Synchronization of different chaotic systems and electronic circuit analysis J.. Park, T.. Lee,.. Ji,.. Jung, S.M. Lee epartment of lectrical ngineering, eungnam University, Kyongsan, Republic of Korea. igital Media and ommunications, Samsung lectronics, Suwon, Republic of Korea. epartment of Information and ommunication ngineering, eungnam University, Republic of Korea. epartment of lectronic ngineering, aegu University, yungsan, Republic of Korea. bstract This paper investigates the problem of synchronization between the hen-lee and Lorenz chaotic systems. ased on the Lyapunov stability theory and active control method, an effective controller is designed for asymptotic stability of the null solution of an error dynamics between master and slave chaotic systems. In order to verify the effectiveness of proposed control scheme, the computer simulation via Matlab software is applied to the hen-lee and Lorenz chaotic systems. Then, the realization model of the hen-lee and Lorenz chaotic systems is revised for electronic circuit simulation. inally, the circuit simulation via NI National Instruments Multisim is performed to confirm the efficiency of our results. Keywords: hen-lee chaotic system, Lorenz chaotic system, haos synchronization, Lyapunov method, ircuit analysis.. Introduction Synchronization of one system with another is very important process in the control of complex physical, chemical and biological systems as well as engineering. Therefore, many researchers have focused on this topic and developed several efficient synchronization techniques for various dynamic systems including chaotic systems, which are very sensitive to variations in the parameters and initial conditions. Since Pecora and arroll [] introduced the concept of synchronization in chaotic systems, the study of chaos synchronization has received increasing interest from scientists and engineers. It makes very big issue in nonlinear society. Up to date, not only various applications of chaos synchronization [-] but also diverse methods for control of the chaos synchronization have been introduced [-]. Originally, the chaos synchronization refers to the state in which the master or drive and the slave or response systems have precisely identical trajectories for time to infinity. We usually regard such a synchronization as complete synchronization or identical synchronization. uring the last several years, for more practical and real applications, the investigation of synchronization between different chaotic systems has been researched. or example, assen [] studied the synchronization problem of different unified chaotic systems such as Lorenz-hen, Lorenz-Lü and Lü-hen. Park [] proposed a method to synchronize between enesio-tesi and Rössler chaotic systems. In more development point of view, uang [] studied the chaos synchronization between hyperchaotic Lorenz system and hyperchaotic Lü system which has more complicated chaotic behavior and more than four Lyapunov exponent. y the way, when the mathematical model of chaotic system is implemented to electronic circuit, some adjustment due to difference of time scale between the mathematical model and electronic circuit model is sure to conduct. owever, this is not easy job. ence, that s why the numerical simulations are only provided to verify their synchronization algorithms without circuit simulation in most of literatures. In addition, most of chaos circuit analysis dealt with only hua s circuit and single chaos system which are a simple electronic circuit that exhibits chaotic behavior [- ]. Sometimes, even though the circuit analysis for chaos synchronization is conducted, these researches only deal with two identical chaotic systems not different chaotic systems. or example, uomo et al. [] and Lian et al. [] presented a solution to the synchronization problem for identical Lorenz systems. They used transformed Lorenz equation because of some errors between theoretical system and practical system. In [], u et al. investigated the synchronization of Qi hyperchaotic master and slave systems with parameters mismatch using high order differentiator. lso, iao et al. [] studied the synchronization problem between two identical Van der Pol oscillators using adaptive control method. s is well-known, some difference between theoretical system parameters and practical system parameters exists. So, it is difficult and significant to materialize theoretical system to real one. In addition, the electrical circuit simulation of different chaotic systems have more complicated problems such as readjustment of time range or difference of limitation in power supply and electronic device and so on. Therefore, in this paper, the synchronization scheme between the revised practical hen-lee chaotic master system and the revised practical Lorenz chaotic slave system will be showed by applying our control law via NI Multisim. To the best of authors knowledge, this is the first circuit analysis between different chaotic systems.
This paper is organized as follows. In Section, system description is given. In Section, the theoretical synchronization scheme between hen-lee and Lorenz chaotic systems is illustrated. In Section, a numerical simulation via Matlab is given to demonstrate the effectiveness of the proposed control method. In Section, the electronic circuit implementations are presented to show real applications of the method. inally, some conclusions are given in Section.. System description onsider the following master drive and slave response chaotic systems ẋt = ft, x, ẏt = gt, y + ut, x, y, where xt = x, x,..., x n T R n and yt = y, y,..., y n T R n are master and slave state vectors, respectively, f : R R n R n and g : R R n R n are continuous nonlinear vector functions and ut, x, y = u, u,..., u n T R n is the control input for synchronization between master and slave system. s previously stated, we deal with the hen-lee master system and Lorenz slave system for synchronization problem. Now let us consider following hen-lee master chaotic system ẋ t = ax t x tx t ẋ t = bx t + x tx t ẋ t = cx t + x tx t, where a =, b =, c =.. In order to see chaotic motion of the system, let us take an initial condition x =,, T. Then, ig. shows chaotic behavior of hen-lee system. Next, the Lorenz chaotic systems as slave system is given as follows ẏ t = a y t y t + u t ẏ t = b y t y ty t y t + u t ẏ t = y ty t c y t + u t, where a =, b =, c = /. The chaotic behavior of system with an initial condition y =,, T is presented in ig... Synchronization between the hen- Lee and Lorenz systems In this section, we design control law for achieving synchronization between the hen-lee and Lorenz systems. efinition. It is said that synchronization occurs between master system and slave system such that lim t y i t x i t =, i =,,. Now, for our synchronization scheme, let us define error signals between the hen-lee chaotic system and Lorenz chaotic system in the sense of efinition as e t = y t x t e t = y t x t e t = y t x t. The time derivative of error signal is ė t = ẏ t ẋ t ė t = ẏ t ẋ t ė t = ẏ t ẋ t. y substituting and into, we have the following error dynamics ė = a y a y + x x ax + u = a e a + ax + a y + x x + u ė = b y y y y x x + bx + u = be + b y + b y y y x x + u ė = y y c y x x + cx + u = c e + c c y + y y x x + u. ere, our goal is to achieve synchronization between the hen-lee and Lorenz systems. or this end, the following theorem shows that chaotic systems and can be synchronized effectively by the following designed controller. Theorem. haotic hen-lee system and Lorenz system can be synchronized asymptotically for any different initial conditions with the following controller: u = x tx t a y t + a + ax t u = y ty t + x tx t b y t b y t u = y ty t + x tx t c c y t. Proof. Let us take the following Lyapunov function candidate y differentiating q., we get V = e + e + e. V = e ė + e ė + e ė.
y x e e e y applying our controller and error dynamics to q., we obtain V = e a e a + ax + a y + x x + u e be + b y + b y y y x x + u e c e + c c y + y y x x + u = a e be c e = e T e e e e e e T P e <, which guarantees the stability of error systems in the sense of Lyapunov theory. This implies that the error signals satisfy lim t e i t = i =,,. This completes the proof.. Numerical simulation In order to demonstrate the validity of proposed ideas, numerical simulation via Matlab software is presented. ourthorder Runge-Kutta method with sampling time.[sec] is used to solve the system of differential equations and. The system parameters are used by a =, b =, c =., a =, b =, c = / in numerical simulation. The initial conditions for master and slave system are given by x =,, T and y =,, T, respectively. ig. shows that error signals go to zero asymptotically. It means synchronization occurs between state of x i t and state of y i t, i =,,................ time ig. : rror signals of numerical example. ircuit design and analysis In this section, we present circuit design and analysis for proposed synchronization scheme. s previously stated, chaotic systems have some errors between theoretical system parameters and practical system parameters. So we will conduct some process for elimination of these errors.. hen-lee circuit or the circuit design of mathematical dynamic model, we use transformed hen-lee chaotic system because of some problems. ased on electronic circuit of q., the range of state variables is over the limit of power supply. So, the reasonable transformation is to multiply by nonlinear term. x x x x x x x x x.. x. x. x hen Lee system of q. hen Lee system of q. ig. : omparing with original hen-lee and modified hen-lee systems x x onsider the following transformed hen-lee equations ig. : haotic motion of hen-lee system y y ig. : haotic motion of Lorenz system ẋ t = ax t x tx t ẋ t = bx t + x tx t ẋ t = cx t + x tx t, where a =, b =, c =.. This system can be more easily operated with analog circuit because all the state variables gave similar dynamic range and circuit voltages remain well within the range of typical power supply limits. In order to present effect of previous process, ig. is given which shows phase to phase portrait of original hen-lee system and modified hen-lee system of x x, x x, x x respectively. In
I I ig., we can note that the state value of modified hen- Lee system is similar the state value of original hen- Lee system divided by but inherent chaotic behavior is not changed. It means we can use the transformed hen- Lee system for our synchronization scheme because this process keep the range of state variables less than the limit of electronic device and transformed equations behave same chaotic motions. The analog circuit of transformed hen-lee q. is shown in ig.. R R R U R U R R R U R U R R U ig. : haotic phase of hen-lee system. Lorenz circuit s the same reason, we transformed Lorenz chaotic system into following equations R R R U R U R R U ẏ t = a y t y t ẏ t = b y t y ty t y t ẏ t = y ty t c y t, ig. : The circuit of hen-lee system where a =, b =, c = /. The electrical equations of the circuit are given by y y y.. y ẋ = R R R x R R R R x x ẋ = R R R R R x + R R x x ẋ = R R R R R x + R R x x, where we can note that q. is equivalent to q. after some calculation and applying the required electrical parameters such as: R, R, R = kω; R = kω; R, R, R = MΩ; R, R, R, R, R, R, R = kω; R = kω; R = kω; i = µ, i =,,. The operational amplifiers are considered to be ideal, the time step is. [s] and the initial condition of master circuit is x =.,.,. [V]. ig. displays phase to phase portrait of master system of x x, x x, x x, respectively, in left side and time to state x, x, x, respectively, in right side. y y y y Lorenz system of q. y y.. y.... y Lorenz system of q. ig. : omparing with original Lorenz and modified Lorenz systems s comparing with q., the transformed equation is changed nonlinear terms which are multiplied by. ig. displays phase to phase portrait of original Lorenz system and modified Lorenz system of x x, x x, x x respectively. Like the preceding, we can note that the state value of modified Lorenz system is similar the state value of original Lorenz system divided by. ut we can also know inherent chaotic behavior is not changed. The analog circuit of transformed Lorenz equation is shown in ig..
The electrical equations of the circuit are given by ẏ = R R R y + y R R R + R R ẏ = R R R y R R R R y R R R R y y ẏ = R R R y y R R R R y, where we can note that q. is equivalent to q. after rescaling time by a factor of. nd the required electrical parameters are as following: R, R, R = kω; R, R, R, R, R, R = kω; R, R, R = MΩ; R = kω; R, R = kω; R, R = kω; R = kω; i = µ, i =,,. The operational amplifiers are considered to be ideal, the time step is. [s] and the initial condition of master circuit is x =.,.,. [V]. ig. displays phase to phase of master system of y y, y y, y y, respectively, in left side and time to state y, y, y, respectively, in right side.. Synchronization circuit s transforming qs. and to qs. and, respectively, the control inputs of Theorem. should be also changed as follows: u = x x a y + a + ax u = y y + x x b y b y u = y y + x x c c y. R R U R R R U R ig. : Simulation results without control R R R R U R R R U R R U R R U R U ig. : The circuit of Lorenz system U To show the effect of control input, first of all, we run the circuit without control inputs. ig. displays phase to phase and time to phase portraits of master and slave systems for this case. One can see that the errors do not approach to zero as expected since the control inputs are not applied. inally, the circuit of the whole synchronizing system is given in ig.. The circuit consists of three parts: master systems, slave systems, and controllers. Then, ig. displays that synchronization between hen-lee chaotic system and Lorenz chaotic system is achieved by control inputs as expected.. onclusion In this paper, we have investigated the synchronization problem for the hen-lee and Lorenz chaotic systems. Our proposed control scheme is verified by numerical simulation of the system. It should be noted that we included circuit analysis for the synchronization between the different chaotic systems for the first time. ig. : haotic phase of Lorenz system cknowledgements This research was supported by asic Science Research Program through the National Research oundation of Korea NR funded by the Ministry of ducation, Science and Technology -.
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