Chaos Synchronization Based on Linear and Adaptive Controls: Theory and Experiment


  • O. I. Olusola University of Lagos, Akoka, Lagos, Lagos State Nigeria
  • R. T. Ogundare University of Lagos, Akoka, Lagos, Lagos State, Nigeria
  • A. I. Egunjobi Federal University of Agriculture, P.M.B.2240, Abeokuta, Nigeria
  • E. O. Odufuwa University of Lagos, Akoka, Lagos, Lagos State. Nigeria
  • M. O. Esan University of Lagos, Akoka, Lagos, Lagos State. Nigeria
  • U. E. Vincent Redeemer’s University, Ede, Nigeria


Chaos, Synchronization, Linear feedback controller, Adaptive controller, single variable


Communication Physical Sciences, 2021, 7(3):  246-262

Authors: I. Olusola*, R. T. Ogundare, A. I. Egunjobi, E. O. Odufuwa, M. O. Esan,and E. Vincent

Received 15 September 2021/Accepted  17 October 2021

In this paper, we report on the theoretical and experimental investigation of chaotic synchronization using a single variable linear feedback and adaptive controllers. Based on the Lyapunov stability theory, theoretical approaches to the design of controls are presented, and the results are validated numerically and by employing electronic circuit experiments. We used two typical oscillators, namely, the Lorenz and Sprott chaotic systems to demonstrate our results; while off-the-shelf components on breadboard were used to experimentally implement the proposed single variable controllers. We specifically show that synchronization of two chaotic systems can be experimentally realized when the strength of the feedback exceeds a theoretically determined threshold.


Download data is not yet available.

Author Biographies

O. I. Olusola, University of Lagos, Akoka, Lagos, Lagos State Nigeria

Nonlinear Dynamics, Research Group, Department of Physics

R. T. Ogundare, University of Lagos, Akoka, Lagos, Lagos State, Nigeria

Department of Physics

A. I. Egunjobi, Federal University of Agriculture, P.M.B.2240, Abeokuta, Nigeria

Department of Physics

E. O. Odufuwa, University of Lagos, Akoka, Lagos, Lagos State. Nigeria

Nonlinear Dynamics,amics Research Group, Department of Physics

M. O. Esan, University of Lagos, Akoka, Lagos, Lagos State. Nigeria

Nonlinear Dynamics,amics Research Group, Department of Physics

U. E. Vincent, Redeemer’s University, Ede, Nigeria

Department of Physical Sciences


Abd, M. H., Tahir, F. R., Al-Suhail, G. A. & Pham, V.-T. (2017). An adaptive observer synchronization using chaotic time-delay system for secure communication. Nonlinear Dynamics, 90, pp. 2583-2598.

Aguilar-Lpez, R., Martnez-Guerra, R. & Perez-Pinacho, C. (2014). Nonlinear observer for synchronization of chaotic systems with application to secure data transmission. Er. Phys. J. Spec. Top. 223, pp. 1541-1548,

Ahmed, H., Ushirobira, R. & Efimov, D. (2017): Experimental study of the robust global synchronization of Brockett oscillators. The European Physical Journal Special Topics 226(15), pp. 3199-3210.

Almatroud Othmana, A., Noorania, M. S. M. & Mossa Al–Sawalha, M. (2016): Adaptive dual synchronization of chaotic and hyperchaotic systems with fully uncertain parameters, Optik – International Journal for Light and Electron Optics 12, 19, pp. 7852–7864 (2016).

Arellano-Delgado, A., Lo´pez-Guti´errez, R. M., Cruz-Hern´andez, C., Posadas-Castillo, C. Cardoza-Avendan˜o, L., & Serrano-Guerrero, H. (2013): Experimental network synchronization via plastic optical fiber. Optical Fiber Technology 19, 2, pp. pp. 93–108.

Bhatnagar, G. & Wu, Q. M. J. (2015). A novel chaos based secure transmission of biometric data. Neurocomputing,147, pp. 444-455.

Hamed T, Saeid S & Hamidreza T. (2018). Chaos synchronization and parameter identification of a finance chaotic system with unknown parameters, a linear feedback controller. Alexandria Engineering Journal, 57, 3, pp. 1519-1524.

Choi, H. & Lee, J. (2017). Principles, Applications, and Challenges of Synchronization in Nature for Future Mobile Communication Systems. Hindawi, Mobile Information Systems,

Cheng, C., Gao, F. Xu, J., Wang, Y. & Yuan, T. (2020). Adaptive Control Design for Arneodo Chaotic System with Uncertain Parameters and Input Saturation. Hindawi Advances in Mathematical Physics, 55/2020/3285414

Egunjobi, A. I., Olusola, O. I., Njah, A. N., Saha, S. & Dana, S. K. (2018). Experimental evidence of chaos synchronization via cyclic coupling. Commun Nonlinear Science and Numerical Simulations, 56, pp. 588-595.

Filali, R. L., Benrejeb, M. & Borne, P. (2014). On observer based secure communication design using discrete-time hyperchaotics systems. Communication in Nonlinear Science Numerical Simulation, 19, 5, pp. 1424–1432.

Guo, R., Vincent, U. E. & Idowu, B. A. (2009): Synchronization of Chaos in RCL- shunted Josephson Junction using a simple adaptive control, Physica Scr. 79, 035801.

Medhaffar, H., Feki, M. & Derbe, N. (2020). Adaptive fuzzy control for the stabilisation of chaotic systems. International Journal of Automation and Control,14, 2, pp.115 – 137.

Hu, M. F., & Xu, Z. Y. (2008): Adaptive feedback controller for projective synchronization. Nonlinear Anal., Real World Appl., 9, 3, 1253-1260.

Hua, C., Li, J., Yang, Y. & Guan, X. (2016). Extended-state-observer-based finite-time synchronization control design of teleoperation with experimental validation. Nonlinear Dynamics, 85, pp. 317-331.

Hua, M. F. & Xua, Z. Y. (2008): Adaptive feedback controller for projective synchronization. Nonlinear Analysis: Real World Applications ,9, pp. 1253 - 1260.

Liu L. & Guo, R. (2017). Control problems of Chen-Lee system by adaptive control method. Nonlinear Dynamics,., 87, pp. 503-510.

Liu, X., Zhai, D., Dong, J. & Zhang, Q. (2018). A Adaptive fault-tolerant control with prescribed performance for switched nonlinear pure-feedback systems. Journal of the Franklin Institute 355, pp. 273-290.

Liu, Y.J., Wang, W., Tong, S.C., & Liu, Y.S. (2010). Robust adaptive tracking control for nonlinear systems based on bounds of fuzzy approximation parameters. IEEE Trans. Syst. Man Cybern., Part A, Syst. Hum. 40, 1, pp. 170-184.

Liu, Y.J. & Zheng, Y.Q. (2009): Adaptive robust fuzzy control for a class of uncertain chaotic systems. Nonlinear Dynamics,. 57(3), pp. 431-439.

Lorenz, E.N. (1964). Deterministic nonperiodic flow. Journal of Atmospheric Science, 20, pp. 130–141.

Mahmoud, E.E., & Abood, F.S (2017): A novel sort of adaptive complex synchronizations of two indistinguishable chaotic complex nonlinear models with uncertain parameters and its applications in secure communications. Results in Physics, 7 pp. 4174-4182.

Machuc, J. L. (2014): Experimental Synchronization by Means of Observers. Journal of Applied Research and Technology, 12, 1, pp. 52–62.

Ni, J. K., Liu, L., Liu, C. X., Hu, X. O. & Li. A. A. (2017). Chaos suppression for a four-dimensional fundamental power system model using adaptive feedback control. Trans. Inst. Measurement and Control, 39, 2, pp. 194-207.

Niu, H., Ma, S. & Fan, T., Chen, C. & He, P. (2014). Linear state feedback stabilization of unified hyperchaotic systems. World Journal of Modelling and Simulation 10, 1, pp. 34–48.

Ojo, K.S. Njah, A. N., Ogunjo, S.T. & Olusola, O.I. (2014). Reduced Order Functional Projective Synchronization of three Josephine junctions using backstepping technique. Nonlinear Dynamics and Systems Theory, 14, 12, pp. 110–133.

Olusola, O.I., U.E. Vincent, & Njah, A.N. (2010). Multistability and basin crisis in synchronized parametrically driven oscillators. Nonlinear Dynamics, 62, pp. 717-727.

Onma, O.S. Olusola, O.I. & Njah, A.N. (2016). Control and synchronization of chaotic and hyperchaotic Lorenz systems via extended adaptive control techniques, Far East Journal of Dynamical systems, 28, 1, pp. 1 – 32.

Pallov Anand & B. B. Sharma (2020). Synchronization of Chain Network of 4-D Lorenz-Stenflo Systems using Contraction based Backstepping. 2020 2nd International Conference on Innovative Mechanisms for Industry Applications (ICIMIA)

Pecora, L. M. & Caroll,T. L. (1990). Synchronization of Chaotic Systems., Chaos, 25, 097611, 1063/1.4917383

Perlikowski, P., Jagiello, B., Stefanski, A. & Kapitaniak, T. (2008). Experimental observation of ragged synchronizability. Phys. Rev. E 78, 017203.

Ping He, & Fei Tan. (2011): Linear State Feedback Stabilization for Controlled Chaotic Systems. International Journal of Nonlinear Science 123, pp. 373–384.

Ren, H. P., Baptista, M. S., & Grebogi, C. (2013). Wireless communication with chaos, Physics Letters and Review, 3:110, 18, doi: 10.1103/PhysRevLett.110.184101.

Ricardo, A. L. & Rafael, M.. G. (2008). Synchronization of a class of chaotic signals via robust observer design. Chaos Solitons Fractals 37, 2, pp. 581-587.

Salarieh, H. & Alasty, A. (2009): Adaptive synchronization of two chaotic systems with stochastic unknown parameters. Communication in Nonlinear Science and Numerical Simulation,. 14, 2, pp. 508-519.

Shaohua Luo, Frank L. Lewis, Yongduan Song & Kyriakos G. Vamvoudakis (2020). Adaptive backstepping optimal control of a fractional-order chaotic magnetic-field electromechanical transducer. Nonlinear Dynamics, 100, pp. 523–540.

Siddique, M. & Rehan, M. (2016): A concept of coupled chaotic synchronous observers for nonlinear and adaptive observers-based chaos synchronization. Nonlinear Dynamics, 84(4), pp. 2251-2272.

Sprott J.C. (2000): A New Class of Chaotic Circuit. Phys. Letts. A 266, 19–23.

Stefanski, A., Wojewoda, J. & Kapitaniak, T. (2004): Simple estimation of synchronization threshold in ensembles of diffusively coupled chaotic systems. Phys. Rev. E 70, 026217.

Vincent, U.E. & Guo, R. (2009): A simple adaptive control for full and reduced–order synchronization of uncertain time-varying chaotic systems, Communication in Nonlinear Science and Numerical Simulation, 14, pp. 3925–3932.

Wang, X.-Q.& Wang, Y.-Q. (2011): Adaptive control for synchronization of a four- dimensional chaotic system via a single variable. Nonlinear Dynamics, 65, pp. 311-316.

Yang, Chi-Ching (2014). Adaptive Single Input Control for Synchronization of a 4D LorenzStenflo Chaotic System. Arabian Journal of Science and Engineering, 39, pp. 2413-2426.

Yao, J.Y., Jiao, Z.X. & Ma, D.W. (2014). Adaptive robust control of DC motors with extended state observer. IEEE Trans. Ind. Electron., 61, 7, pp. 3630-3637.

Yaping Hu, Yaru Zhang, & Rongwei Guo (2020). Coexistence of anti-phase and complete synchronization in the Chen-Lee system. Conference: 2020 39th Chinese Control Conference (CCC).