Synchronization of Two Chaotic Oscillators Through Threshold Coupling A. Chithra and I. Raja Mohamed
Abstract In this paper, the dynamic modeling of two identical oscillators which are coupled through threshold controller is proposed. Until now, most of the synchronization of chaotic systems found in literature is based on common coupling methods (unidirectional and bidirectional) that attracted the attention of researchers. To strengthen this, the idea illustrated here is to show the effectiveness of a new kind of coupling called threshold controller coupling. Using this, complete and anticipatory synchronization could be achieved. The system used is of second-order non-autonomous type. The coupled system is investigated using MATLAB– Simulink technique. The result shows that based on coupling strength, coupled system is switched among the basic synchronization, viz. lead and complete.
Keywords Modeling Synchronization Chaotic MATLAB–Simulink
Threshold controller
1 Introduction Chaos theory has one of the greatest achievement and application in secure communication over two decades. Chaotic system can produce infinite number of chaotic signals which are non-periodic and is characterized by high sensitive to parameter value and initial condition. Due to this property and broadband nature of chaotic signal, which makes particular interest in the concept of synchronization for the application of secure communication to mask the embedded message or signal becomes possible. Synchronized chaos is a phenomenon that occurs when two or more chaotic systems adjust to common behavior due to coupling. The idea of chaos synchroA. Chithra I. Raja Mohamed (&) Department of Physics, B. S. Abdur Rahman University, Chennai, India e-mail:
[email protected] A. Chithra e-mail:
[email protected] © Springer Nature Singapore Pte Ltd. 2018 S. K. Muttoo (ed.), System and Architecture, Advances in Intelligent Systems and Computing 732, https://doi.org/10.1007/978-981-10-8533-8_24
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nization is where one separate identical drive system drives another system under suitable threshold coupling parameter; then the response system follows the drive asymptotically. So far, many types of synchronization and coupling methods have been reported in the literature. In this paper, a new method of achieving complete and anticipatory synchronization is proposed. The two chaotic systems with threshold nonlinearity are coupled via threshold controller coupling. To enforce synchronization, one of the signals is fed to the response side. By simply altering the coupling threshold value and coupling parameter value, the system goes through complete synchronization state from unsynchronized state.
2 Literature Overview The concept of synchronization of chaotic oscillation is extensively studied [1]. Dutch Physicist Huygens discovered the first observation of synchronization in pendulum clock [2]. Edward Appleton and Balthasar van der Pol showed the experimental and theoretical study for this phenomenon. Then the synchronization of chaotic system which used in secure communication systems was first reported by Yamada and Fujisaka [3] followed by Pecora and Carroll [4]. Since then several interesting synchronization phenomena have been observed and reported in the literature. To initiate synchronization, the system should be coupled in a proper way. Eventually by the nature of coupling, different types of synchronization are found, viz. generalized synchronization [5], complete synchronization [6], lag and anticipatory synchronization [7], phase synchronization [8], and global synchronization [9]. Hence to achieve these types of synchronization, the commonly used coupling methods are unidirectional [10] and bidirectional [11], but rarely cascaded [12] and delay [13]. The design of flexible nonlinearity exhibiting chaos has not been much explored in the literature, and only few studies are found in this direction [14, 15]. This specific type of nonlinearity is used as coupling element in the system. Complete synchronization has been observed in Chua’s oscillator [16], Lorenz and Rossler system [17], Duffing system [18], and so on. In this paper, threshold controller-based second-order chaotic system reported in Int. J. Bifuracat. Chaos; V 20 (2010) is coupled through threshold controller. The advantage of this coupling is to control the drive and response separately by altering the threshold level of threshold controller. The system will exhibit complete and anticipatory synchronization when coupling parameter value is increased. The paper is organized as follows: The dynamic modeling of single and coupled system through threshold controller is discussed in Sect. 3. Section 4 describes the simulation result of coupled system. Summary of the result and conclusion is given finally.
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3 Modeling and Simulation of Chaotic System 3.1
Dynamic Modeling of Single System
Threshold controller-based chaotic oscillator [14] is used to design dynamic modeling of coupled systems for synchronization. The dynamical differential equation of second-order non-autonomous system is as follows: d2 x dx ¼ a bx þ dGðxÞ þ f sin xt 2 dt dt
ð1Þ
Here a, b, d are constants, x; ddxt are the state variables of the oscillators, and G (x) is the nonlinear term used in the system called threshold controller. The control will be triggered whenever the value of dynamical variable exceeds positive ( þ x ) as well as negative threshold (x ), then it will be reset to threshold value (x = 0.7), where 8 0, the system transits from unsynchronized to anticipatory and to complete or identical synchronization. Figure 2 shows the MATLAB–Simulink modeling of two identical oscillators’ drive (D) and response (R) through threshold coupling with threshold level at x = 1.4, and other parameters values are fixed as same as drive system. When є1 = 0.55 and є2 = 1.06, the system is in unsynchronized state. Then by gradually increasing the coupling parameter beyond є1 = 0.95 and є2 = 1.06, the system exhibits anticipatory synchronization. By further increasing the coupling parameter value, it shows complete synchronized state at є1 = 1.2 and є2 = 1.06. Figure 3 shows the time series, phase difference plot of ðx x0 Þ, and the simulation result is confirmed with time series analysis.
Fig. 2 Simulink modeling of two coupled systems through threshold coupling
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Fig. 3 Time series plot and phase difference plot. a and b are anticipatory synchronization at є1 = 0.98 and є2 = 1.06, x = 1.4. c and d are complete synchronization at є1 = 1.2 and є2 = 1.06
5 Conclusion In this work, the dynamic modeling of two identical oscillators through threshold controller using MATLAB–Simulink technique is reported. The proposed system is based on new kind of coupling called threshold controller. The system exhibits various types of synchronization like complete and anticipatory synchronization for increasing value of coupling parameter. Further work regarding experimental proof with electronic circuit is currently being investigated. Acknowledgements This research work is supported by SERB under project No: SR/S2/ HEP-042/2012, and authors thank SERB for providing financial support.
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