcation diagram with the variable parameter d, initial conditions (x(0),y(0),z(0))=(1,1,0) (blue) and (x(0),y(0),z(0))=(-1,-1,0) (red).

Dynamic map of Largest finite-time local LEs on the time interval t[0,1500] with parameters a and b by selecting initial conditions (x(0),y(0),z(0))=(1,1,0).

3.3. Influences of Fractional-Order q and Initial Conditions

In Figure 9, keeping the system parameters as a=1.2, b=1, c=1, and d=-1.3, the bifurcation of system (2) with the change of fractional-order q illuminates the system could undergo the period-doubling bifurcation and end up in chaos as the order is set on q>0.982. And there exists diverse coexisting attractors under different orders. For q=0.99, a symmetric pair of strange attractors coexist with different initial conditions as shown in Figures 6(c) and 11, and its basins of attractions in the z=0 plane are shown in Figure 10, where the one kind of chaotic attractor is distinguished from another by red and orange and the yellow region indicates initial conditions in it can reach unbounded orbits. For q=0.98, there coexists a symmetric pair of limit cycles with period-4, as shown in Figure 12, and its basins in the z=0 plane are provided in Figure 13. Corresponding to the above analysis of basic dynamic properties, the basins have the similar symmetry about the z-axis.

Bifurcation diagram with the variable fractional-order q, initial conditions (x(0),y(0),z(0))=(1,1,0) (blue) and (x(0),y(0),z(0))=(-1,-1,0) (red).

Cross-section for z=0 of the basins of attraction for the symmetric pair of strange attractors (red and orange) with fractional-order q=0.99 (the yellow region can reach unbounded orbits).

Coexisting chaotic attractors of the fractional-order system (2) with q=0.99, initial conditions (x(0),y(0),z(0))=(1,1,0) (blue) and (x(0),y(0),z(0))=(-1,-1,0) (red): (a) x-y plane; (b) x-z plane; (c) y-z plane.

Coexisting limit cycles of the fractional-order system (2) with q=0.98, initial conditions (x(0),y(0),z(0))=(1,1,0) (blue) and (x(0),y(0),z(0))=(-1,-1,0) (red).

Cross-section for z=0 of the basins of attraction for the symmetric pair of limit cycles (blue and cyan) with fractional-order q=0.98 (the yellow region can reach unbounded orbits).

Nevertheless, there are other cases of coexisting attractors. With order chosen as q=0.999, a strange attractor coexists with a limit cycle as shown in Figures 14 and 16, whose basins of attractions can be seen in Figure 15. The two attractors are respectively symmetric with respect to the z-axis and so do their basins. From the presented numerical results, the yellow region where initial values will lead to unbounded orbits largely remains invariant with different fractional orders, and the region of oscillation displays multitudinous and diverse dynamical behaviors.

Coexistence of a chaotic attractor and a limit cycle of system (2) with q=0.999, initial conditions (x(0),y(0),z(0))=(1,1,0) (blue) and (x(0),y(0),z(0))=(-0.5,5,0) (red).

Cross-section for z=0 of the basins of attraction for the chaotic attractor (red) and the limit cycle (blue) with fractional-order q=0.999 (the yellow region can reach unbounded orbits).

Coexistence of a chaotic attractor and a limit cycle of system (2) with q=0.999, initial conditions (x(0),y(0),z(0))=(1,1,0) (blue) and (x(0),y(0),z(0))=(-0.5,5,0) (red): (a) x-y plane; (b) x-z plane; (c) y-z plane.

4. Circuit Implementation Results of the Fractional-Order System

Circuit implementation of the fractional-order chaotic system plays a crucial role in actual application, which can also verify whether the results obtained by theory analysis and numerical computation are consistent and correct or not. Based on PSpice software, the schematic of the fractional-order chaotic system with order q=0.992 and parameters a=1.2, b=1, c=1, and d=-1.3 is designed and implemented as shown in Figure 17. In Figure 17, AD633 and OP-07 are adopt as all analogue multipliers and operational amplifiers.

The electronic circuit schematic of the fractional-order system (2) in PSpice.

Therein, with the proposed transfer function approximations method  utilized and ωmax=100, PT=0.01 assumed, the fractional integrator is derived by (4). In engineering practice it can be realized by three chain fractances composed of three resistor-capacitor pairs, as shown by (5). With comparing (4) with (5), the values of these circuit components are determined as Ra=95957.169kΩ, Ca=1.002μF, Rb=1.153kΩ, Cb=13.893μF and Rc=0.205Ω, Cc=12.959μF.(4)1s0.99=1.1471s+58.2265s+3.5105×105s+0.0104s+62.4256s+3.7636×105(5)Fs=1/Cas+1/RaCa+1/Cbs+1/RbCb+1/Ccs+1/RcCc=1C0C0/Ca+C0/Cb+C0/Ccs2+Cb+Cc/Ra+Ca+Cc/Rb+Ca+Cb/Rc/CaCb+CbCc+CaCcs+Ra+Rb+Rc/RaRbRc/CaCb+CbCc+CaCcs+1/RaCas+1/RbCbs+1/RcCc

With the Kirchhoff’s circuit laws, the mathematical model of system (2) is described by the circuit equations as follow:(6)dqxdtq=R7R1R6C0ydqydtq=-1R3C0x-1ηR2C0yzdqzdtq=R5ηR8R4C0x2+R7ηR9R6C0y2+R7ηR10R6C0xy+R12R13R11C0VCC-where C0=1μF and 1/η=1/10 hinged on the characteristic of AD633. The parameters of electronic components are calculated as: VCC+=-VCC-=15VDC, R1=R3=R11=1000kΩ, R2=R4=R5=R6=R7=R9=R10=100kΩ, R8=83.33kΩ, R12=10kΩ, R13=115.38kΩ. Figure 18 demonstrates circuit implementation results in which hidden strange attractors can been observed distinctly. And it also indicates that the circuital results are comparatively consistent with the numerical simulations.

Hidden chaotic attractors of the fractional-order system (2) in PSpice: (a) on the x-y plane; (b) on the x-z plane; (c) on the y-z plane.

5. Finite-Time Synchronization of the Fractional-Order System with Hidden Coexisting Attractors

There is a wide variety of literature [24, 3337] on synchronization of fractional-order chaotic systems by reason of comprehensive practical application in many fields, such as aerospace, power system, secure communication and so on. Research about finite-time synchronization in allusion to fractional-order chaotic system with hidden attractors, are quite limited. In this section, based on the finite-time stable theorem , in consideration of the disturbances, the modified controller is devised for synchronization of the fractional-order system with hidden chaotic attractors and achieves the anticipated effect.

Theorem 2 (see [<xref ref-type="bibr" rid="B38">38</xref>]).

For the fractional-order system in general, if it satisfies(7)xaCDtqxT=Γ2Γ2+qaCDtqxxqT-kxxTα,α<q+q22,k>0,where x=[x1,x2,xn], xq=[x1q,x2q,xnq]. Then the state variables x of the given fractional-order system will converge to zero within a finite-time:(8)t=v0q-2α/1+qΓ1+q-2α/1+qΓ1+qΓ1+2q-2α/1+qkΓ2+q1/q,where v=xxqT.

Lemma 3.

a , b > 0 , 0<c<1: (9)a+bcac+bc

Making allowances for the internal disturbances Δfim(X,t) and Δfis(X1,t) (i=1,2,3 and x,y,zX, x1,y1,z1X1) in the fractional-order chaotic system (2), the master system and the slave system can be obtained as(10)dqxdtq=y+Δf1mX,tdqydtq=-x-yz+Δf2mX,tdqzdtq=ax2+by2+cxy+d+Δf3mX,t(11)dqx1dtq=y1+Δf1sX1,t-u1dqy1dtq=-x1-y1z1+Δf2sX1,t-u2dqz1dtq=ax12+by12+cx1y1+d+Δf3sX1,t-u3where a=1.2, b=1, c=1, d=-1.3, and q=0.99; u=[u1,u2,u3] is the designed controller.

Assumption 4.

The disturbances Δfim(X,t), Δfis(X1,t) are bounded. In other words, positive constants Fim and Fis can be found to satisfy(12)ΔfimX,tFim,ΔfisX1,tFis

Afterwards, we define the error system as e1=x1-x, e2=y1-y, e3=z1-z, which can ultimately be derived as(13)dqe1dtq=e2+Δf1sX1,t-Δf1mX,t-u1dqe2dtq=-e1-y1e3-ze2+Δf2sX1,t-Δf2mX,t-u2dqe3dtq=1.2x1e1+1.2xe1+y1e2+ye2+x1e2+ye1+Δf3sX1,t-Δf3mX,t-u3

Theorem 5.

It can realize the synchronization of the master system (10) and the slave system (11) within a finite-time, while the control input is provided as(14)u1=1.2x1e3+ye3+ke1β+k1signe1u2=-ze2+x1e3+ke2β+k2signe2u3=1.2xe1+ye2+ke3β+k3signe3where ki=Fis+Fim, i=1,2,3.

Proof.

According to (7) and (12), the corresponding function with the error system (13) and the control input (14) can be constructed as(15)Γ2Γ2+qdqdtqe1e2e3e1qe2qe3qT=e1e2e3dqdtqe1dqdtqe2dqdtqe3T=e1e2-e1e2-y1e2e3-ze22+1.2x1e1e3+1.2xe1e3+y1e2e3+ye2e3+x1e2e3+ye1e3+Δf1sX1,t-Δf1mX,te1+Δf2sX1,t-Δf2mX,te2+Δf3sX1,t-Δf3mX,te3-u1e1-u2e2-u3e3=-ke1β+1-ke2β+1-ke3β+1+i=13ΔfisX1,t-ΔfimX,tei-kieisignei-ke1β+1-ke2β+1-ke3β+1+i=13Fis+Fimei-kieisignei=-ke1β+1-ke2β+1-ke3β+1+i=13kiei-eisignei-ke1β+1-ke2β+1-ke3β+1where ei-eisign(ei)=ei-ei0.

By reference to Lemma 3, (15) can be derived as(16)-ke1β+1-ke2β+1-ke3β+1=-ke12β+1/2-ke22β+1/2-ke32β+1/2-ke12+e22+e32β+1/2=-keeTβ+1/2

Therefore, from Theorem 2, the error system (13) states will converge to zero within a finite-time.

The numerical simulations are carried out to verify the effectiveness of the proposed scheme. The disturbances Δfim(X,t), Δfis(X1,t) are chosen as (17). The parameters for controllers are set as k=1.5, β=0.5 and k1=1.0, k2=1.4, k3=4.4. Initial states of the master system and the slave system are selected as (x(0),y(0),z(0))=(-0.5,-0.5,2) and (x1(0),y1(0),z1(0))=(1,1,0). As shown in Figure 19, the error system (13) can stabilize to zero within 1s. Meanwhile, time response of the master system (10) and the slave system (11) under control input (14) is presented in Figure 20 and the system states of the slave system will track the trajectories of the master system within 1s. In comparison with the original controllers which do not take into account disturbance factors in , the improved controllers in this paper can synchronize the master system and the slave system under disturbances. The simulation results in Figure 20 also indicate that the proposed method is fast and can achieve chaos synchronization within finite-time. All the results verify the effectiveness of the proposed scheme in synchronization of fractional-order chaotic system with hidden coexisting attractors.(17)Δf1mX,t=0.15xcos2t,Δf1sX1,t=-0.2x1sin2tΔf2mX,t=-0.2ysin3t,Δf2sX1,t=0.1y1sin2tΔf3mX,t=0.1zcost,Δf3sX1,t=0.15z1cost

Synchronization errors between the master and slave systems.

Synchronization results of the master and slave systems: (a) x-x1; (b) y-y1; (c) z-z1.

6. Conclusions

In this paper, a novel three-dimensional symmetric fractional-order system without equilibrium, which can generate various hidden coexisting attractors, has been introduced. We have analyzed dynamical characteristics of the fractional-order system in detail by numerical simulations and found the system presents abundant complex dynamic characteristics. Especially, through bifurcation diagram, the basins of attraction, and phase diagram, the influence of the order on dynamic behavior of the fractional-order system has been explored in depth. With the approximations of fractional-order integrator q=0.992 derived, circuit implementation of the fractional-order system with hidden coexisting attractors has been achieved in PSpice and it has a good agreement with the numerical simulations. Furthermore, the modified controller has been devised and proved, which can realize the finite-time synchronization of the fractional-order system without equilibrium under the disturbances. Finally, simulation results also illustrate the robustness and speediness of the proposed controller. In order to simplify control inputs, fractional-order finite-time backstepping method and dynamic surface method will be considered in the future. Furthermore, secure communication and image encryption based on the novel fractional-order chaotic system with hidden coexisting attractors will also be further investigated.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

There are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This project was supported by the National Natural Science Foundation of China (Grant no. 51877162).

Lorenz E. N. Deterministic nonperiodic flow Journal of the Atmospheric Sciences 1963 20 2 130 141 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 Zbl06851676 Rössler O. E. An equation for continuous chaos Physics Letters A 1976 57 5 397 398 2-s2.0-49549126801 10.1016/0375-9601(76)90101-8 Zbl1371.37062 Chen G. Ueta T. Yet another chaotic attractor International Journal of Bifurcation and Chaos 1999 9 7 1465 1466 10.1142/S0218127499001024 MR1729683 Zbl0962.37013 2-s2.0-62249200922 J. Chen G. A new chaotic attractor coined International Journal of Bifurcation and Chaos 2002 12 3 659 661 10.1142/S0218127402004620 MR1894886 Zbl1063.34510 2-s2.0-0036011505 Liu C. Liu T. Liu L. Liu K. A new chaotic attractor Chaos, Solitons & Fractals 2004 22 5 1031 1038 10.1016/j.chaos.2004.02.060 MR2078830 Zbl1060.37027 2-s2.0-2942653227 Leonov G. A. Kuznetsov N. V. Vagaitsev V. I. Localization of hidden Chua's attractors Physics Letters A 2011 375 23 2230 2233 10.1016/j.physleta.2011.04.037 MR2800438 Zbl1242.34102 2-s2.0-79956081226 Leonov G. A. Kuznetsov N. V. Mokaev T. N. Hidden attractor and homoclinic orbit in Lorenz-like system describing convective fluid motion in rotating cavity Communications in Nonlinear Science and Numerical Simulation 2015 28 1–3 166 174 10.1016/j.cnsns.2015.04.007 MR3348101 2-s2.0-84929618582 Wang X. Chen G. R. A chaotic system with only one stable equilibrium Communications in Nonlinear Science and Numerical Simulation 2012 17 3 1264 1272 10.1016/j.cnsns.2011.07.017 MR2843793 Molaie M. Jafari S. Sprott J. C. Golpayegani S. M. R. H. Simple chaotic flows with one stable equilibrium International Journal of Bifurcation and Chaos 2013 23 11 10.1142/s0218127413501885 1350188 Zbl1284.34064 2-s2.0-84890497588 Jafari S. Sprott J. C. Simple chaotic flows with a line equilibrium Chaos, Solitons & Fractals 2013 57 4 79 84 10.1016/j.chaos.2013.08.018 2-s2.0-84884765978 Jafari S. Ahmadi A. Khalaf A. J. M. Abdolmohammadi H. R. Pham V.-T. Alsaadi F. E. A new hidden chaotic attractor with extreme multi-stability AEÜ - International Journal of Electronics and Communications 2018 89 131 135 10.1016/j.aeue.2018.03.037 2-s2.0-85044609056 Wei Z. C. Dynamical behaviors of a chaotic system with no equilibria Physics Letters A 2011 376 2 102 108 10.1016/j.physleta.2011.10.040 Zbl1255.37013 Jafari S. Sprott J. C. Golpayegani S. M. R. H. Elementary quadratic chaotic flows with no equilibria Physics Letters A 2013 377 9 699 702 10.1016/j.physleta.2013.01.009 2-s2.0-84873119136 Li C. B. Sprott J. C. Xing H. Hypogenetic chaotic jerk flows Physics Letters A 2016 380 11-12 1172 1177 10.1016/j.physleta.2016.01.045 MR3457319 Hu X. Y. Liu C. X. Liu L. Yao Y. P. Zheng G. C. Multi-scroll hidden attractors and multi-wing hidden attractors in a 5-dimensional memristive system Chinese Physics B 2017 26 11 110502 2-s2.0-85033792577 Danca M. F. Hidden chaotic attractors in fractional-order systems Nonlinear Dynamics 2017 89 1 577 586 10.1007/s11071-017-3472-7 Li C. B. Sprott J. C. Coexisting hidden attractors in a 4-D simplified Lorenz system International Journal of Bifurcation & Chaos 2014 24 3 1450034 Zbl1296.34111 Ruan Y J. Sun H K. Mou J. Memristor-based Lorenz hyper-chaotic system and its circuit implementation Acta Physica Sinica 2016 65 19 190502 Bao B. C. Bao H. Wang N. Chen M. Xu Q. Hidden extreme multistability in memristive hyperchaotic system Chaos, Solitons & Fractals 2017 94 102 111 2-s2.0-85002170676 10.1016/j.chaos.2016.11.016 Zbl1373.34069 Zhou W. Wang G. Y. Shen Y. R. Yuan F. Yu S. M. Hidden coexisting attractors in a chaotic system International Journal of Bifurcation & Chaos 2018 28 10 1830033 Zbl1401.34022 Danca M. F. Fečkan M. Kuznetsov N. V. Chen G. R. Complex dynamics, hidden attractors and continuous approximation of a fractional-order hyperchaotic PWC system Nonlinear Dynamics 2018 91 4 2523 2540 10.1007/s11071-017-4029-5 Danca M. Fečkan M. Kuznetsov N. V. Chen G. Fractional-order PWC systems without zero Lyapunov exponents Nonlinear Dynamics 2018 92 3 1061 1078 10.1007/s11071-018-4108-2 Zbl1398.37028 Borah M. On coexistence of fractional-order hidden attractors Journal of Computational and Nonlinear Dynamics 2018 13 9 090906 2-s2.0-85050995404 Pham V. Ouannas A. Volos C. Kapitaniak T. A simple fractional-order chaotic system without equilibrium and its synchronization AEÜ - International Journal of Electronics and Communications 2018 86 69 76 10.1016/j.aeue.2018.01.023 Yang N. N. Liu C. X. Wu C. J. Modeling and dynamics analysis of the fractional-order Buck-Boost converter in continuous conduction mode Chinese Physics B 2012 21 8 080503 10.1088/1674-1056/21/8/080503 2-s2.0-84864650944 Li C.-L. Wu L. Sliding mode control for synchronization of fractional permanent magnet synchronous motors with finite time Optik - International Journal for Light and Electron Optics 2016 127 6 3329 3332 2-s2.0-84955583185 10.1016/j.ijleo.2015.12.102 Zheng G. C. Liu C. X. Wang Y. Dynamic analysis and finite time synchronization of a fractional-order chaotic system with hidden attractors Acta Physica Sinica 2018 67 5 050502 Sun H. G. A new collection of real world applications of fractional calculus in science and engineering Communications in Nonlinear Science & Numerical Simulation 2018 64 213 231 10.1016/j.cnsns.2018.04.019 Podlubny I. Fractional Differential Equations 1999 San Diego, Calif, USA Academic Press MR1658022 Khalil R. Al Horani M. Yousef A. Sababheh M. A new definition of fractional derivative Journal of Computational and Applied Mathematics 2014 264 5 65 70 10.1016/j.cam.2014.01.002 MR3164103 Sprott J. C. Some simple chaotic flows Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 1994 50 2 R647 R650 10.1103/physreve.50.r647 2-s2.0-33751555569 Charef A. Sun H. H. Tsao Y. Onaral B. Fractal system as represented by singularity function IEEE Transactions on Automatic Control 1992 37 9 1465 1470 10.1109/9.159595 MR1183117 Zbl0825.58027 2-s2.0-0026928376 Li C. G. Liao X. F. Yu J. B. Synchronization of fractional order chaotic systems Physical Review E 2003 68 6 067203 Wang D.-F. Zhang J.-Y. Wang X.-Y. Synchronization of uncertain fractional-order chaotic systems with disturbance based on a fractional terminal sliding mode controller Chinese Physics B 2013 22 4 10.1088/1674-1056/22/4/040507 040507 2-s2.0-84876124726 Zhang L. G. Yan Y. Robust synchronization of two different uncertain fractional-order chaotic systems via adaptive sliding mode control Nonlinear Dynamics 2014 76 3 1761 1767 10.1007/s11071-014-1244-1 2-s2.0-84899618074 Wang X. Ouannas A. Pham V. Abdolmohammadi H. R. A fractional-order form of a system with stable equilibria and its synchronization Advances in Difference Equations 2018 2018 1 20 10.1186/s13662-018-1479-0 Zbl07002831 Wei Z. C. Akgul A. Kocamaz U. E. Moroz I. Zhang W. Control, electronic circuit application and fractional-order analysis of hidden chaotic attractors in the self-exciting homopolar disc dynamo Chaos, Solitons & Fractals 2018 111 157 168 10.1016/j.chaos.2018.04.020 2-s2.0-85045705711 Zhao L. D. Hu J. B. Bao Z. H. A finite-time stable theorem about fractional systems and finite-time synchronizing fractional super chaotic Lorenz systems Acta Physica Sinica 2011 60 10 100507