There are many recent investigations on chaotic hidden attractors although hyperchaotic hidden attractor systems and their relationships have been less investigated. In this paper, we introduce a hyperchaotic system which can change between hidden attractor and self-excited attractor depending on the values of parameters. Dynamic properties of these systems are investigated. Fractional order models of these systems are derived and their bifurcation with fractional orders is discussed. Field programmable gate array (FPGA) implementations of the systems with their power and resource utilization are presented.
1. Introduction
Many recent works on dynamical systems are categorized into self-excited and hidden attractors [1–3]. A self-excited attractor has a basin of attraction that is associated with an unstable equilibrium, while a hidden attractor has a basin of attraction that does not intersect with small neighborhoods of any equilibrium points. Hidden attractors are important in most of the engineering problems as they allow chaotic responses [4, 5]. Control of such hidden oscillations is a big challenge because of the multistability nature of the systems [6, 7]. Chaotic attractors are with no equilibrium points [8–15], with only stable equilibria [16–19], and with curves of equilibria [20]. Fractional order with no equilibrium systems with its FPGA implementation has also been reported recently [21, 22].
Recently many researchers have discussed fractional order calculus and its applications [23–25]. Fractional order nonlinear systems with different control approaches are investigated [26–28]. Fractional order memristor based with no equilibrium chaotic system is proposed by Rajagopal et al. [21, 22]. A novel fractional order with no equilibrium chaotic system is investigated by Li and Chen [29]. Cafagna and Grassi investigated a fractional order hyperchaotic system without equilibrium points [30]. Memristor based fractional order system with a capacitor and an inductor is discussed [31]. Numerical analysis and methods for simulating fractional order nonlinear system are proposed by Petras [32] and MATLAB solutions for fractional order chaotic systems are discussed by Trzaska Zdzislaw [33]. A FPGA implementation of fractional order chaotic system using approximation method is investigated recently for the first time [21, 22].
Recently Jafari et al. announced a 3D chaotic system [34] which can belong to three famous categories of hidden attractors plus systems with self-excited attractors. Motivated by this, in this paper we announce a hyperchaotic chameleon which can be self-excited or a hidden attractor depending on the values of the parameters. This system helps us to better understand the hidden chaotic flows of higher dimension.
2. Novel Chaotic System (NCS)
In this section we introduce a class of novel chaotic and hyperchaotic systems with flux controlled memristor [35, 36], derived from the hyperchaotic system [37] by including parameter a which governs the equilibrium of the system and parameter b controls the number of Lyapunov exponents in the system, that is, chaotic and hyperchaotic case. The novel hyperchaotic system’s dimensionless state equations are(1)x˙=15y-x+13yz,y˙=18x-xz-bxWϕ-a,z˙=xy-cz,w˙=-dx,where W(ϕ)=α+3βϕ2 is the memductance of the flux controller memristor where the flux element ϕ is defined by the fourth state w with α=4 and β=0.01. The parameters c and d are fixed at 2 and 1.4, respectively. We investigate four different choices for the parameters a and b as in Table 1.
Different cases for the parameters a and b.
Name of the system
Parameters a and b
Type of system
NCS1
a=0, b≠0
Hyperchaotic system with single equilibrium at origin
NCS2
a≠0, b≠0
Hyperchaotic system with no equilibrium
NCS3
a=0, b=0
Hyperchaotic system with single equilibrium at origin
NCS4
a≠0, b=0
Chaotic system with no equilibrium
Figures 1–4 show the 3D phase portraits of the systems NCS1, NCS2, NCS3, and NCS4, respectively.
3D phase portraits of NCS1 system.
3D phase portraits of NCS2 system.
3D phase portraits of the NCS3 system.
3D phase portraits of the NCS4 system.
3. Dynamic Properties of the NCS 3.1. Equilibrium Points
The equilibrium points for the NCS can be calculated by equating the state equations to 0. It can be seen that 18x-xz-bxW(ϕ)-a=0 shows two cases of equilibrium points; that is, when a=0 the system has origin as the only defined equilibrium point and when a≠0 the system has no defined equilibrium and hence exhibits hidden attractors. Table 2 shows the equilibrium points for different choices of a and b. The characteristic equation of NCS1 and NCS3 is 2λ4+19λ3-270λ2-600λ and the Eigenvalues are λ1=0; λ2=-16.55; λ3=9.06; λ4=-2 and λ3 is an unstable focus and thus the two systems are self-excited attractors. As investigated by many researchers [8–15], chaotic attractors with no equilibrium are hidden thus making NCS2 and NCS4 hidden attractors.
Equilibrium points of the NCS systems.
Name of the system
Parameters a and b
Equilibrium points
NCS1
a=0, b=0.5
[0,0,0,0]
NCS2
a≠0, b=0.5
No equilibrium
NCS3
a=0, b=0
[0,0,0,0]
NCS4
a≠0, b=0
No equilibrium
3.2. Lyapunov Exponents and Kaplan-Yorke Dimension
Lyapunov exponents of a nonlinear system define the convergence and divergence of the states. The existence of a positive Lyapunov exponent confirms the chaotic behavior of the system [38, 39]. Lyapunov exponents (LEs) are necessary and more convenient for detecting hyperchaos in fractional order hyperchaotic system. A definition of LEs for fractional differential systems was given in [40] based on frequency-domain approximations, but the limitations of frequency-domain approximations are highlighted by Tavazoei and Haeri [41]. Time series based LEs calculation methods like Wolf algorithm [11], Jacobian method [12], and neural network algorithm [13] are popularly known ways of calculating Lyapunov exponents for integer and fractional order systems. Hence we use the Jacobian method to calculate the LEs. Table 3 shows the Lyapunov exponents of the NCS.
Lyapunov exponents and KY dimension of NCS systems.
System
Lyapunov exponents
Kaplan-Yorke (KY) dimension
NCS1
L1=19.3050, L2=1.0627, L3=-0.1084,L4=-37.2591
3.5465
NCS2
L1=19.7244;L2=1.2240;L3=0;L4=-38.0565
3.5506
NCS3
L1=3.5123;L2=0.0026;L3=0;L4=-20.5149
3.1714
NCS4
L1=3.1029;L2=0;L3=0;L4=-20.1029
3.1543
3.3. Bifurcation
In this section we derive the bifurcation contours for the NCS. The chaotic behavior of the system largely depends on the parameters a, b, c, and d. As discussed in [42], the transient behaviors occurring in memristor based nonlinear systems may result in longer simulation times to reach steady states. Hence we used the ode45 solver for numerical simulations. Four different cases of bifurcations are investigated for the NCS. In case 1 the parameter a is varied and the bifurcation of the attractor is investigated. Figure 5 shows the bifurcation plot for a. In the second case parameter b is varied and the bifurcation of the NCS is studied which is shown in Figure 6. In case 3 the parameter c is varied and bifurcation analysis of the system is investigated. Finally bifurcation plot for parameter d is derived in case 4. Figures 7 and 8 show the bifurcation contours of c and d, respectively. As can be seen from the figures the NCS shows strange attractor, hyperchaos, chaos, and quasiperiodic systems. For 0≤a≤0.25, 0.51≤b≤0.72, 0.8≤c≤1.2, and 1.34≤d≤1.45 the NCS shows strange attractor and the transient behavior of the memristor prevents the bifurcation plots to show the period doubling property even after reducing the transients to 70%. The system exhibits hyperchaotic attractor with hidden oscillations for 0.26≤a≤0.8, 0.2≤b≤0.5, 1.8≤c≤2.3, and 1.04≤d≤1.25. For a small band of 0≤b≤0.1 with 0.8≤a≤1.4, 2.4≤c≤2.8, and 1.31≤d≤1.45 the system shows chaotic oscillations with hidden attractors. A quasichaotic system is seen for 1.5≤a≤1.8, 0.8≤b≤2, 0.42≤c≤0.7, and 1.47≤d≤1.5.
Bifurcation plots for parameter a.
Bifurcation plots for parameter b.
Bifurcation plots for parameter c.
Bifurcation plots for parameter d.
3.4. Bicoherence
The motivation to study the bicoherence is twofold. First, the bicoherence can be used to extract information due to deviations from Gaussianity and suppress additive (colored) Gaussian noise. Second, the bicoherence can be used to detect and characterize asymmetric nonlinearity in signals via quadratic phase coupling or identify systems with quadratic nonlinearity. The bicoherence is the third-order spectrum. Whereas the power spectrum is a second-order statistic, formed from X′f∗Xf, where Xf is the Fourier transform of xt, the bispectrum is a third-order statistic formed from Xfj∗Xfk∗X′fj+fk. The bispectrum is therefore a function of a pair of frequencies fj,fk. It is also a complex-valued function. The (normalized) square amplitude is called the bicoherence (by analogy with the coherence from the cross-spectrum). The bispectrum is calculated by dividing the time series into M segments of length N_seg, calculating their Fourier transforms and biperiodogram and then averaging over the ensemble. Although the bicoherence is a function of two frequencies the default output of this function is a one-dimensional output, the bicoherence refined as a function of only the sum of the two frequencies. The autobispectrum of a chaotic system is given by Pezeshki et al. [43]. They derived the autobispectrum with the Fourier coefficients.(2)Bω1,ω2=EAω1Aω2A∗ω1+ω2,where ωn is the radian frequency and A is the Fourier coefficients of the time series. The normalized magnitude spectrum of the bispectrum known as the squared bicoherence is given by(3)bω1,ω2=Bω1,ω22Pω1Pω2Pω1+ω2,where P(ω1) and P(ω2) are the power spectrums at f1 and f2.
Figures 9 and 10 show the bicoherence contours of the NCS for a≠0 and b≠0 and a=0 and b=0, respectively. As can be seen from the figures, the NCS shows wider band of power spectrum when a≠0 and b≠0 because of the hyperchaotic hidden attractor as when b≠0 the memristor element introduces a quadratic nonlinearity resulting in the cross-bicoherence. Shades in yellow represent the multifrequency components contributing to the power spectrum. From Figures 9 and 10 the cross-bicoherence is significantly nonzero and nonconstant, indicating a nonlinear relationship between the states. The yellow shades and nonsharpness of the peaks, as well as the presence of structure around the origin in figures (cross-bicoherence), indicate that the nonlinearity between the states x, y, z, and w is not of the quadratic nonlinearity and hence may be because of nonlinearity of higher dimensions. The most two dominant frequencies (f1,f2) are taken for deriving the contour of bicoherence. The sampling frequency (fs) is taken as the reference frequency. Direct FFT is used to derive the power spectrum for individual frequencies and Hankel operator is used as the frequency mask. Hanning window is used as the FIR filter to separate the frequencies.
Bicoherence plots of NCS (a≠0,b≠0).
Bicoherence plot of NCS (a=0,b=0).
4. Fractional Order NCS (FONCS)
In this section we derive the fractional order model novel chaotic system (FONCS). There are three commonly used definitions of the fractional order differential operator, namely, those of Grunwald–Letnikov, Riemann–Liouville, and Caputo [23–25].
In this section, we will study the dynamical behavior of fractional order system derived from the NCS with the Grunwald-Letnikov (GL) definition, which is defined as(4)Datqft=limh→01hq∑j=0t-q/h-1jqjft-jh=limh→01hqΔhqft,where a and t are limits of the fractional order equation, Δhqf(t) is generalized difference, h is the step size, and q is the fractional order of the differential equation.
For numerical calculations the above equation is modified as(5)Dt-Ltqft=limh→0h-q∑j=0Ntbjft-jh.Theoretically fractional order differential equations use infinite memory. Hence when we want to numerically calculate or simulate the fractional order equations we have to use finite memory principal, where L is the memory length and h is the time sampling.(6)Nt=minth,Lh.The binomial coefficients required for the numerical simulation are calculated as(7)bj=1-a+qjbj-1.Using (4)–(6) the fractional order NCS is defined as(8)Dqxx=15y-x+13yz,Dqyy=18x-xz-bxWϕ-a,Dqzz=xy-cz,Dqww=-dx.As discussed in Section 1, the FONCS also shows chaotic and hyperchaotic systems with no equilibrium and single equilibrium points for a choice of parameter values a and b as shown in Table 4.
Choice of parameters and type of the system.
Name of the system
Parameters a and b
Type of system
FONCS1
a=0, b=0.5
Hyperchaotic system with single equilibrium at origin
FONCS2
a≠0, b=0.5
Hyperchaotic system with no equilibrium
FONCS3
a=0, b=0
Hyperchaotic system with single equilibrium at origin
FONCS4
a≠0, b=0
Chaotic system with no equilibrium
The 3D phase portraits of the FONCS are shown in Figures 11–14. The commensurate fractional orders of the system for a=0, b≠0, a≠0, b≠0, a=0, b=0, a≠0, and b=0 are taken as q=0.991, q=0.995, q=0.989, and q=0.990, respectively.
3D phase portraits of NCS (a≠0,b≠0).
3D phase portraits of NCS (a≠0,b=0).
3D phase portraits of NCS (a=0,b=0).
3D phase portraits of NCS (a=0,b≠0).
4.1. Dynamic Analysis of the FONCS
Most of the dynamic properties of the NCS like the Lyapunov exponents and bifurcation with parameters are preserved in the FONCS [44] if qi>0.98, where i=x,y,z,andw. The most important analysis of interest when investigating a fractional order system is the bifurcation with fractional order. The largest positive Lyapunov exponents (L1=19.8942,L2=1.1436) of the NCS for a=0 and b≠0 appear when q=0.991 against their largest integer order Lyapunov exponents (L1=19.3050,L2=1.0627), largest positive Lyapunov exponents (L1=20.1613,L2=1.4108) of the NCS for a≠0 and b≠0 appear when q=0.995 against their largest integer order Lyapunov exponents (L1=19.7244,L2=1.2208), largest positive Lyapunov exponents (L1=3.8407,L2=0.0134) of the NCS for a=0 and b=0 appear when q=0.989 against their largest integer order Lyapunov exponents (L1=3.51,L2=0.0026) and largest positive Lyapunov exponent (L1=3.1187) of the NCS for a≠0 and b=0 appear when q=0.990 against its largest integer order Lyapunov exponent (L1=3.1029). It can also be seen that as the fractional order q decreases, the FONCS starts losing its largest positive Lyapunov exponent. When q≤0.90 the positive Lyapunov exponents of the system become negative and thus the chaotic oscillation in the system disappears. Figures 15 and 16 show the bifurcation of the FONCS for variation in fractional orders qx=qy=qz=qw=q for the two distinctive cases a≠0 and b≠0 and a=0 and b=0.
Bifurcation of NCS versus fractional order q (a≠0,b≠0).
Bifurcation of NCS versus fractional order q (a=0,b=0).
4.2. Stability Analysis of FONCS 4.2.1. Commensurate Order
For commensurate FONCS of order q, the system is stable and exhibits chaotic oscillations if arg(eig(JE))=arg(λi)>qπ/2, where JE is the Jacobian matrix at the equilibrium E and λi are the Eigenvalues of the FONCS, where i=1,2,3,and4. As seen from the FONCS, the Eigenvalues should remain in the unstable region and the necessary condition for the FONCS to be stable is q>2/πtan-1Imλ/Reλ. The Eigenvalues of FONCS1 and FONCS3 are λ1=0; λ2=-16.55; λ3=9.06; λ4=-2 and λ3 is an unstable focus contributing to the existence of chaotic oscillations.
4.2.2. Incommensurate Order
The necessary condition for the FONCS to exhibit chaotic oscillations in the incommensurate case is π/2M-miniarg(λi)>0, where M is the LCM of the fractional orders. If qx=0.9, qy=0.9, qz=0.8, and qw=0.8, then M=10. The characteristic equation of the system evaluated at the equilibrium is det(diag[λMqx,λMqy,λMqz,λMqw]-JE)=0 and by substituting the values of M and the fractional orders, det(diag[λ9,λ9,λ8,λ8]-JE)=0, the characteristic equation is λ34+2λ27+4λ26+15λ25+λ20+ 6λ19+35λ18+45λ17-300λ16+2λ12+21λ11+62λ10-570λ9-600λ8+λ4+17λ3-270λ2-600λ and is the same for both FONCS1 and FONCS3. The approximated solution of the characteristic equation is λ34=0.912 whose argument is zero and which is the minimum argument and hence the stability necessary condition becomes π/20-0>0 which solves for 0.0785>0 and hence the FONCS is stable and chaos exists in the incommensurate system.
5. FPGA Implementation of the Fractional Order Novel Cubic Nonlinear Systems
In this section we discuss the implementation of the proposed FONCS in FPGA [21, 22, 38, 45–48] using the Xilinx (Vivado) System Generator toolbox in Simulink. The challenge while implementing the systems in FPGAs is to design the fractional order integrator which is not a readily available block in the System Generator [21, 22]. Hence we implement the fractional integrators using the mathematical relation [32, 33] discussed in (4), (5), and (6) and the value of h is taken as 0.001 with the initial conditions as described in Table 1 and the commensurate fractional order taken as q=0.991 for FONCS-1, q=0.995 for FONCS-2, q=0.989 for FONCS-3, and q=0.99 for FONCS-4. Figures 17, 18(a), and 18(b) show the Xilinx RTL schematics of the FONCS-1 system implemented in Kintex-7 (device = 7k160t, package = fbg484 S), power utilized by the system, and power utilized for various fractional orders, respectively. Table 5 shows the resources utilized by the FONCS-1 system including the clock frequency. Figure 19 shows the 2D state portraits of the FPGA implemented FONCS-1 system. Figures 20, 21(a), and 21(b) show the Xilinx schematics of the FONCS-2 system implemented in Kintex-7 (Device = 7k160t Package = fbg484 S), power utilized by the system and power utilization for various fractional orders, respectively. Table 6 shows the resources utilized by the FONCS-2 system including the clock frequency. Figure 22 shows the 2D state portraits of the FPGA implemented FONCS-2 system. Figures 23, 24(a), and 24(b) show the Xilinx schematics of the FONCS-3 system implemented in Kintex-7 (device = 7k160t, package = fbg484 S), power utilized by the system, and power utilization for various fractional orders, respectively. Table 7 shows the resources utilized by the FONCS-3 system including the clock frequency. Figure 25 shows the 2D state portraits of the FPGA implemented FONCS-3 system. Figures 26, 27(a), and 27(b) show the Xilinx schematics of the FONCS-4 system implemented in Kintex-7 (device = 7k160t, package = fbg484 S), power utilized by the system, and power utilization for various fractional orders, respectively. Table 8 shows the resources utilized by the FONCS-3 system including the clock frequency. Figure 28 shows the 2D state portraits of the FPGA implemented FONCS-3 system. The sampling rates of the FPGA blocks play a crucial role in the existence of Lyapunov exponents and also increasing the sampling time period in some implementations may lead to a clock frequency mismatch. Maximum power will be consumed by the system when the FONCS shows largest Lyapunov exponents (FONCS-1 q=0.991, FONCS-2 q=0.995, and FONCS-3 q=0.996). To utilize the power of FPGA, the computation needs to be divided into several independent blocks of threads that can be executed simultaneously [39, 49]. The FONCS power efficiency also depends on the parameter b as can be observed from the power efficiency Figures 18, 21, 24, and 27. When b≠0 the FONCS uses a maximum power of 299 w against 280 w when b=0. This is because of the reason that the system shows their largest positive Lyapunov exponents when b≠0. The effect of parameter a on the power efficiency is quiet minimal and negligible. The performance on FPGA is directly related to the number of threads and its performances and hence the FONCS are designed as four parallel threads. The fractional order operators are implemented as building blocks and the so-called “frame delay” is not noticeable in the FPGA hardware implementation due to its parallel data structure, unlike a microprocessor-based implementation.
Resource utilization of FONCS-1 system.
Resource
Utilization
Available
Utilization %
Clock frequency
fmax
Used
LUT
1165
101400
1.15
300 Mhz
167 Mhz
FF
256
202800
0.13
500 Mhz
226 Mhz
DSP
20
600
3.33
250 Mhz
140 Mhz
IO
129
285
45.26
300 Mhz
108 Mhz
BUFG
1
32
3.13
300 Mhz
47 Mhz
Resource utilization of FONCS-2 system.
Resource
Utilization
Available
Utilization %
Clock frequency
fmax
Used
LUT
1196
101400
1.18
300 Mhz
172 Mhz
FF
256
202800
0.13
500 Mhz
226 Mhz
DSP
20
600
3.33
250 Mhz
123 Mhz
IO
129
285
45.26
300 Mhz
108 Mhz
BUFG
1
32
3.13
300 Mhz
47 Mhz
Resource utilization of FONCS-3 system.
Resource
Utilization
Available
Utilization %
Clock frequency
fmax
Used
LUT
880
101400
0.87
300 Mhz
142 Mhz
FF
256
202800
0.13
500 Mhz
226 Mhz
DSP
12
600
2.00
250 Mhz
123 Mhz
IO
129
285
45.26
300 Mhz
108 Mhz
BUFG
1
32
3.13
300 Mhz
47 Mhz
Resource utilization of FONCS-4 system.
Resource
Utilization
Available
Utilization %
Clock frequency
fmax
Used
LUT
911
101400
0.90
300 Mhz
151 Mhz
FF
256
202800
0.13
500 Mhz
226 Mhz
DSP
12
600
2.00
250 Mhz
123 Mhz
IO
129
285
45.26
300 Mhz
108 Mhz
BUFG
1
32
3.13
300 Mhz
47 Mhz
RTL schematics of FONCS-1 system.
(a) Power utilized and (b) power utilization versus fractional order of FONCS-1 system.
2D phase portraits of the FPGA implemented FONCS-1 system (XY, XW, YZ, and ZW).
RTL schematics of FONCS-2 system.
(a) Power utilized and (b) power utilization versus fractional order of FONCS-2 system.
2D phase portraits of the FPGA implemented FONCS-2 system (XY, XW, YZ, and ZW).
RTL schematics of FONCS-3 system.
(a) Power utilized and (b) power utilization versus fractional order of FONCS-1 system.
2D phase portraits of the FPGA implemented FONCS-3 system.
RTL schematics of FONCS-4 system.
(a) Power utilized and (b) power utilization versus fractional order of FONCS-4 system.
2D phase portraits of the FPGA implemented FONCS-4 system.
6. Conclusion
This paper introduces a new hyperchaotic system which when changing the values of parameters exhibits self-excited and hidden attractors. Dynamic analyses of the proposed hyperchaotic system are investigated. Fractional order model of the hyperchaotic system is derived and implemented in FPGA. Power efficiency analyses for various fractional orders are derived and it is shown that the system uses maximum power when exhibiting its largest Lyapunov exponent.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors wish to extend their sincere thanks and gratitude to Dr. Sajad Jafari, Department of Biomedical Engineering, Amirkabir University of Technology, for guiding in achieving crucial results.
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