Journal of Xidian University
https://doi.org/10.37896/jxu15.12/036
ISSN No:1001-2400
LabVIEW implementation for Design of a multiterm hyperchaotic system
with quadratic nonlinearities.
Miretab Tesfayohanis1, Deepak Tyagi*2, Workneh Girma Gelalcha2
1
2
Infomation Technology, Dambi Dollo University, Dambi Dollo Ethiopia.
College of Buisness and economics, Dambi Dollo University, Dambi Dollo Ethiopia.
* Corresponding Author, Deepak Tyagi.
Abstract: The design of a 12-term novel four-dimensional hyperchaotic system with four quadratic
nonlinearities. The Lyapunov exponents of the dissipative novel hyperchaotic system are obtained as L 1 =
4.1043, L2 = 0.1571, L3 = 0 and L4 = 34.2529. The maximal Lyapunov exponent (MLE) of the novel
hyperchaotic system is L1 = 4.1043. The Lyapunov dimension of the hyperchaotic system has been obtained
as DL = 3.1244. The qualitative properties of the novel hyperchaotic system are described in detail and
MATLAB plots have been shown to describe the 3-D projections of the phase portraits of the novel
hyperchaotic system. Moreover, the Field Programmable Gate Array (FPGA) technology has been applied
to implement the novel hyperchaotic system. FPGA results have been given in detail for the novel
hyperchaotic system.
Keyword: Chaos, hyperchaos, hyperchaotic systems, Lyapunov exponents, Lyapunov dimension, LabVIEW
implementation.
1. Introduction
Chaotic systems are nonlinear dynamical systems which are irregular, aperiodic, unpredictable and have a sensitive
dependence on the initial conditions. The first experimentally verified chaotic system is a 3-D weather model (Lorenz,
1963). This was followed by several important chaotic systems in the literature such as Rössler system (Rössler, 1976),
Leipnik system (Leipnik and Newton, 1981), Sprott systems (Sprott, 1994), Chen system (Chen, 1999), Lü system (Lü
and Chen, 2002), Harb system (Harb and Zohdy, 2002), Liu system (Liu et al., 2004), Cai system (Cai and Tan, 2007),
Elhadj system (Elhadj, 2008), Tigan system (Tigan and Opris, 2008), Sundarapandian systems (Sundarapandian, 2013a,
2013b, 2013c), Li-Wu system (Li et al., 2013), etc. Mathematically, a chaotic system can be defined as a nonlinear
dissipative dynamical system having at least one positive Lyapunov exponent, where the Lyapunov exponent of a
dynamical system is a measure of the divergence of points which are initially very close and this can be used to quantify
chaotic systems (Leonov, 2008).
Chaotic systems have many applications in science and engineering such as cryptosystems (Usama et al., 2010; Rhouma
and Belghith, 2010), secure communications (Chen and Min, 2008; Fallahi and Leung, 2010), oscillators (Shimizu et
al., 2011; Ueta and Tamura, 2012), lasers (Antonelli and Mecozzi, 2009; Hu et al., 2011), vibrations (Liu et al., 2008;
Luo and Lv, 2009), thermal systems (Yuda and Zhiqiang, 2011; Peng et al., 2012), turbines (Miyano et al., 2012; Asgari
et al., 2013), power systems (Widyan, 2013), robotics (Volos et al., 2012), cardiology (Denton et al., 1990), neurology
(Sarbadhikari and Chakrabarty, 2001), etc.
Synchronization of chaotic systems has many applications in secure communications (Abdullah, 2013; Yang and Zhu
2013). Chaos synchronization problem has been studied by a variety of methods such as active control method (Ho and
Hung, 2002; Chen, 2005; Njah and Vincent, 2009; Chen et al., 2009; Salarieh and Alasty, 2009; Sundarapandian, 2011a;
Zhang et al., 2012), adaptive control method (Liao and Tsai, 2000; Sundarapandian and Karthikeyan, 2011;
Sundarapandian, 2011b; Sundarapandian and Pehlivan, 2012; Sundarapandian, 2013d), sampled-data control (Zhao and
Lu, 2008; Gan and Liang, 2012), backstepping method (Yu and Zhang, 2004; Park, 2006; Sundarapandian, 2013e),
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Journal of Xidian University
https://doi.org/10.37896/jxu15.12/036
ISSN No:1001-2400
delayed feedback control (Batista et al., 2010; Yu et al., 2013), sliding mode control (Sundarapandian and Sivaperumal,
2011, 2012; Qian et al., 2011; Zarrabi et al., 2012; Shang and Wang, 2013; Zhen et al., 2013), etc.
A hyperchaotic system is mathematically defined as a chaotic system having more than one positive Lyapunov
exponent implying that its dynamics are expended in many different directions simultaneously. Thus, a hyperchaotic
system has more complex dynamical behaviours than a chaotic system. The German scientist, O.E. Rössler was the first
to discover a hyperchaotic system (Rössler, 1979). There are many well-known hyperchaotic systems such as
hyperchaotic Lorentz system (Jia, 2007), hyperchaotic Chen system (Li, 2009), hyperchaotic Liu system (Liu et al.,
2008), hyperchaotic Lü system (Loría, 2010), hyperchaotic Cai system (Zheng et al., 2010), hyperchaotic Li system (Li
et al., 2005), etc.
Hyperchaotic systems have special characteristics such as high capacity, high security and high efficiency and they
find broad applications in nonlinear circuits (Elwakil and Kennedy, 1999; Yujun et al., 2010; Banerjee et al., 2012),
neural networks (Li et al., 2005; Huang et al., 2012), lasers (Pu et al., 2013), optics (Grygiel, 1998), secure
communications (Wu et al., 2012), biological systems (Ibrahim and Elnashaie, 1997) and synchronization (Wang et al.,
2008; Buscarino et al., 2009; Mahmoud and Mahmoud, 2010).
The earlier implementations of Chaotic systems were in Pspice (Buscarino et al., 2009;Zhaeng 2010). Then after the
integration of LabVIEW with various platforms its easier tomimplement the chaotic systems in LabVIEW. Earlier works
on LabVIEW digital implementation (Zhang et al.,2007;Bao et al., 2011;Cai et al.,2008;Zhao et al., 2003) were on
interfacing Simulink with LabVIEW and PSpice with LabVIEW.
The rest of the paper is organized as follows. Section 2 describes the 12-term novel hyperchaotic system with four
quadratic nonlinearities. In this section, phase portraits of the novel hyperchaotic system are described. Section 3
describes the qualitative properties of the novel hyperchaotic system. In this section, it is shown that the novel
hyperchaotic system is a dissipative system with an unstable equilibrium point. Also, the Lyapunov exponents and the
Lyapunov dimension of the novel hyperchaotic system have been obtained. Section 4 describes the FPGA
implementation of the novel hyperchaotic system. Section 5 contains the conclusions of this work.
2. A 12-term novel hyperchaotic system with four quadratic nonlinearities.
In this section, we describe a 12-term novel hyperchaotic system with four quadratic nonlinearities given by the 4-D
dynamics
x1  a ( x2  x1 )  x2 x3
x2  bx1  cx2  x4  x1 x3
(1)
x3  x1 x2  cx3  x22
x4   x1  rx2
Where x1 , x2 , x3 , x4 are state variables and a, b, c , r are constant, positive, parameters of the system.
The 4-D system (1) displays a hyper chaotic attractor when the parameter values are taken as
(2)
a  30, b  44, c  12.6, r  23
For numerical simulations, we take the initial values of the 4-D system (1) as
(3)
x1 (0)  1.5, x2 (0)  0.6, x3 (0)  1.8, x4 (0)  2.4
Figures 1-4 depict a 3-D projection of the strange hyper chaotic attractor of the system (1) in ( x1 , x2 , x3 ), ( x1 , x2 , x4 ),
( x1 , x3 , x4 ) and ( x2 , x3 , x4 ) spaces, respectively.
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Figure 1 Projection of the 4-D system in
( x1 , x2 , x3 ) space.
Figure 2 Projection of the 4-D system in
( x1 , x2 , x4 )
Figure 3 Projection of the 4-D system in
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( x1 , x3 , x4 ) space
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https://doi.org/10.37896/jxu15.12/036
Figure 4 Projection of the 4-D system in
( x2 , x3 , x 4 )
ISSN No:1001-2400
space
3. Qualitative properties of the novel hyper chaotic system
3.1. Symmetry
The novel 4-D hyperchaotic system (1) has rotation symmetry about the x3  axis since the system (1) is invariant under
the coordinates transformation
( x1 , x2 , x3 , x4 )  ( x1 ,  x2 , x3 ,  x4 )
(4)
Since the transformation (4) persists for all values of the system parameters a, b, c and r , any non-trivial trajectory
of the system (1) must have a twin trajectory.
3.2. Dissipativity
We can write the four-dimensional system (1) in vector notation as
 f1 ( x ) 
 f ( x) 
2

x  f ( x )  
 f3 ( x) 


 f 4 ( x) 
(5)
where
f1 ( x )  a ( x2  x1 )  x2 x3
f 2 ( x )  bx1  cx2  x4  x1 x3
f 3 ( x )  x1 x2  cx3  x22
(6)
f 4 ( x )   x1  rx2
The divergence of the vector field f on R 4 is defined by
 f 
4
f1 f 2 f 3 f 4
f



 i
x1 x2 x3 x4 i 1 xi
(7)
Let  be any region in R 4 with smooth boundary.
Let (t )  t    , where  t is the flow of f .
Let V (t ) denote the hypervolume of the region  (t ) in R 4 .
By Liouville’s theorem, we have
dV

(  f ) dx1dx2 dx3 dx4
dt  
 (t )
(8)
For the system (1), the divergence of the vector field f is calculated as
 f 
f1 f 2 f3 f 4



 a  0
x1 x2 x3 x4
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ISSN No:1001-2400
since a  0.
Substituting the value of divergence of f in (8), we get
dV
  a   dx1dx2 dx3 dx4 =  aV (t )
dt
 (t )
(10)
Solving the linear ordinary differential equation (10), we get the solution as
(11)
V (t )  exp( at ) V (0)
From (11), we find that V (t ) shrinks to zero exponentially as t  .
Hence, the asymptotic motion of the system (1) settles exponentially onto a set of measure zero, i.e. a strange attractor.
3.3. Equilibrium points
The equilibrium points of the 4-D system (1) are obtained by solving the nonlinear system of equations
f1 ( x )  a( x2  x1 )  x2 x3  0
f 2 ( x)  bx1  cx2  x4  x1 x3  0
(12)
f 3 ( x )  x1 x2  cx3  x22  0
f 4 ( x)   x1  rx2  0
We take the parameter values as
(13)
a  30, b  44, c  12.6, r  23
Solving the nonlinear system (12) with the parameter values (13), we obtain the system equilibrium point as
0
0
(14)
E0   
0
 
0
The Jacobian matrix of the system (1) is calculated as
a  x3
x2 0 
 a
b  x
c
 x1 1 
3
J ( x)  
 x2
x1  2 x2 c 0 


r
0 0
 1
Thus, the Jacobian matrix of the system (1) at E0 is obtained as
0
0
 30 30
 44 12.6
0
1 

J0 
 0
0
12.6 0 


0
0
 1 23
The eigenvalues of J 0 are numerically calculated using MATLAB as
(15)
(16)
1  50.7106, 2  12.6, 3  32.8787, 4  0.4318
(17)
This shows that the equilibrium E0 of the novel 4-D system (1) is a saddle-node, which is unstable.
3.4. Lyapunov Exponents
We take the initial values of the 4-D system as
x1 (0)  1.5, x2 (0)  0.6, x3 (0)  1.8, x4 (0)  2.4
(18)
We take the parameter values of the 4-D system (1) as
a  30, b  44, c  12.6, r  23
(19)
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https://doi.org/10.37896/jxu15.12/036
ISSN No:1001-2400
The Lyapunov exponents of the system (1) are found numerically using MATLAB as
(20)
L1  4.1043, L2  0.1571, L3  0, L4  34.2519
Since there are two positive Lyapunov exponents in (20), the nonlinear system (1) is a hyperchaotic system. The
maximal Lyapunov exponent (MLE) of the system (1) is
L1  4.1043
We also note that the sum of the Lyapunov exponents of the system (1) is given by
(21)
L1  L2  L3  L4  29.9905  0
This confirms our results in Section 3.1 that the 4-D system (1) is dissipative.
The dynamics of the Lyapunov exponents of the system (1) is shown in Figure 5.
Figure 5 Dynamics of the Lyapunov exponents
3.5. Lyapunov Dimension
The Lyapunov dimension of the novel hyper chaotic system (1) is calculated as
L  L2  L3
DL  3  1
 3.1244
| L4 |
(22)
Eq. (22) shows that (1) is a hyper chaotic system with fractional dimension.
4. LabVIEW implementation of the novel hyper chaotic system
The proposed new 4D hyperchaotic system is then implemented in LabVIEW. The state equations of the new
hyperchaotic system are implemented in CSD tool book of LabVIEW. Fixed step simulation method is applied for the
iteration process. The configuration parameters are altered to match the simulation environment. Simulation time
waveform converters are deployed to generate time based signals for waveform plotting. Reshape array blocks are used
to get 1D array elements from double precision data. The LabVIEW VI model is shown in Figure 6. The front panel of
the VI block diagram is shown in Figure 7. The Phase portraits of the proposed system are shown in Figure 8.
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ISSN No:1001-2400
Figure 6 VI Model for the new Hyperchaotic system.
Figure 7 VI Model – Front Panel.
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Figure 8 Time waveforms of the state variables
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Figure 9: 2D Phase Portraits of the Attractors.
5. Conclusions
In this research work, we have introduced a 12-term novel 4-D hyperchaotic system with four quadratic
nonlinearities and discussed its qualitative properties. The system is dissipative and has an unstable,
saddle-node, equilibrium point at the origin. Lyapunov exponents and Lyapunov dimension for the
novel hyperchaotic system have also been obtained. Explicitly, the Lyapunov exponents are obtained
as L1 = 4.1043, L2 = 0.1571, L3 = 0 and L4 = 34.2519. It was found that maximal Lyapunov exponent
(MLE) for the novel chaotic system is L1 = 4.1043 and the Lyapunov dimension is DL = 3.1244.
Moreover, the LabVIEW model is generated for the digital implementation of the novel hyperchaotic
system. The VI results were discussed in detail for the proposed 4-D novel hyperchaotic system.
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