This paper presents a finite-time adaptive synchronization strategy for a class of new hyperchaotic systems with unknown slave system’s parameters. Based on the finite-time stability theory, an adaptive control law is derived to make the states of the new hyperchaotic systems synchronized in finite-time. Numerical simulations are presented to show the effectiveness of the proposed finite time synchronization scheme.
1. Introduction
Chaos synchronization has attracted increasing attention since the pioneering work of Pecora and Carroll [1] for its potential applications in secure communications, biological systems, chemical reactions, biological networks, and so on. Different notations of synchronization have been proposed, such as complete synchronization [2], generalized synchronization [3, 4], phase synchronization [5, 6], lag synchronization [7, 8], antisynchronization [9, 10], and projective synchronization [11, 12]. The idea of synchronization is to use the output of the master system to control the slave system so that the output of the slave system follows the output of the master system asymptotically. A wide variety of synchronization approaches have been developed such as impulsive control [13], feedback control [14], active control [15], adaptive control [16, 17], sliding mode control [18], model predictive control [19–22], and impulsive control [23, 24] and others [25, 26]. In the last thirty years, as hyperchaos has more than one positive Lyapunov exponent and has more complex dynamical behavior than chaos, many researches have focused their attention on the synchronization of hyperchaotic systems [27–31].
However, in real-world application, it is usually expected that two systems can synchronize as quickly as possible and the finite-time control is an efficient technique [32–37]. Furthermore, the finite-time control techniques have demonstrated better robustness and disturbance rejection properties [38]. However, most of the results are derived based on the hypothesis that the system’s parameters are precisely known. But in practice, most of the system’s parameters cannot be exactly known in advance. The designed synchronization will be destroyed with the effects of these uncertainties. To the best of our knowledge, there is no work on the problem of finite-time adaptive synchronization of hyperchaotic systems with uncertain parameters. Motivated by the above discussions, in this paper, we are concerned with the finite-time adaptive synchronization for hyperchaotic systems. Via adaptive control method, finite-time adaptive synchronization between two identical hyperchaotic systems with unknown parameters is achieved and we prove that the suggested approach can realize finite-time synchronization. Simulation results show the effectiveness of the proposed method.
2. The Dadras System and Lemmas
Recently, a new 4D dynamical system is proposed [39], which can generate a four-ring hyperchaotic attractor and a four-wing chaotic attractor. The system is called Dadras system in this paper and it is described by
(1)x˙=ax-yz+wy˙=xz-byz˙=xy-cz+xww˙=-y,
where [x,y,z,w]T∈R4 is the state vector and a, b, and c are positive constant parameters of the system. When a=8, b=40, c=14.9, and the initial condition is set to [10,1,10,1]T, the system has generated a four-wing hyperchaotic attractor which is shown in Figure 1.
Phase portraits of the four-wing hyperchaotic attractor for a=8, b=40, and c=14.9.
Lemma 1 (see [40]).
Assume that a continuous, positive-definite function V(t) satisfies the following differential inequality:
(2)V˙(t)≤-γVη(t)∀t≥t0,V(t0)≥0,
where γ>0, 0<η<1, are all constants. Then, for any given t0, V(t) satisfies the following inequality:
(3)V1-η(t)≤V1-η(t0)-γ(1-η)(t-t0),t0≤t≤t1,(4)V(t)≡0∀t≥t1
with t1 given by
(5)t1=t0+V1-η(t0)γ(1-η).
3. The Proposed Synchronization Method
In order to achieve master-slave synchronization of Dadras system, the master and slave systems are defined with the subscripts m and s below, respectively:
(6)x˙m=axm-ymzm+wmy˙m=xmzm-bymz˙m=xmym-czm+xmwmw˙m=-ym,(7)x˙s=asxs-yszs+ws+u1y˙s=xszs-bsys+u2z˙s=xsys-cszs+xsws+u3w˙s=-ys+u4,
where as, bs, and cs are uncertain parameters, which need to be estimated in the slave system, and u1, u2, u3, and u4 are the designed controllers to realize the two hyperchaotic systems’ finite-time synchronization.
Let
(8)e1=xs(t)-xm(t)e2=ys(t)-ym(t)e3=zs(t)-zm(t)e4=ws(t)-wm(t).
Then the error dynamical system between (2) and (3) is
(9)e˙1=asxs-yszs+ws-axm+ymzm-wm+u1e˙2=xszs-bsys-xmzm+bym+u2e˙3=xsys-cszs+xsws-xmym+czm-xmwm+u3e˙4=-ys+ym+u4.
Our goal is to design controllers ui(i=1,2,3,4) to realize finite-time synchronization between the master system (6) and the slave system (7) with the uncertain slave system parameters; that is, ∥e(t)∥=0 when t>T0, where e=[e1,e2,e3,e4]T.
Define
(10)ea=as-aeb=bs-bec=cs-c.
Then (5) can be converted to the following form:
(11)e˙1=eaxs+ae1-yme3-zse2+e4+u1e˙2=xse3+zme1-bse2-ymeb+u2e˙3=xse2+yme1-cse3-zmec+xse4+wme1+u3e˙4=-e2+u4.
In order to achieve the synchronization, we select the following control laws and the update rules for three uncertain parameters as, bs, and cs:
(12)u1=-ae1+yme3+zse2-e4-e1βu2=-xse3-zme1+bse2-e2βu3=-xse2-yme1+cse3-xse4-wme1-e3βu4=e2-e4β,(13)e˙a=-e1xs-eaβe˙b=yme2-ebβe˙c=zme3-ecβ.
Then, the following result is obtained.
Theorem 2.
For any initials, the two systems (6) and (7) realize finite-time adaptive synchronization under the control laws (12) and the parameters’ update laws (13).
Proof.
Choose the following Lyapunov function candidate:
(14)V=12(e12+e22+e32+e42+ea2+eb2+ec2).
The differential of the Lyapunov function along the trajectory of the error system (13) is
(15)dVdt=e1(eaxs-e1β)+e2(-ymeb-e2β)+e3(-zmec-e3β)+e4(-e4β)+ea(-e1xs-eaβ)+eb(yme2-ebβ)+ec(zme3-ecβ)=-e11+β-e21+β-e31+β-e41+β-ea1+β-eb1+β-ec1+β=-2(β+1)/2((12e12)(β+1)/2+(12e22)(β+1)/200000000000+(12e32)(β+1)/2+(12e42)(β+1)/200000000000+(12ea2)(β+1)/2+(12eb2)(β+1)/200000000000+(12ec2)(β+1)/2)≤-2(β+1)/2(12e12+12e22+12e32+12e42+12ea2+12eb2000000000000+12ec2)(β+1)/2=-2(β+1)/2V(β+1)/2.
From Lemma 1, it follows that the error system (11) is finite-time stabilized. Then the uncertain slave system (7) can synchronize the master system (6) in finite time.
Remark 3.
From the proof of Theorem 2, it is found that dV/dt≤-2(β+1)/2V(β+1)/2. Using Lemma 1, we can get V(t)≡0,∀t≥t1, where t1=2(1-β)/2·(1/(1-β))[V(0)](1-β)/2. So the synchronization time is influenced by not only the parameters and initial values’ mismatch but also the control parameter β.
Remark 4.
Although the synchronization scheme is designed for Dadras systems, it can be used in two other identical hyperchaotic systems and even two different hyperchaotic systems.
4. Simulation Results
Numerical simulation results are presented to show the effectiveness of the proposed finite-time synchronization method. Fourth-order Runge-Kutta method is used and the time step size is 0.001 s. The master system’s parameters and initial conditions are the same as in Figure 1. The initial states of the response system are xs(0)=1, ys(0)=-5, zs(0)=-8, and ws(0)=12. Furthermore, the initial values of estimated parameters are a=9, b=38, and c=12 and the parameter β=0.8. The simulations of the two Dadras systems without control are shown in Figure 2, followed by the simulation with the designed finite-time adaptive control shown in Figure 3. Figure 4 shows that the trajectories of e1(t), e2(t), e3(t), and e4(t) tended to zero in finite time. The changes of parameters of as, bs, and cs are shown in Figure 5. Obviously, the synchronization errors converge to zero and the estimations of parameters converge to some constants in finite time.
States of the master and slave systems without control.
States of the master and slave systems with the designed controller.
Synchronization errors between the master and slave systems with the designed controller.
Adaptive parameters as, bs, and cs.
5. Conclusion
This paper has addressed the finite-time adaptive synchronization of Dadras hyperchaotic systems. Based on finite-time stability theory, the proposed scheme can assure the states of slave system to track the states of the master system in finite time. From the process of proof we can see that this method can be extended to other hyperchaotic systems such as Rössler hyperchaotic system and Lü hyperchaotic system.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
PecoraL. M.CarrollT. L.Synchronization in chaotic systems199064882182410.1103/PhysRevLett.64.821MR1038263ZBL0938.370192-s2.0-0343689904LiC.LiaoX.Complete and lag synchronization of hyperchaotic systems using small impulses200422485786710.1016/j.chaos.2004.03.0062-s2.0-2442601193RulkovN. F.SushchikM. M.TsimringL. S.AbarbanelH. D. I.Generalized synchronization of chaos in directionally coupled chaotic systems199551298099410.1103/PhysRevE.51.9802-s2.0-33744833457YangS. S.DuanC. K.Generalized synchronization in chaotic systems19989101703170710.1016/S0960-0779(97)00149-5ZBL0946.340402-s2.0-0032182417RosenblumM. G.PikovskyA. S.KurthsJ.Phase synchronization of chaotic oscillators199676111804180710.1103/PhysRevLett.76.18042-s2.0-4243262725OsipovG. V.PikovskyA. S.RosenblumM. G.KurthsJ.Phase synchronization effects in a lattice of nonidentical Rössler oscillators19975532353236110.1103/PhysRevE.55.2353MR14387222-s2.0-0000655970RosenblumM. G.PikovskyA. S.KurthsJ.From phase to lag synchronization in coupled chaotic oscillators199778224193419610.1103/PhysRevLett.78.4193YanZ.Q-S (lag or anticipated) synchronization backstepping scheme in a class of continuous-time hyperchaotic systems—a symbolic-numeric computation approach200515202390210.1063/1.1876612MR21502282-s2.0-31144438685KimC.-M.RimS.KyeW.-H.RyuJ.-W.ParkY.-J.Anti-synchronization of chaotic oscillators20033201394610.1016/j.physleta.2003.10.0512-s2.0-0346154825WangZ.Anti-synchronization in two non-identical hyperchaotic systems with known or unknown parameters20091452366237210.1016/j.cnsns.2008.06.0272-s2.0-56049107047MainieriR.RehacekJ.Projective synchronization in three-dimensional chaotic systems199982153042304510.1103/PhysRevLett.82.30422-s2.0-0000777715XuD.LiZ.BishopS. R.Manipulating the scaling factor of projective synchronization in three-dimensional chaotic systems200111343944210.1063/1.1380370ZBL0996.370752-s2.0-0035457991WangY.-W.GuanZ.-H.XiaoJ.-W.Impulsive control for synchronization of a class of continuous systems200414119920310.1063/1.16443512-s2.0-1842587881RafikovM.BalthazarJ. M.On control and synchronization in chaotic and hyperchaotic systems via linear feedback control20081371246125510.1016/j.cnsns.2006.12.011MR2369454ZBL1221.932302-s2.0-38049079764AgizaH. N.YassenM. T.Synchronization of Rossler and Chen chaotic dynamical systems using active control2001278419119710.1016/S0375-9601(00)00777-5MR1827067ZBL0972.370192-s2.0-0035139439ChenS.LüJ.Synchronization of an uncertain unified chaotic system via adaptive control200214464364710.1016/S0960-0779(02)00006-1ZBL1005.930202-s2.0-0036722179ZhangL.YanY.Robust synchronization of two different uncertain fractional-order chaotic systems via adaptive sliding mode control20147631761176710.1007/s11071-014-1244-12-s2.0-84899618074YangT.ShaoH. H.Synchronizing chaotic dynamics with uncertainties based on a sliding mode control design2002654704621010.1103/PhysRevE.65.0462102-s2.0-41349103836ZhangL.LiuX.The synchronization between two discrete-time chaotic systems using active robust model predictive control201374490591010.1007/s11071-013-1009-2MR31271002-s2.0-84888644493DingB.HuangB.Reformulation of LMI-based stabilisation conditions for non-linear systems in Takagi-Sugeno's form200839548749610.1080/00207720701832671MR24007512-s2.0-40849134086DingB.XiY.CychowskiM. T.O'MahonyT.Improving off-line approach to robust MPC based-on nominal performance cost200743115816310.1016/j.automatica.2006.07.022MR22667822-s2.0-33751205591DingB.XiY.CychowskiM. T.O'MahonyT.A synthesis approach for output feedback robust constrained model predictive control200844125826410.1016/j.automatica.2007.04.005MR25304912-s2.0-37549016744LiC.ChenG.LiaoX.FanZ.Chaos quasisynchronization induced by impulses with parameter mismatches200616202310210.1063/1.2179648MR22438042-s2.0-33748741723LiC.LiaoX.YangX.HuangT.Impulsive stabilization and synchronization of a class of chaotic delay systems200515404310310.1063/1.2102107MR21949102-s2.0-29844457886LiC.LiaoX.Lag synchronization of Rossler system and Chua circuit via a scalar signal20043294-530130810.1016/j.physleta.2004.06.077MR20953302-s2.0-4243073993HanQ.LiC.HuangT.Anticipating synchronization of a class of chaotic systems20091921002310510.1063/1.3125755MR25487462-s2.0-67651123093LiY.TangW. K. S.ChenG.Generating hyperchaos via state feedback control200515103367337510.1142/S02181274050139882-s2.0-29244467721CafagnaD.GrassiG.New 3D-scroll attractors in hyperchaotic Chua's circuits forming a ring200313102889290310.1142/S0218127403008284MR2020987ZBL1057.370262-s2.0-0346910376RosslerO. E.An equation for hyperchaos1979712-315515710.1016/0375-9601(79)90150-6MR5889512-s2.0-0000548789WangF.LiuC.A new criterion for chaos and hyperchaos synchronization using linear feedback control2006360227427810.1016/j.physleta.2006.08.0372-s2.0-33750448275YangN.LiuC.A novel fractional-order hyperchaotic system stabilization via fractional sliding-mode control201374372173210.1007/s11071-013-1000-yMR31176542-s2.0-84886292642YingY.GuopeiC.Finite time control of a class of time-varying unified chaotic systems201323303314310.1063/1.48237272-s2.0-84900382677AghababaM. P.Finite-time chaos control and synchronization of fractional-order nonautonomous chaotic (hyperchaotic) systems using fractional nonsingular terminal sliding mode technique2012691-224726110.1007/s11071-011-0261-6MR29298692-s2.0-84861726301AghababaM. P.AghababaH. P.A general nonlinear adaptive control scheme for finite-time synchronization of chaotic systems with uncertain parameters and nonlinear inputs20126941903191410.1007/s11071-012-0395-1MR29455282-s2.0-84866122006CaiN.LiW.JingY.Finite-time generalized synchronization of chaotic systems with different order201164438539310.1007/s11071-010-9869-1MR28032182-s2.0-79959532972LiuY.Circuit implementation and finite-time synchronization of the 4D Rabinovich hyperchaotic system2012671899610.1007/s11071-011-9960-2ZBL1242.930562-s2.0-82255163644VincentU. E.GuoR.Finite-time synchronization for a class of chaotic and hyperchaotic systems via adaptive feedback controller2011375242322232610.1016/j.physleta.2011.04.041ZBL1242.340782-s2.0-79957442815YuW.Finite-time stabilization of three-dimensional chaotic systems based on CLF2010374303021302410.1016/j.physleta.2010.05.040MR26606012-s2.0-77955429793DadrasS.MomeniH. R.QiG.WangZ. l.Four-wing hyperchaotic attractor generated from a new 4D system with one equilibrium and its fractional-order form20126721161117310.1007/s11071-011-0060-0MR28705792-s2.0-84855761900WangH.HanZ.-Z.XieQ.-Y.ZhangW.Finite-time chaos synchronization of unified chaotic system with uncertain parameters200914522392247