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Chaotic analysis and combination-combination synchronization of a novel hyperchaotic system without any equilibria
Accepted Manuscript
Chaotic Analysis and Combination-Combination Synchronization of a Novel Hyperchaotic System without any Equilibria Ayub Khan, Shikha Singh PII: DOI: Reference:
S0577-9073(17)30783-9 10.1016/j.cjph.2017.12.023 CJPH 426
To appear in:
Chinese Journal of Physics
Received date: Revised date: Accepted date:
24 June 2017 19 December 2017 23 December 2017
Please cite this article as: Ayub Khan, Shikha Singh, Chaotic Analysis and Combination-Combination Synchronization of a Novel Hyperchaotic System without any Equilibria, Chinese Journal of Physics (2017), doi: 10.1016/j.cjph.2017.12.023
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Highlights • A novel hyperchaotic system with no equilibrium point is constructed. • It is observed that as the parameter values varies the system displays different behavior. • Various properties of novel hyperchaotic system are investigated.
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• Using this novel system combination-combination synchronization is performed.
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Chaotic Analysis and Combination-Combination Synchronization of a Novel Hyperchaotic System without any Equilibria Ayub Khana , Shikhab,∗ a Department
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of Mathematics, Jamia Millia Islamia, New Delhi - 110025, India. Email : [email protected] b Department of Mathematics, Jamia Millia Islamia, New Delhi - 110025, India. Email : [email protected]
Abstract
This manuscript examines a novel 4D continuous autonomous hyperchaotic system with two scroll attractor. This system does not produce any equilibrium point so, it generates hidden hyperchaotic attractor. The Lyapunov exponent, bi-
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furcation diagram, Poincar´e section, Kaplan-Yorke dimension and phase portraits are given to justify the hyperchaotic nature of the system. The novel system displays hyperchaotic orbit, chaotic orbit, periodic orbit, quasi-periodic orbit as the parameters values varies. Furthermore, the combination-combination synchronization is performed by considering four identical 4D novel hyperchaotic systems.
Keywords: Chaotic system, Lyapunov exponent, bifurcation diagram, Poincar´e section, Kaplan-Yorke dimension, 2010 MSC: 34H10, 34H05, 34D05
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1. Introduction
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combination-combination synchronization
In the most recent decades researchers from all fields of natural sciences have studied phenomena that include nonlinear systems exhibiting chaotic behavior [1], [2, 3, 4, 5, 6]. This is because of the way that nonlinear systems
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demonstrate rich dynamics and have sensitivity on initial conditions. Chaotic systems are third order or higher order nonlinear differential equations with at least one positive lyapunov exponent. Hyperchaos is characterized as a chaotic
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system with more than one positive Lyapunov exponent [7], this infer that its dynamics are expended in several different directions simultaneously. Thus, hyperchaotic systems have more complex dynamical behaviors than ordinary chaotic systems. In 1976 [8] Rossler led imperative work that revived the enthusiasm in 3D autonomous chaotic sys-
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tem. Since Lorenz proposed a astounding chaotic system for portraying atmospheric convention in 1963 [9], hordes of consideration have been committed to the examination to the autonomous system. As of late, there has been increasing enthusiasm in creating chaos particularly since Chen and Ueta [2] found a new chaotic system, called the Chen system, ✩ Fully
documented templates are available in the elsarticle package on CTAN. author Email address: [email protected] (Shikha )
∗ Corresponding
Preprint submitted to Chinese Journal of Physics
December 29, 2017
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by adding a simple state feedback to the second equation of the Lorenz system, which is not topologically equivalent to the Lorenz system. After that, Lu and Chen [3] further developed a new chaotic system, bearing the name of the Lu system, which represents the transition between the Lorenz system and Chen system. From then on, an exceptionally 15
rich family of the so-called unified system or Lorenz system family [4] was presented as a association of the Lorenz, Chen, and Lu systems. In chaos theory, the equilibrium of a dynamical system is significant for understanding its complex dynamics. A
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hyperchaotic system producing no equilibrium points has been a important point of discussion in non-linear dynamical system because such systems displays unique properties [10, 11, 12, 13, 14]. It is noted that to verify the presence 20
of chaos in systems producing no equilibrium points Shilnikov’s method fails because such systems have neither homoclinic nor heteroclinic orbits. The system producing no equilibrium point belongs to a class of hidden attractor. Hidden attractors are important in engineering applications because they allow unexpected and potential disastrous responses to perturbations in a structure like a bridge or an airplane wing. Very few systems with no equilibrium
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points have been investigated till now. Analysis of the dynamics of hyperchaotic systems received a great deal of interest in the recent past due to its applications in secure communication [6], information processing [15], biological systems [16], chemical reactions [17], neural network [18] etc. So, more and more chaotic or hyper chaotic systems showing wide dynamical behavior were found [19, 20, 21, 22, 23, 24, 25, 26, 27]. For example Wu and Li [19] presented a new chaotic system and studied its basic dynamical properties and then synchronize it using different control techniques. Kingni et. al. [20] introduced a novel chaotic system with a circular equilibrium. Analysis, circuit simulation and its fractional order form is also discussed. S.Vaidyanathan [10] announced a novel highly hyperchaotic
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system with no equilibrium point and analysed its dynamical behaviour. Furthermore synchronization by adaptive
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control is also achieved. X.Wang et. al. [21] constructed few chaotic systems having no equilibrium, one stable equilibria, two equilibria, three equilibria and any number of equilibria. V.T.Pham et. al [22] introduces a chaotic system with different shapes of equilibria like chaotic attractor with circular equilibrium, elliptical equilibrium and square shaped equilibrium. Deng et. al. [23] introduces a hyper-chaotic attractor formed by adding one additional
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state variable and modifying nonlinear terms and parameters of Chen system having only two parameters and the
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system shows hyper-chaotic behavior for the wide range of one parameter establishing the more chaotic nature of the system. Chen et. [24] al. constructed a hyper-chaotic system by adding a feedback controller to the classical Lorenz system which shows abundant dynamics having a curve of equilibria. Wang et. al. [25] describes a 4D hyper-chaotic system and analyse it by studying its local bifurcation dynamics and by computing its ultimate bound. In 2017, Vishal,
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K., and Saurabh K. Agrawal. [26]have studied the dynamics of a novel complex chaotic system with fractional order derivative and found the existence of chaos. The novel complex system is simulated for integer as well as fractional orders which shows some unusual phenomena. In 2017, Gholamin, P., and AH Refahi Sheikhani. [27] recomends a new three-dimensional autonomous chaotic system with six terms including three multipliers, which is different
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from the Lorenz system and other existing systems and the dynamical properties are studied. Motivated by the above
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researches and based on the 3D chaotic system introduced by Q. Yang et al. [28], a novel 4D continuous autonomous hyperchaotic system with no equilibrium point by adding a feedback controller to the 3D chaotic system is constructed. Further, some basic dynamical properties including Lyapunov exponent, bifurcations, Poincar´e section , Kaplan-Yorke dimension and phase portraits are illustrated by both theoretical analysis and computer simulations. 50
Due to the vast practical applications of chaotic dynamical systems in fields stated above, numerous investigations have been done theoretically and experimentally on controlling chaos and synchronization. In 1990, Pecora
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and Carroll [29] gave the synchronization of chaotic systems using the concept of master and slave system. Also, in 1990 Ott et al. [30] introduced the OGY method for controlling chaos. Looking for better strategies for chaos control and synchronization different types of methods have been developed for controlling chaos and synchroniza55
tion of non identical and identical systems for instance linear feedback [31], optimal control [32], adaptive control [33, 34, 35, 36, 37, 38, 39, 40], active control [41, 42, 43], active sliding control [44], backstepping control [45, 46, 47], hybrid control [48] etc. As an aftereffect of quickly developing enthusiasm in chaos control and syn-
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chronization various synchronization types and schemes have been proposed and reported, for instance generalized synchronization [49], projective synchronization [50], modied projective synchronization [51], function projective 60
synchronization [52], modied-function projective synchronization [53], hybrid synchronization [54] and hybrid function projective synchronization [55]. It is noted that most of the researches mainly focused on the previous master-slave synchronization scheme within one master and one response system. Only a few research papers have been published on combination-combination synchronization where four chaotic systems were taken into account [56, 57, 58, 59].
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Combination-combination synchronization scheme used in this paper is generalized in such a way that other forms of synchronization scheme can be achieved form it. As a result, combination-combination synchronization scheme
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is more flexible and applicable to the real world systems. In addition, the combination-combination synchronization also gives better insight into the complex synchronization and several pattern formations that take place in real world systems since synchronization in real world systems are complex. This manuscript is categorize as follows : In section 3 combination-combination synchronization of four 4D novel hyperchaotic system is achieved. Finally in section 4
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concluding remarks are given.
2. Formulation of 4D Novel Hyper-Chaotic System and its Fundamental Dynamical Properties In 2010 Q. Yang et al. [28] formulated a 3D autonomous chaotic Lorenz-type system which is given by u˙1 = a(u2 − u1 ) u˙2 = −u1 u3 − cu2 u˙ = −b + u u 3 1 2
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2 formulation of 4D novel hyperchaotic system and its fundamental dynamical properties are discussed. In section
(2.1)
where a, b, c ∈ R are parameters. For the parameter values a = 10, b = 100 and c = 4.7466 system has two saddle focus and unstable equilibria (±10, ± 10, − 4.7466). The eigenvalues of the Jacobian evaluated at the equilibrium points 4
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are λ1 = − 16.1965, λ2,3 = 0.7250 ± 11.0886ι. For the initial condition (u1 (0), u2 (0), u3 (0)) = (10, − 0.2, 0.75) and the parameter values (a, b, c) = (10, 100, 4.7466) , the Lyapunov exponents are (γ1 , γ2 , γ3 ) = (1.0054, − 0.0018, − 15.7308). So, the system shows chaotic attractor displayed in figure (1). The detailed description of system (2.1) is given in [28].
100 u2(t)
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Figure 1: Phase Portrait of the 3D chaotic Lorenz type system
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Now adding a feedback controller to system (2.1), a novel 4D hyperchaotic system is formed given by u˙1 = a(u2 − u1 ) − mu4 u˙2 = −u1 u3 − cu2 u˙3 = −b + u1 u2 u˙4 = −ku1 5
(2.2)
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where k, m ∈ R are parameters. When b 6= 0, system (2.2) has no equilibrium point. Thus, the phenomena like
pitchfork, Hopf or homoclinic bifurcations does not take place while these phenomena are common in systems with equilibria. 2.1. Chaotic Attractor of the System When the parameters a = 10, b = 25, c = −2.5, k = 1 and m = 3 and initial condition (10, −0.2, 0.05, −3) are
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chosen then the system displays two scroll chaotic attractor as shown in figure (2). Also the time series of the novel hyperchaotic attractor is shown in figure (3). The corresponding Lyapunov exponents of the novel hyperchaotic attractor are γ1 = 0.88503, γ2 = 0.048124, γ3 = −0.00028044, γ4 = −8.4325 as shown in figure (4).
The Kaplyan-Yorke dimension is defined by
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γi |γ i=1 j+1 |
D = j+∑
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= 3.1106
j j+1 where j is the largest integer satisfying ∑i=1 γ j ≥ 0 and ∑i=1 γ j < 0. Therefore Kaplan-Yorke dimension of the chaotic
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attractor is D = 3.1106 which means that the Lyapunov dimension of the chaotic attractor is fractional.
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Figure 2: Phase Portrait of the 4D novel hyperchaotic system
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−10
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Figure 3: Time series of the 4D novel hyperchaotic system
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1
λ = 0.88503 1 λ = 0.048124
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λ = −0.00028044 3
Lyapunov Exponents
−1 −2 −3 −4 −5
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Figure 4: Lyapunov exponents of the 4D novel hyperchaotic system
2.2. Dissipation and existence of Chaotic Attractor
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The divergence of the novel 4D hyperchaotic system (2.2) is ∇.V =
∂ u˙1 ∂ u˙2 ∂ u˙3 ∂ u˙4 + + + ∂ u1 ∂ u2 ∂ u3 ∂ u4
= −a − c = −7.5 < 0
So, the system (2.2) is dissipative system and converges by the index rate of e−7.5t .
It means that a volume element V0 is contracted into V0 e−7.5t at time t. That is, each volume containing the system
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orbit shrinks to zero as t → 0, at an exponential rate ∇V , which is independent of u1 , u2 , u3 , u4 . Consequently, all
system orbits will ultimately be confined to a specific subset of zero volume and the asymptotic motion settles onto an attractor. Then the existence of attractor is proved. 2.3. Symmetry and Invariance
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The novel hyperchaotic system (2.2) is invariant under the transformation (u1 , u2 , u3 , u4 ) → (−u1 , − u2 , u3 , −
u4 ) this means the system (2.2) is symmetric about u3 -axis.
As an important analysis technique, the Poincar´e map reflects the periodic, quasi-periodic, chaotic and hyperchaotic behavior of the system. When a = 10, b = 25, c = −2.5, k = 1 and m = 3 one may take u2 = 0 and u3 = 0 as
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2.4. Poincar´e map, Bifurcation diagram and Impact of system parameters
the crossing sections respectively as shown in figure (5).
The bifurcation diagram of |u1 | with respect to parameters c is shown in figure (6) which shows abundant and complex dynamical behaviors.
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Figure 5: Poincar´e map on the crossing section u2 =0 and u3 = 0 respectively
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Figure 6: Bifurcation diagram of |u1 | versus c
Denoting the four Lyapunov exponents of system (2.2) by γ1 , γ2 , γ3 , γ4 which satisfy γ1 > γ2 > γ3 > γ4 . The
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behavior of the system depends on the parameter values. It is also depicted in bifurcation diagram. For different values of parameters system exhibits hyperchaotic attractor, chaotic attractor, quasi-periodic orbit and periodic orbit. Chaotic analysis of the system (2.2) is carried out by varying one parameter value and fixing other parameters. Vary parameter
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a and fix other parameters. For the parameter values (12,25,-2.5,1,3), (10,27,-2.5,1,3), (10,25,-1,1,3), (10,25,-2.5,2,3) and (10,25,-2.5,1,4) two largest Lyapunov exponents γ1 and γ2 are positive, which indicate that the system (2.2) shows the hyperchaotic behavior. For the parameters values (7,25,-2.5,1,3), (10,140,-2.5,1,3), (10,25,-4.9,1,3) and (10,25,2.5,1,-12) the largest Lyapunov exponent γ1 is positive, which indicate the chaotic behavior of the system (2.2). For
the parameter values (22,25,-2.5,1,3), (10,0,-2.5,1,3), (10,25,-2.5,-1,3) and (10,25,-2.5,1,-28) the largest Lyapunov exponent γ1 tends to zero, which indicate that the orbit of the system (2.2) is periodic in nature. For the parameter 110
values (10,-12,-2.5,1,3), (10,25,-2.5,-150,3) and (10,25,-2.5,1,-600) two largest Lyapunov exponents γ1 tends to zero, 10
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(γ1 , γ2 , γ3 , γ4 )
Lyapunov Dimension
Orbital mode
(12,25,-2.5,1,3)
(0.6167,0.0601,-0.0150,-10.1605)
3.0651
hyperchaotic
(7,25,-2.5,1,3)
(0.8662,0.0010,-0.0195,-5.3476)
3.1585
chaotic
(22,25,-2.5,1,3)
(0.0081,-0.0171,-1.8235,-17.5757)
2.9743
periodic
(10,27,-2.5,1,3)
(0.9775,0.0306,-0.0044,-8.5032)
3.1180
hyperchaotic
(10,140,-2.5,1,3)
(0.9287,0.000949,-0.08895,-8.3355)
3.1008
chaotic
(10,0,-2.5,1,3)
(-0.00264,-0.1054,-3.6626,-3.7292)
1.9888
periodic
(10,-3,-2.5,1,3)
(-0.0059,-0.0037,-3.7307,-3.7663)
2.0068
quasi-periodic
(10,25,-1,1,3)
(0.6815,0.0558,3.82×10− 6,-9.7364)
3.0757
hyperchaotic
(10,25,-4.9,1,3)
(1.0706,0.00601,-0.0896,-6.0869)
3.1621
chaotic
(10,25,-2.5,2,3)
(0.8077,0.1006,-0.00643,-8.4015)
3.1073
hyperchaotic
(10,25,-2.5,-1,3)
(-0.0048,-0.2742,-3.4966,-3.7291)
2.0054
periodic
(10,25,-2.5,-150,3)
(0.00087,-0.0140,-3.7439,-3.7778)
1.9874
quasi-periodic
(10,25,-2.5,1,4)
(0.8174,0.0678,-0.0039,-8.381)
3.1051
hyperchaotic
(10,25,-2.5,1,-12)
(0.0951,-0.0026,-3.6588,-3.7515)
2.0493
chaotic
(10,25,-2.5,1,-28)
(-0.00042,-0.0554,-3.7071,-3.7482)
1.9960
periodic
(10,25,-2.5,1,-600)
(-0.0041,-0.0064,-3.7625,-3.7653)
1.9979
quasi-periodic
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Values of parameters (a,b,c,k,m)
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Table 1: Lyapunov exponents, Lyapunov dimension and orbital mode for the different values of parameters
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which indicate that the orbit of the system (2.2) is quasi-periodic in nature. In table (1) all the Lyapunov exponents, Lyapunov dimension and orbital mode for the different values of parameters are given. The phase portraits for the
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parameter values listed in table (1), displaying the fixed orbit, periodic orbit and chaotic orbit is shown in figure (7)
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Figure 7: Phase portraits showing hyperchaotic, chaotic, periodic and quasi-periodic behavior in u1 − u4 plane
3. Methodology of combination-combination synchronization
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Methodology for combination-combination synchronization with two master systems and two slave systems is
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described in this section. The two master systems are described as follows: u˙1 = f1 (u1 )
(3.1)
u˙2 = f2 (u2 )
(3.2)
while the two slave systems are described as follows: v˙1 = g1 (v1 )
(3.3)
v˙2 = g2 (v2 )
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where u1 = (u11 , u12 , ..., u1n ), u2 = (u21 , u22 , ..., u2n ), u1 = (v11 , v12 , ..., v1n ) and v1 = (v21 , v22 , ..., v2n ) are the state vectors of the master and slave systems respectively and f1 , f2 , g1 , g2 : Rn → Rn are the controllers added to the slave system which are to be determined.
Definition 3.1. If there exists four constant matrices A, B, C, D ∈ Rn and C 6= 0 or D 6= 0 such that (3.5)
lim kAu1 + Bu2 −Cv1 − Dv2 k = 0
t→∞
system (3.3) and (3.4). Here k.k represents the matrix norm. 120
The following observation arises :
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then the master systems (3.1) and (3.2) are said to achieve combination-combination synchronization with the slave
1. The constant matrices A, B, C, D are called the scaling matrices. In addition A, B, C, D can be extended to functional matrices of state variables u1 , u2 , v1 and v2 .
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2. If C = 0 or D = 0, then combination-combination synchronization problem becomes the combination synchronization problem.
3. If A = 0, C = I, D = 0 or A = C = 0, D = I or B = 0, C = I, D = 0 or B = C = 0, D = I, then the combination synchronization becomes projective synchronization, where I is the n × n matrix.
4. If A = B = C = 0 or A = B = D = 0, then the combination synchronization becomes the chaos control problem.
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4. Combination-combination synchronization among four novel identical hyper-chaotic systems
described above.
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The two master systems are
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To perform the combination-combination synchronization we choose the four identical novel hyperchaotic system
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u˙11 = a(u12 − u11 ) − mu14 u˙12 = −u11 u13 − cu12 u˙13 = −b + u11 u12 u˙14 = −ku11 u˙21 = a(u22 − u21 ) − mu24 u˙22 = −u21 u23 − cu22 u˙23 = −b + u21 u22 u˙24 = −ku21
13
(4.1)
(4.2)
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while the two slave systems are given as v˙11 = a(v12 − v11 ) − mv14 + u1 v˙12 = −v11 v13 − cv12 + u2
v˙21 = a(v22 − v21 ) − mv24 + u∗1 v˙22 = −v21 v23 − cv22 + u∗2 v˙23 = −b + v21 v22 + u∗3 v˙24 = −kv21 + u∗4
(4.3)
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v˙13 = −b + v11 v12 + u3 v˙14 = −kv11 + u4
(4.4)
where u1 , u2 , u3 , u4 , u∗1 , u∗2 , u∗3 and u∗4 are the controllers to be determined. Let us choose A = diag(α1 , α2 , α3 , α4 ),
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B = (β1 , β2 , β3 , β4 ), C = (γ1 , γ2 , γ3 , γ4 ) and D = (δ1 , δ2 , δ3 , δ4 ) in this synchronization scheme. The error system is obtained as follows: e1 = α1 u11 + β1 u21 − γ1 v11 − δ1 v21 e2 = α2 u12 + β2 u22 − γ2 v12 − δ2 v22
(4.5)
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e3 = α3 u13 + β3 u23 − γ3 v13 − δ3 v23 e4 = α4 u14 + β4 u24 − γ4 v14 − δ4 v24
Denote
U3 = γ3 u3 + δ3 u∗3 U4 = γ4 u4 + δ4 u∗4
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U1 = γ1 u1 + δ1 u∗1 U2 = γ2 u2 + δ2 u∗2
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Theorem 4.1. For the controllers: U1 = α1 (a(u12 − u11 ) − mu14 ) + β1 (a(u22 − u21 ) − mu24 ) − γ1 (a(v12 − v11 ) − mv14 ) − δ1 (a(v22 − v21 ) − mv24 ) + α1 u11 + β1 u21 − γ1 v11 − δ1 v21 + α2 u12 + β2 u22 − γ2 v12 − δ2 v22 + α3 u13 + β3 u23 − γ3 v13 − δ3 v23 + α4 u14 + β4 u24 − γ4 v14 − δ4 v24 U2 = α2 (−u11 u13 − cu12 ) + β2 (−u21 u23 − cu22 ) − γ2 (−v11 v13 − cv12 )
4 14
4 24
4 14
4 24
(4.7)
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− δ2 (−v21 v23 − cv22 ) + α2 u12 + β2 u22 − γ2 v12 − δ2 v22 − α1 u11 + β1 u21 − γ1 v11 − δ1 v21 U3 = α3 (−b + u11 u12 ) + β3 (−b + u21 u22 ) − γ3 (−b + v11 v12 ) − δ3 (−b + v21 v22 ) + α3 u13 + β3 u23 − γ3 v13 − δ3 v23 − α1 u11 + β1 u21 − γ1 v11 − δ1 v21 U = α4 (−ku11 ) + β4 (−ku21 ) − γ4 (−kv11 ) − δ4 (−kv21 ) 4 +α u +β u −γ v −δ v −α u +β u −γ v −δ v 1 11
1 21
1 11
1 21
the master systems (3.1) and (3.2) will attain the combination-combination synchronization with the slave systems (3.3) and (3.4).
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Proof The error dynamical system using (4.5) is obtained as follows:
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∗ e˙1 = α1 (a(u12 − u11 ) − mu14 ) + β1 (a(u22 − u21 ) − mu24 ) − γ1 (a(v12 − v11 ) − mv14 + u1 ) − δ1 (a(v22 − v21 ) − mv24 + u1 ) e˙2 = α2 (−u11 u13 − cu12 ) + β2 (−u21 u23 − cu22 ) − γ2 (−v11 v13 − cv12 + u2 ) − δ2 (−v21 v23 − cv22 + u∗2 )
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e˙3 = α3 (−b + u11 u12 ) + β3 (−b + u21 u22 ) − γ3 (−b + v11 v12 + u3 ) − δ3 (−b + v21 v22 + u∗3 ) e˙4 = α4 (−ku11 ) + β4 (−ku21 ) − γ4 (−kv11 + u4 ) − δ4 (−kv21 + u∗4 )
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Choosing a candidate Lyapunov function as follows:
1 V (t) = (e21 + e22 + e23 + e24 ) 2
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(4.8)
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Obviously, V (t) > 0. The time derivative of V(t) along the trajectories of the error system (4.8) is V˙ (t) = e1 e˙1 + e2 e˙2 + e3 e˙3 + e4 e˙4 = e1 (α1 (a(u12 − u11 ) − mu14 ) + β1 (a(u22 − u21 ) − mu24 ) − γ1 (a(v12 − v11 ) − mv14 + u1 ) − δ1 (a(v22 − v21 ) − mv24 + u∗1 ) + e2 (α2 (−u11 u13 − cu12 ) + β2 (−u21 u23 − cu22 ) − γ2 (−v11 v13 − cv12 + u2 )
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− δ2 (−v21 v23 − cv22 + u∗2 ) + e3 (α3 (−b + u11 u12 ) + β3 (−b + u21 u22 ) − γ3
(4.9)
(−b + v11 v12 + u3 ) − δ3 (−b + v21 v22 + u∗3 ) + e4 (α4 (−ku11 ) + β4 (−ku21 ) − γ4 (−kv11 + u4 ) − δ4 (−kv21 + u∗4 )
= e1 (−e1 − e2 − e3 − e4 ) + e2 (−e2 + e1 ) + e3 (−e3 + e1 ) + e4 (−e4 + e1 ) = −e21 − e22 − e23 − e24
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where k is the positive constant which represents feedback gain.
Since V˙ ≤ 0, according to Lyapunov theorem we know ei → 0 (i=1,2) as t → ∞ which means that the required synchronization is achieved.
The following Corollaries can easily be obtained from Theorem 1, the proofs of these Corollaries are similar to Theorem 1. So, the proofs are omitted.
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Corollary 4.1.1. For δ1 = δ2 = δ3 = δ4 = 0 and controllers : 1 u1 = [α1 (a(u12 − u11 ) − mu14 ) + β1 (a(u22 − u21 ) − mu24 ) − γ1 (a(v12 − v11 ) − mv14 ) γ 1 − δ1 (a(v22 − v21 ) − mv24 ) + α1 u11 + β1 u21 − γ1 v11 + α2 u12 + β2 u22 − γ2 v12 + α3 u13 + β3 u23 − γ3 v13 + α4 u14 + β4 u24 − γ4 v14 ] 1 u2 = [α2 (−u11 u13 − cu12 ) + β2 (−u21 u23 − cu22 ) − γ2 (−v11 v13 − cv12 ) γ 2 + α2 u12 + β2 u22 − γ2 v12 − α1 u11 + β1 u21 − γ1 v11 ] 1 u3 = [α3 (−b + u11 u12 ) + β3 (−b + u21 u22 ) − γ3 (−b + v11 v12 ) γ 3 + α3 u13 + β3 u23 − γ3 v13 − α1 u11 + β1 u21 − γ1 v11 ] 1 u4 = [α4 (−ku11 ) + β4 (−ku21 ) − γ4 (−kv11 ) γ 4 +α u +β u −γ v −α u +β u −γ v ]
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4 14
4 24
4 14
1 11
1 21
1 11
the master systems (3.1) and (3.2) will achieve combination synchronization with the slave system (3.3).
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(4.10)
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Corollary 4.1.2. For γ1 = γ2 = γ3 = γ4 = 0 and controllers :
(4.11)
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1 u∗1 = [α1 (a(u12 − u11 ) − mu14 ) + β1 (a(u22 − u21 ) − mu24 ) δ 1 − δ1 (a(v22 − v21 ) − mv24 ) + α1 u11 + β1 u21 − δ1 v21 + α2 u12 + β2 u22 − δ2 v22 + α3 u13 + β3 u23 − δ3 v23 + α4 u14 + β4 u24 − δ4 v24 ] u∗2 = 1 [α2 (−u11 u13 − cu12 ) + β2 (−u21 u23 − cu22 ) δ2 − δ2 (−v21 v23 − cv22 ) + α2 u12 + β2 u22 − δ2 v22 − α1 u11 + β1 u21 − δ1 v21 ] 1 u∗3 = [α3 (−b + u11 u12 ) + β3 (−b + u21 u22 ) − δ3 (−b + v21 v22 ) δ 3 + α3 u13 + β3 u23 − δ3 v23 − α1 u11 + β1 u21 − δ1 v21 ] 1 u∗4 = [α4 (−ku11 ) + β4 (−ku21 ) − δ4 (−kv21 ) δ 4 + α4 u14 + β4 u24 − δ4 v24 − α1 u11 + β1 u21 − δ1 v21 ]
the master systems (3.1) and (3.2) will achieve combination synchronization with the slave system (3.4). Corollary 4.1.3. For β1 = β2 = β3 = β4 = 0, γ1 = γ2 = γ3 = γ4 = 1, δ1 = δ2 = δ3 = δ4 = 0 and u1 = α1 (a(u12 − u11 ) − mu14 ) − (a(v12 − v11 ) − mv14 ) + α1 u11 − v11 + α2 u12 − v12 + α3 u13 − v13 + α4 u14 − v14 u2 = α2 (−u11 u13 − cu12 ) − (−v11 v13 − cv12 ) + α2 u12 − v12 − α1 u11 − v11 u3 = α3 (−b + u11 u12 ) − (−b + v11 v12 ) + α3 u13 − v13 − α1 u11 − v11 u4 = α4 (−ku11 ) − (−kv11 ) +α u −v −α u −v
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controllers :
4 14
14
1 11
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the master system (3.1) will attain the projective synchronization with the slave systems (3.4). 140
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(4.12)
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4 24
(4.13)
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+ β2 u22 − v12 + β1 u21 − v11 u3 = β3 (−b + u21 u22 ) − (−b + v11 v12 ) + β3 u23 − v13 + β1 u21 − v11 u4 = β4 (−ku21 ) − (−kv11 ) +β u −v +β u −v
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Corollary 4.1.4. For α1 = α2 = α3 = α4 = 0, γ1 = γ2 = γ3 = γ4 = 1, δ1 = δ2 = δ3 = δ4 = 0 and controllers : u1 = β1 (a(u22 − u21 ) − mu24 ) − (a(v12 − v11 ) − mv14 ) + β1 u21 − v11 + β2 u22 − v12 + β3 u23 − v13 + β4 u24 − v14 u2 = β2 (−u21 u23 − cu22 ) − (−v11 v13 − cv12 )
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1 21
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the master system (3.2) will achieve the projective synchronization with the slave system (3.3). Corollary 4.1.5. For β1 = β2 = β3 = β4 = 0, γ1 = γ2 = γ3 = γ4 = 0, δ1 = δ2 = δ3 = δ4 = 1 and u∗1 = α1 (a(u12 − u11 ) − mu14 ) − (a(v22 − v21 ) − mv24 ) + α1 u11 − v21 + α2 u12 − v22 + α3 u13 − v23 + α4 u14 − v24 ∗ u2 = α2 (−u11 u13 − cu12 ) − (−v21 v23 − cv22 ) + α2 u12 − v22 − α1 u11 − v21 u∗3 = α3 (−b + u11 u12 ) − (−b + v21 v22 ) + α3 u13 − v23 − α1 u11 − v21 u∗ = α4 (−ku11 ) − (−kv21 ) 4 +α u −v −α u −v
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controllers :
4 14
24
1 11
(4.14)
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the master system (3.1) will achieve the projective synchronization with the slave system (3.4). Corollary 4.1.6. (iv) For α1 = α2 = α3 = α4 = 0, γ1 = γ2 = γ3 = γ4 = 0, δ1 = δ2 = δ3 = δ4 = 1 and
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controllers :
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u∗1 = β1 (a(u22 − u21 ) − mu24 ) − (a(v22 − v21 ) − mv24 ) + β1 u21 − v21 + β2 u22 − v22 + β3 u23 − v23 +β u −v 4 24 24 u∗2 = β2 (−u21 u23 − cu22 ) − (−v21 v23 − cv22 ) + β2 u22 − v22 + β1 u21 − v21 u∗3 = β3 (−b + u21 u22 ) − (−b + v21 v22 ) + β3 u23 − v23 + β1 u21 − v21 u∗4 = β4 (−ku21 ) − (−kv21 ) +β u −v +β u −v 24
1 21
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the master system (3.2) will achieve the projective synchronization with the slave system (3.4). 4.1. Numerical simulations 145
Numerical simulations are carried out in Matlab to verify the efficiency of the designed controllers. The pa-
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rameters values are chosen so that system shows chaotic behavior in the absence of controllers as shown in figure (2) . The initial conditions of the master systems and slave system are chosen as (u11 , u12 , u13 , u14 ) = (10, −
0.2, 0.05, − 3), (u21 , u22 , u23 , u24 ) = (0.2, 0.1, 0.75, − 2), (v11 , v12 , v13 , v14 ) = (5, 2, − 1, 3), (v21 , v22 , v23 , v24 )
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= (3, 1, − 2, 1).
Suppose that α1 = α2 = α3 = α4 = 1, β1 = β2 = β3 = β4 = 1, γ1 = γ2 = γ3 = γ4 = 1 and δ1 = δ2 = δ3 = δ4 = 1. The corresponding initial conditions for error system is obtained as (e1 , e2 , e3 , e4 ) = (2.2, − 3.1, 3.8, − 9). The
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convergence of error state variables in figure (8) shows that the combination-combination synchronization among systems (3.1), (3.2), (3.3) and (3.4) is achieved when controllers are activated at t > 0. Figure (9) shows the trajectories of synchronization among systems (3.1), (3.2), (3.3) and (3.4).
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master and slave state variables when controller are activated at t > 0, this again confirms combination-combination
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4 e1 e2 e3 e4
2
e1,e2,e3,e4
0
−2
−4
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−6
−8
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10 Time(sec)
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u11 + u21 v11 + v21 20
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u12+u22,v12+v22
10 u11+u21,v11+v21
u12 + u22 v12 + v22
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0
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Figure 8: Error dynamics among the drives and the response system with controllers deactivated for t > 0
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10 Time(sec)
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15 u14 + u24 v14 + v24
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u13 + u23 v13 + v23
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u13+u23,v13+v23
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u14+u24,v14+v24
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Figure 9: Dynamics of the drives and the response state variables with controllers are activated for t > 0
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5. Conclusion A 4D novel hyperchaotic system has been investigated which generates two scroll attractors. Some fundamental dynamical properties such as Lyapunov exponent, bifurcation diagram, Poincar´e section, Kaplan-Yorke dimension, equilibria and phase portraits are analyzed both theoretically and numerically. It is observed that the systems shows 160
different chaotic behavior as the parameter values varies. Furthermore, combination-combination synchronization is achieved by considering four identical 4D novel hyperchaotic systems. Numerical simulations are performed to justify
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the validity of the theoretical results discussed. Computational and analytical results are in excellent agrement. We have shown from the theoretical analysis that various controllers which are suitable for different type of synchronization scheme can be obtained from the general combination-combination synchronization.
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Chaotic Analysis and Combination-Combination Synchronization of a Novel Hyperchaotic System without any Equilibria Ayub Khan, Shikha Singh PII: DOI: Reference:
S0577-9073(17)30783-9 10.1016/j.cjph.2017.12.023 CJPH 426
To appear in:
Chinese Journal of Physics
Received date: Revised date: Accepted date:
24 June 2017 19 December 2017 23 December 2017
Please cite this article as: Ayub Khan, Shikha Singh, Chaotic Analysis and Combination-Combination Synchronization of a Novel Hyperchaotic System without any Equilibria, Chinese Journal of Physics (2017), doi: 10.1016/j.cjph.2017.12.023
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Highlights • A novel hyperchaotic system with no equilibrium point is constructed. • It is observed that as the parameter values varies the system displays different behavior. • Various properties of novel hyperchaotic system are investigated.
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• Using this novel system combination-combination synchronization is performed.
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Chaotic Analysis and Combination-Combination Synchronization of a Novel Hyperchaotic System without any Equilibria Ayub Khana , Shikhab,∗ a Department
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of Mathematics, Jamia Millia Islamia, New Delhi - 110025, India. Email : [email protected] b Department of Mathematics, Jamia Millia Islamia, New Delhi - 110025, India. Email : [email protected]
Abstract
This manuscript examines a novel 4D continuous autonomous hyperchaotic system with two scroll attractor. This system does not produce any equilibrium point so, it generates hidden hyperchaotic attractor. The Lyapunov exponent, bi-
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furcation diagram, Poincar´e section, Kaplan-Yorke dimension and phase portraits are given to justify the hyperchaotic nature of the system. The novel system displays hyperchaotic orbit, chaotic orbit, periodic orbit, quasi-periodic orbit as the parameters values varies. Furthermore, the combination-combination synchronization is performed by considering four identical 4D novel hyperchaotic systems.
Keywords: Chaotic system, Lyapunov exponent, bifurcation diagram, Poincar´e section, Kaplan-Yorke dimension, 2010 MSC: 34H10, 34H05, 34D05
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1. Introduction
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combination-combination synchronization
In the most recent decades researchers from all fields of natural sciences have studied phenomena that include nonlinear systems exhibiting chaotic behavior [1], [2, 3, 4, 5, 6]. This is because of the way that nonlinear systems
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demonstrate rich dynamics and have sensitivity on initial conditions. Chaotic systems are third order or higher order nonlinear differential equations with at least one positive lyapunov exponent. Hyperchaos is characterized as a chaotic
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system with more than one positive Lyapunov exponent [7], this infer that its dynamics are expended in several different directions simultaneously. Thus, hyperchaotic systems have more complex dynamical behaviors than ordinary chaotic systems. In 1976 [8] Rossler led imperative work that revived the enthusiasm in 3D autonomous chaotic sys-
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tem. Since Lorenz proposed a astounding chaotic system for portraying atmospheric convention in 1963 [9], hordes of consideration have been committed to the examination to the autonomous system. As of late, there has been increasing enthusiasm in creating chaos particularly since Chen and Ueta [2] found a new chaotic system, called the Chen system, ✩ Fully
documented templates are available in the elsarticle package on CTAN. author Email address: [email protected] (Shikha )
∗ Corresponding
Preprint submitted to Chinese Journal of Physics
December 29, 2017
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by adding a simple state feedback to the second equation of the Lorenz system, which is not topologically equivalent to the Lorenz system. After that, Lu and Chen [3] further developed a new chaotic system, bearing the name of the Lu system, which represents the transition between the Lorenz system and Chen system. From then on, an exceptionally 15
rich family of the so-called unified system or Lorenz system family [4] was presented as a association of the Lorenz, Chen, and Lu systems. In chaos theory, the equilibrium of a dynamical system is significant for understanding its complex dynamics. A
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hyperchaotic system producing no equilibrium points has been a important point of discussion in non-linear dynamical system because such systems displays unique properties [10, 11, 12, 13, 14]. It is noted that to verify the presence 20
of chaos in systems producing no equilibrium points Shilnikov’s method fails because such systems have neither homoclinic nor heteroclinic orbits. The system producing no equilibrium point belongs to a class of hidden attractor. Hidden attractors are important in engineering applications because they allow unexpected and potential disastrous responses to perturbations in a structure like a bridge or an airplane wing. Very few systems with no equilibrium
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points have been investigated till now. Analysis of the dynamics of hyperchaotic systems received a great deal of interest in the recent past due to its applications in secure communication [6], information processing [15], biological systems [16], chemical reactions [17], neural network [18] etc. So, more and more chaotic or hyper chaotic systems showing wide dynamical behavior were found [19, 20, 21, 22, 23, 24, 25, 26, 27]. For example Wu and Li [19] presented a new chaotic system and studied its basic dynamical properties and then synchronize it using different control techniques. Kingni et. al. [20] introduced a novel chaotic system with a circular equilibrium. Analysis, circuit simulation and its fractional order form is also discussed. S.Vaidyanathan [10] announced a novel highly hyperchaotic
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system with no equilibrium point and analysed its dynamical behaviour. Furthermore synchronization by adaptive
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control is also achieved. X.Wang et. al. [21] constructed few chaotic systems having no equilibrium, one stable equilibria, two equilibria, three equilibria and any number of equilibria. V.T.Pham et. al [22] introduces a chaotic system with different shapes of equilibria like chaotic attractor with circular equilibrium, elliptical equilibrium and square shaped equilibrium. Deng et. al. [23] introduces a hyper-chaotic attractor formed by adding one additional
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state variable and modifying nonlinear terms and parameters of Chen system having only two parameters and the
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system shows hyper-chaotic behavior for the wide range of one parameter establishing the more chaotic nature of the system. Chen et. [24] al. constructed a hyper-chaotic system by adding a feedback controller to the classical Lorenz system which shows abundant dynamics having a curve of equilibria. Wang et. al. [25] describes a 4D hyper-chaotic system and analyse it by studying its local bifurcation dynamics and by computing its ultimate bound. In 2017, Vishal,
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K., and Saurabh K. Agrawal. [26]have studied the dynamics of a novel complex chaotic system with fractional order derivative and found the existence of chaos. The novel complex system is simulated for integer as well as fractional orders which shows some unusual phenomena. In 2017, Gholamin, P., and AH Refahi Sheikhani. [27] recomends a new three-dimensional autonomous chaotic system with six terms including three multipliers, which is different
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from the Lorenz system and other existing systems and the dynamical properties are studied. Motivated by the above
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researches and based on the 3D chaotic system introduced by Q. Yang et al. [28], a novel 4D continuous autonomous hyperchaotic system with no equilibrium point by adding a feedback controller to the 3D chaotic system is constructed. Further, some basic dynamical properties including Lyapunov exponent, bifurcations, Poincar´e section , Kaplan-Yorke dimension and phase portraits are illustrated by both theoretical analysis and computer simulations. 50
Due to the vast practical applications of chaotic dynamical systems in fields stated above, numerous investigations have been done theoretically and experimentally on controlling chaos and synchronization. In 1990, Pecora
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and Carroll [29] gave the synchronization of chaotic systems using the concept of master and slave system. Also, in 1990 Ott et al. [30] introduced the OGY method for controlling chaos. Looking for better strategies for chaos control and synchronization different types of methods have been developed for controlling chaos and synchroniza55
tion of non identical and identical systems for instance linear feedback [31], optimal control [32], adaptive control [33, 34, 35, 36, 37, 38, 39, 40], active control [41, 42, 43], active sliding control [44], backstepping control [45, 46, 47], hybrid control [48] etc. As an aftereffect of quickly developing enthusiasm in chaos control and syn-
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chronization various synchronization types and schemes have been proposed and reported, for instance generalized synchronization [49], projective synchronization [50], modied projective synchronization [51], function projective 60
synchronization [52], modied-function projective synchronization [53], hybrid synchronization [54] and hybrid function projective synchronization [55]. It is noted that most of the researches mainly focused on the previous master-slave synchronization scheme within one master and one response system. Only a few research papers have been published on combination-combination synchronization where four chaotic systems were taken into account [56, 57, 58, 59].
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Combination-combination synchronization scheme used in this paper is generalized in such a way that other forms of synchronization scheme can be achieved form it. As a result, combination-combination synchronization scheme
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is more flexible and applicable to the real world systems. In addition, the combination-combination synchronization also gives better insight into the complex synchronization and several pattern formations that take place in real world systems since synchronization in real world systems are complex. This manuscript is categorize as follows : In section 3 combination-combination synchronization of four 4D novel hyperchaotic system is achieved. Finally in section 4
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concluding remarks are given.
2. Formulation of 4D Novel Hyper-Chaotic System and its Fundamental Dynamical Properties In 2010 Q. Yang et al. [28] formulated a 3D autonomous chaotic Lorenz-type system which is given by u˙1 = a(u2 − u1 ) u˙2 = −u1 u3 − cu2 u˙ = −b + u u 3 1 2
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2 formulation of 4D novel hyperchaotic system and its fundamental dynamical properties are discussed. In section
(2.1)
where a, b, c ∈ R are parameters. For the parameter values a = 10, b = 100 and c = 4.7466 system has two saddle focus and unstable equilibria (±10, ± 10, − 4.7466). The eigenvalues of the Jacobian evaluated at the equilibrium points 4
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are λ1 = − 16.1965, λ2,3 = 0.7250 ± 11.0886ι. For the initial condition (u1 (0), u2 (0), u3 (0)) = (10, − 0.2, 0.75) and the parameter values (a, b, c) = (10, 100, 4.7466) , the Lyapunov exponents are (γ1 , γ2 , γ3 ) = (1.0054, − 0.0018, − 15.7308). So, the system shows chaotic attractor displayed in figure (1). The detailed description of system (2.1) is given in [28].
100 u2(t)
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0 −50 −100 −50
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u2(t)
Figure 1: Phase Portrait of the 3D chaotic Lorenz type system
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Now adding a feedback controller to system (2.1), a novel 4D hyperchaotic system is formed given by u˙1 = a(u2 − u1 ) − mu4 u˙2 = −u1 u3 − cu2 u˙3 = −b + u1 u2 u˙4 = −ku1 5
(2.2)
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where k, m ∈ R are parameters. When b 6= 0, system (2.2) has no equilibrium point. Thus, the phenomena like
pitchfork, Hopf or homoclinic bifurcations does not take place while these phenomena are common in systems with equilibria. 2.1. Chaotic Attractor of the System When the parameters a = 10, b = 25, c = −2.5, k = 1 and m = 3 and initial condition (10, −0.2, 0.05, −3) are
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chosen then the system displays two scroll chaotic attractor as shown in figure (2). Also the time series of the novel hyperchaotic attractor is shown in figure (3). The corresponding Lyapunov exponents of the novel hyperchaotic attractor are γ1 = 0.88503, γ2 = 0.048124, γ3 = −0.00028044, γ4 = −8.4325 as shown in figure (4).
The Kaplyan-Yorke dimension is defined by
j
γi |γ i=1 j+1 |
D = j+∑
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= 3.1106
j j+1 where j is the largest integer satisfying ∑i=1 γ j ≥ 0 and ∑i=1 γ j < 0. Therefore Kaplan-Yorke dimension of the chaotic
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attractor is D = 3.1106 which means that the Lyapunov dimension of the chaotic attractor is fractional.
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0 −50 −20
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10 u4(t)
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u4(t)
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Figure 2: Phase Portrait of the 4D novel hyperchaotic system
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20 0 u (t) 1
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−10
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Time(sec)
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Figure 3: Time series of the 4D novel hyperchaotic system
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1
λ = 0.88503 1 λ = 0.048124
0
2
λ = −0.00028044 3
Lyapunov Exponents
−1 −2 −3 −4 −5
−7
λ = −8.4325 4
−8 −9
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−6
0
200
400
600 Time
800
1000
1200
Figure 4: Lyapunov exponents of the 4D novel hyperchaotic system
2.2. Dissipation and existence of Chaotic Attractor
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The divergence of the novel 4D hyperchaotic system (2.2) is ∇.V =
∂ u˙1 ∂ u˙2 ∂ u˙3 ∂ u˙4 + + + ∂ u1 ∂ u2 ∂ u3 ∂ u4
= −a − c = −7.5 < 0
So, the system (2.2) is dissipative system and converges by the index rate of e−7.5t .
It means that a volume element V0 is contracted into V0 e−7.5t at time t. That is, each volume containing the system
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orbit shrinks to zero as t → 0, at an exponential rate ∇V , which is independent of u1 , u2 , u3 , u4 . Consequently, all
system orbits will ultimately be confined to a specific subset of zero volume and the asymptotic motion settles onto an attractor. Then the existence of attractor is proved. 2.3. Symmetry and Invariance
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The novel hyperchaotic system (2.2) is invariant under the transformation (u1 , u2 , u3 , u4 ) → (−u1 , − u2 , u3 , −
u4 ) this means the system (2.2) is symmetric about u3 -axis.
As an important analysis technique, the Poincar´e map reflects the periodic, quasi-periodic, chaotic and hyperchaotic behavior of the system. When a = 10, b = 25, c = −2.5, k = 1 and m = 3 one may take u2 = 0 and u3 = 0 as
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2.4. Poincar´e map, Bifurcation diagram and Impact of system parameters
the crossing sections respectively as shown in figure (5).
The bifurcation diagram of |u1 | with respect to parameters c is shown in figure (6) which shows abundant and complex dynamical behaviors.
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10 10
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-10 -10
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Figure 5: Poincar´e map on the crossing section u2 =0 and u3 = 0 respectively
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Figure 6: Bifurcation diagram of |u1 | versus c
Denoting the four Lyapunov exponents of system (2.2) by γ1 , γ2 , γ3 , γ4 which satisfy γ1 > γ2 > γ3 > γ4 . The
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behavior of the system depends on the parameter values. It is also depicted in bifurcation diagram. For different values of parameters system exhibits hyperchaotic attractor, chaotic attractor, quasi-periodic orbit and periodic orbit. Chaotic analysis of the system (2.2) is carried out by varying one parameter value and fixing other parameters. Vary parameter
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a and fix other parameters. For the parameter values (12,25,-2.5,1,3), (10,27,-2.5,1,3), (10,25,-1,1,3), (10,25,-2.5,2,3) and (10,25,-2.5,1,4) two largest Lyapunov exponents γ1 and γ2 are positive, which indicate that the system (2.2) shows the hyperchaotic behavior. For the parameters values (7,25,-2.5,1,3), (10,140,-2.5,1,3), (10,25,-4.9,1,3) and (10,25,2.5,1,-12) the largest Lyapunov exponent γ1 is positive, which indicate the chaotic behavior of the system (2.2). For
the parameter values (22,25,-2.5,1,3), (10,0,-2.5,1,3), (10,25,-2.5,-1,3) and (10,25,-2.5,1,-28) the largest Lyapunov exponent γ1 tends to zero, which indicate that the orbit of the system (2.2) is periodic in nature. For the parameter 110
values (10,-12,-2.5,1,3), (10,25,-2.5,-150,3) and (10,25,-2.5,1,-600) two largest Lyapunov exponents γ1 tends to zero, 10
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(γ1 , γ2 , γ3 , γ4 )
Lyapunov Dimension
Orbital mode
(12,25,-2.5,1,3)
(0.6167,0.0601,-0.0150,-10.1605)
3.0651
hyperchaotic
(7,25,-2.5,1,3)
(0.8662,0.0010,-0.0195,-5.3476)
3.1585
chaotic
(22,25,-2.5,1,3)
(0.0081,-0.0171,-1.8235,-17.5757)
2.9743
periodic
(10,27,-2.5,1,3)
(0.9775,0.0306,-0.0044,-8.5032)
3.1180
hyperchaotic
(10,140,-2.5,1,3)
(0.9287,0.000949,-0.08895,-8.3355)
3.1008
chaotic
(10,0,-2.5,1,3)
(-0.00264,-0.1054,-3.6626,-3.7292)
1.9888
periodic
(10,-3,-2.5,1,3)
(-0.0059,-0.0037,-3.7307,-3.7663)
2.0068
quasi-periodic
(10,25,-1,1,3)
(0.6815,0.0558,3.82×10− 6,-9.7364)
3.0757
hyperchaotic
(10,25,-4.9,1,3)
(1.0706,0.00601,-0.0896,-6.0869)
3.1621
chaotic
(10,25,-2.5,2,3)
(0.8077,0.1006,-0.00643,-8.4015)
3.1073
hyperchaotic
(10,25,-2.5,-1,3)
(-0.0048,-0.2742,-3.4966,-3.7291)
2.0054
periodic
(10,25,-2.5,-150,3)
(0.00087,-0.0140,-3.7439,-3.7778)
1.9874
quasi-periodic
(10,25,-2.5,1,4)
(0.8174,0.0678,-0.0039,-8.381)
3.1051
hyperchaotic
(10,25,-2.5,1,-12)
(0.0951,-0.0026,-3.6588,-3.7515)
2.0493
chaotic
(10,25,-2.5,1,-28)
(-0.00042,-0.0554,-3.7071,-3.7482)
1.9960
periodic
(10,25,-2.5,1,-600)
(-0.0041,-0.0064,-3.7625,-3.7653)
1.9979
quasi-periodic
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Values of parameters (a,b,c,k,m)
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Table 1: Lyapunov exponents, Lyapunov dimension and orbital mode for the different values of parameters
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which indicate that the orbit of the system (2.2) is quasi-periodic in nature. In table (1) all the Lyapunov exponents, Lyapunov dimension and orbital mode for the different values of parameters are given. The phase portraits for the
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parameter values listed in table (1), displaying the fixed orbit, periodic orbit and chaotic orbit is shown in figure (7)
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Figure 7: Phase portraits showing hyperchaotic, chaotic, periodic and quasi-periodic behavior in u1 − u4 plane
3. Methodology of combination-combination synchronization
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Methodology for combination-combination synchronization with two master systems and two slave systems is
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described in this section. The two master systems are described as follows: u˙1 = f1 (u1 )
(3.1)
u˙2 = f2 (u2 )
(3.2)
while the two slave systems are described as follows: v˙1 = g1 (v1 )
(3.3)
v˙2 = g2 (v2 )
(3.4)
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where u1 = (u11 , u12 , ..., u1n ), u2 = (u21 , u22 , ..., u2n ), u1 = (v11 , v12 , ..., v1n ) and v1 = (v21 , v22 , ..., v2n ) are the state vectors of the master and slave systems respectively and f1 , f2 , g1 , g2 : Rn → Rn are the controllers added to the slave system which are to be determined.
Definition 3.1. If there exists four constant matrices A, B, C, D ∈ Rn and C 6= 0 or D 6= 0 such that (3.5)
lim kAu1 + Bu2 −Cv1 − Dv2 k = 0
t→∞
system (3.3) and (3.4). Here k.k represents the matrix norm. 120
The following observation arises :
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then the master systems (3.1) and (3.2) are said to achieve combination-combination synchronization with the slave
1. The constant matrices A, B, C, D are called the scaling matrices. In addition A, B, C, D can be extended to functional matrices of state variables u1 , u2 , v1 and v2 .
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2. If C = 0 or D = 0, then combination-combination synchronization problem becomes the combination synchronization problem.
3. If A = 0, C = I, D = 0 or A = C = 0, D = I or B = 0, C = I, D = 0 or B = C = 0, D = I, then the combination synchronization becomes projective synchronization, where I is the n × n matrix.
4. If A = B = C = 0 or A = B = D = 0, then the combination synchronization becomes the chaos control problem.
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4. Combination-combination synchronization among four novel identical hyper-chaotic systems
described above.
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The two master systems are
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To perform the combination-combination synchronization we choose the four identical novel hyperchaotic system
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u˙11 = a(u12 − u11 ) − mu14 u˙12 = −u11 u13 − cu12 u˙13 = −b + u11 u12 u˙14 = −ku11 u˙21 = a(u22 − u21 ) − mu24 u˙22 = −u21 u23 − cu22 u˙23 = −b + u21 u22 u˙24 = −ku21
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(4.1)
(4.2)
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while the two slave systems are given as v˙11 = a(v12 − v11 ) − mv14 + u1 v˙12 = −v11 v13 − cv12 + u2
v˙21 = a(v22 − v21 ) − mv24 + u∗1 v˙22 = −v21 v23 − cv22 + u∗2 v˙23 = −b + v21 v22 + u∗3 v˙24 = −kv21 + u∗4
(4.3)
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v˙13 = −b + v11 v12 + u3 v˙14 = −kv11 + u4
(4.4)
where u1 , u2 , u3 , u4 , u∗1 , u∗2 , u∗3 and u∗4 are the controllers to be determined. Let us choose A = diag(α1 , α2 , α3 , α4 ),
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B = (β1 , β2 , β3 , β4 ), C = (γ1 , γ2 , γ3 , γ4 ) and D = (δ1 , δ2 , δ3 , δ4 ) in this synchronization scheme. The error system is obtained as follows: e1 = α1 u11 + β1 u21 − γ1 v11 − δ1 v21 e2 = α2 u12 + β2 u22 − γ2 v12 − δ2 v22
(4.5)
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e3 = α3 u13 + β3 u23 − γ3 v13 − δ3 v23 e4 = α4 u14 + β4 u24 − γ4 v14 − δ4 v24
Denote
U3 = γ3 u3 + δ3 u∗3 U4 = γ4 u4 + δ4 u∗4
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U1 = γ1 u1 + δ1 u∗1 U2 = γ2 u2 + δ2 u∗2
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Theorem 4.1. For the controllers: U1 = α1 (a(u12 − u11 ) − mu14 ) + β1 (a(u22 − u21 ) − mu24 ) − γ1 (a(v12 − v11 ) − mv14 ) − δ1 (a(v22 − v21 ) − mv24 ) + α1 u11 + β1 u21 − γ1 v11 − δ1 v21 + α2 u12 + β2 u22 − γ2 v12 − δ2 v22 + α3 u13 + β3 u23 − γ3 v13 − δ3 v23 + α4 u14 + β4 u24 − γ4 v14 − δ4 v24 U2 = α2 (−u11 u13 − cu12 ) + β2 (−u21 u23 − cu22 ) − γ2 (−v11 v13 − cv12 )
4 14
4 24
4 14
4 24
(4.7)
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− δ2 (−v21 v23 − cv22 ) + α2 u12 + β2 u22 − γ2 v12 − δ2 v22 − α1 u11 + β1 u21 − γ1 v11 − δ1 v21 U3 = α3 (−b + u11 u12 ) + β3 (−b + u21 u22 ) − γ3 (−b + v11 v12 ) − δ3 (−b + v21 v22 ) + α3 u13 + β3 u23 − γ3 v13 − δ3 v23 − α1 u11 + β1 u21 − γ1 v11 − δ1 v21 U = α4 (−ku11 ) + β4 (−ku21 ) − γ4 (−kv11 ) − δ4 (−kv21 ) 4 +α u +β u −γ v −δ v −α u +β u −γ v −δ v 1 11
1 21
1 11
1 21
the master systems (3.1) and (3.2) will attain the combination-combination synchronization with the slave systems (3.3) and (3.4).
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Proof The error dynamical system using (4.5) is obtained as follows:
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∗ e˙1 = α1 (a(u12 − u11 ) − mu14 ) + β1 (a(u22 − u21 ) − mu24 ) − γ1 (a(v12 − v11 ) − mv14 + u1 ) − δ1 (a(v22 − v21 ) − mv24 + u1 ) e˙2 = α2 (−u11 u13 − cu12 ) + β2 (−u21 u23 − cu22 ) − γ2 (−v11 v13 − cv12 + u2 ) − δ2 (−v21 v23 − cv22 + u∗2 )
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e˙3 = α3 (−b + u11 u12 ) + β3 (−b + u21 u22 ) − γ3 (−b + v11 v12 + u3 ) − δ3 (−b + v21 v22 + u∗3 ) e˙4 = α4 (−ku11 ) + β4 (−ku21 ) − γ4 (−kv11 + u4 ) − δ4 (−kv21 + u∗4 )
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Choosing a candidate Lyapunov function as follows:
1 V (t) = (e21 + e22 + e23 + e24 ) 2
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(4.8)
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Obviously, V (t) > 0. The time derivative of V(t) along the trajectories of the error system (4.8) is V˙ (t) = e1 e˙1 + e2 e˙2 + e3 e˙3 + e4 e˙4 = e1 (α1 (a(u12 − u11 ) − mu14 ) + β1 (a(u22 − u21 ) − mu24 ) − γ1 (a(v12 − v11 ) − mv14 + u1 ) − δ1 (a(v22 − v21 ) − mv24 + u∗1 ) + e2 (α2 (−u11 u13 − cu12 ) + β2 (−u21 u23 − cu22 ) − γ2 (−v11 v13 − cv12 + u2 )
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− δ2 (−v21 v23 − cv22 + u∗2 ) + e3 (α3 (−b + u11 u12 ) + β3 (−b + u21 u22 ) − γ3
(4.9)
(−b + v11 v12 + u3 ) − δ3 (−b + v21 v22 + u∗3 ) + e4 (α4 (−ku11 ) + β4 (−ku21 ) − γ4 (−kv11 + u4 ) − δ4 (−kv21 + u∗4 )
= e1 (−e1 − e2 − e3 − e4 ) + e2 (−e2 + e1 ) + e3 (−e3 + e1 ) + e4 (−e4 + e1 ) = −e21 − e22 − e23 − e24
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where k is the positive constant which represents feedback gain.
Since V˙ ≤ 0, according to Lyapunov theorem we know ei → 0 (i=1,2) as t → ∞ which means that the required synchronization is achieved.
The following Corollaries can easily be obtained from Theorem 1, the proofs of these Corollaries are similar to Theorem 1. So, the proofs are omitted.
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Corollary 4.1.1. For δ1 = δ2 = δ3 = δ4 = 0 and controllers : 1 u1 = [α1 (a(u12 − u11 ) − mu14 ) + β1 (a(u22 − u21 ) − mu24 ) − γ1 (a(v12 − v11 ) − mv14 ) γ 1 − δ1 (a(v22 − v21 ) − mv24 ) + α1 u11 + β1 u21 − γ1 v11 + α2 u12 + β2 u22 − γ2 v12 + α3 u13 + β3 u23 − γ3 v13 + α4 u14 + β4 u24 − γ4 v14 ] 1 u2 = [α2 (−u11 u13 − cu12 ) + β2 (−u21 u23 − cu22 ) − γ2 (−v11 v13 − cv12 ) γ 2 + α2 u12 + β2 u22 − γ2 v12 − α1 u11 + β1 u21 − γ1 v11 ] 1 u3 = [α3 (−b + u11 u12 ) + β3 (−b + u21 u22 ) − γ3 (−b + v11 v12 ) γ 3 + α3 u13 + β3 u23 − γ3 v13 − α1 u11 + β1 u21 − γ1 v11 ] 1 u4 = [α4 (−ku11 ) + β4 (−ku21 ) − γ4 (−kv11 ) γ 4 +α u +β u −γ v −α u +β u −γ v ]
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4 14
4 24
4 14
1 11
1 21
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the master systems (3.1) and (3.2) will achieve combination synchronization with the slave system (3.3).
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Corollary 4.1.2. For γ1 = γ2 = γ3 = γ4 = 0 and controllers :
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1 u∗1 = [α1 (a(u12 − u11 ) − mu14 ) + β1 (a(u22 − u21 ) − mu24 ) δ 1 − δ1 (a(v22 − v21 ) − mv24 ) + α1 u11 + β1 u21 − δ1 v21 + α2 u12 + β2 u22 − δ2 v22 + α3 u13 + β3 u23 − δ3 v23 + α4 u14 + β4 u24 − δ4 v24 ] u∗2 = 1 [α2 (−u11 u13 − cu12 ) + β2 (−u21 u23 − cu22 ) δ2 − δ2 (−v21 v23 − cv22 ) + α2 u12 + β2 u22 − δ2 v22 − α1 u11 + β1 u21 − δ1 v21 ] 1 u∗3 = [α3 (−b + u11 u12 ) + β3 (−b + u21 u22 ) − δ3 (−b + v21 v22 ) δ 3 + α3 u13 + β3 u23 − δ3 v23 − α1 u11 + β1 u21 − δ1 v21 ] 1 u∗4 = [α4 (−ku11 ) + β4 (−ku21 ) − δ4 (−kv21 ) δ 4 + α4 u14 + β4 u24 − δ4 v24 − α1 u11 + β1 u21 − δ1 v21 ]
the master systems (3.1) and (3.2) will achieve combination synchronization with the slave system (3.4). Corollary 4.1.3. For β1 = β2 = β3 = β4 = 0, γ1 = γ2 = γ3 = γ4 = 1, δ1 = δ2 = δ3 = δ4 = 0 and u1 = α1 (a(u12 − u11 ) − mu14 ) − (a(v12 − v11 ) − mv14 ) + α1 u11 − v11 + α2 u12 − v12 + α3 u13 − v13 + α4 u14 − v14 u2 = α2 (−u11 u13 − cu12 ) − (−v11 v13 − cv12 ) + α2 u12 − v12 − α1 u11 − v11 u3 = α3 (−b + u11 u12 ) − (−b + v11 v12 ) + α3 u13 − v13 − α1 u11 − v11 u4 = α4 (−ku11 ) − (−kv11 ) +α u −v −α u −v
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controllers :
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1 11
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the master system (3.1) will attain the projective synchronization with the slave systems (3.4). 140
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4 24
(4.13)
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+ β2 u22 − v12 + β1 u21 − v11 u3 = β3 (−b + u21 u22 ) − (−b + v11 v12 ) + β3 u23 − v13 + β1 u21 − v11 u4 = β4 (−ku21 ) − (−kv11 ) +β u −v +β u −v
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Corollary 4.1.4. For α1 = α2 = α3 = α4 = 0, γ1 = γ2 = γ3 = γ4 = 1, δ1 = δ2 = δ3 = δ4 = 0 and controllers : u1 = β1 (a(u22 − u21 ) − mu24 ) − (a(v12 − v11 ) − mv14 ) + β1 u21 − v11 + β2 u22 − v12 + β3 u23 − v13 + β4 u24 − v14 u2 = β2 (−u21 u23 − cu22 ) − (−v11 v13 − cv12 )
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1 21
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the master system (3.2) will achieve the projective synchronization with the slave system (3.3). Corollary 4.1.5. For β1 = β2 = β3 = β4 = 0, γ1 = γ2 = γ3 = γ4 = 0, δ1 = δ2 = δ3 = δ4 = 1 and u∗1 = α1 (a(u12 − u11 ) − mu14 ) − (a(v22 − v21 ) − mv24 ) + α1 u11 − v21 + α2 u12 − v22 + α3 u13 − v23 + α4 u14 − v24 ∗ u2 = α2 (−u11 u13 − cu12 ) − (−v21 v23 − cv22 ) + α2 u12 − v22 − α1 u11 − v21 u∗3 = α3 (−b + u11 u12 ) − (−b + v21 v22 ) + α3 u13 − v23 − α1 u11 − v21 u∗ = α4 (−ku11 ) − (−kv21 ) 4 +α u −v −α u −v
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controllers :
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1 11
(4.14)
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the master system (3.1) will achieve the projective synchronization with the slave system (3.4). Corollary 4.1.6. (iv) For α1 = α2 = α3 = α4 = 0, γ1 = γ2 = γ3 = γ4 = 0, δ1 = δ2 = δ3 = δ4 = 1 and
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controllers :
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u∗1 = β1 (a(u22 − u21 ) − mu24 ) − (a(v22 − v21 ) − mv24 ) + β1 u21 − v21 + β2 u22 − v22 + β3 u23 − v23 +β u −v 4 24 24 u∗2 = β2 (−u21 u23 − cu22 ) − (−v21 v23 − cv22 ) + β2 u22 − v22 + β1 u21 − v21 u∗3 = β3 (−b + u21 u22 ) − (−b + v21 v22 ) + β3 u23 − v23 + β1 u21 − v21 u∗4 = β4 (−ku21 ) − (−kv21 ) +β u −v +β u −v 24
1 21
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the master system (3.2) will achieve the projective synchronization with the slave system (3.4). 4.1. Numerical simulations 145
Numerical simulations are carried out in Matlab to verify the efficiency of the designed controllers. The pa-
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rameters values are chosen so that system shows chaotic behavior in the absence of controllers as shown in figure (2) . The initial conditions of the master systems and slave system are chosen as (u11 , u12 , u13 , u14 ) = (10, −
0.2, 0.05, − 3), (u21 , u22 , u23 , u24 ) = (0.2, 0.1, 0.75, − 2), (v11 , v12 , v13 , v14 ) = (5, 2, − 1, 3), (v21 , v22 , v23 , v24 )
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= (3, 1, − 2, 1).
Suppose that α1 = α2 = α3 = α4 = 1, β1 = β2 = β3 = β4 = 1, γ1 = γ2 = γ3 = γ4 = 1 and δ1 = δ2 = δ3 = δ4 = 1. The corresponding initial conditions for error system is obtained as (e1 , e2 , e3 , e4 ) = (2.2, − 3.1, 3.8, − 9). The
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convergence of error state variables in figure (8) shows that the combination-combination synchronization among systems (3.1), (3.2), (3.3) and (3.4) is achieved when controllers are activated at t > 0. Figure (9) shows the trajectories of synchronization among systems (3.1), (3.2), (3.3) and (3.4).
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master and slave state variables when controller are activated at t > 0, this again confirms combination-combination
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4 e1 e2 e3 e4
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u11 + u21 v11 + v21 20
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Figure 8: Error dynamics among the drives and the response system with controllers deactivated for t > 0
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15 u14 + u24 v14 + v24
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u13 + u23 v13 + v23
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Figure 9: Dynamics of the drives and the response state variables with controllers are activated for t > 0
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5. Conclusion A 4D novel hyperchaotic system has been investigated which generates two scroll attractors. Some fundamental dynamical properties such as Lyapunov exponent, bifurcation diagram, Poincar´e section, Kaplan-Yorke dimension, equilibria and phase portraits are analyzed both theoretically and numerically. It is observed that the systems shows 160
different chaotic behavior as the parameter values varies. Furthermore, combination-combination synchronization is achieved by considering four identical 4D novel hyperchaotic systems. Numerical simulations are performed to justify
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the validity of the theoretical results discussed. Computational and analytical results are in excellent agrement. We have shown from the theoretical analysis that various controllers which are suitable for different type of synchronization scheme can be obtained from the general combination-combination synchronization.
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