Active Control-Based Synchronization of a Lorenz Hyperchaotic System with an Application to Secure Signal Masking

Öz

This study investigates the dynamics of the Lorenz hyperchaotic system and presents an active control-based synchronization approach for two identical systems. To confirm the hyperchaotic nature of the system, an examination of its dynamic structure is conducted, emphasizing Lyapunov exponents, phase-space evolution, and sensitivity to initial conditions. Subsequently, a synchronization scheme operating as a driver-response is introduced. The associated error dynamics are employed to evaluate the stability of the synchronization process, and the influence of control parameters on convergence conditions and system eigenvalues is analyzed. It is examined that a signal masking and recovery application to show that the proposed method goes beyond just theoretical study. Before and after the controller is turned on, the system's behavior is checked. After synchronization, the masked signal can be used to accurately reconstruct the transmitted signal. The use of different input signals also evaluates the consistency of the proposed method. The proposed method maintains effective performance under various conditions, as each case exhibits similar dynamical behavior. These results collectively demonstrate that the synchronization framework based on active control offers a feasible and dependable solution for the Lorenz hyperchaotic system.

Anahtar Kelimeler

• Chaos
• Hyperchaotic systems
• Active synchronization
• Signal masking

Etik Beyan

Ethics committee permission is not required for the prepared article. There is no conflict of interest with any person/institution in the prepared article.

Lorenz Hiperkaotik Sisteminin Aktif Kontrol Tabanlı Senkronizasyonu ve Sinyal Maskeleme Uygulaması

Öz

Bu çalışma, Lorenz hiperkaotik sisteminin nasıl çalıştığını ve iki özdeş sistemin birlikte çalışmasını sağlamak için aktif kontrolün nasıl kullanılabileceğini incelemektedir. Sistemin hiperkaotik doğasını doğrulamak amacıyla, Lyapunov üsleri, faz uzayı diyagramı ve başlangıç koşullarına duyarlılık üzerinde durularak sistemin dinamik yapısı incelenmiştir. Ardından, sürücü-yanıt gibi çalışan bir senkronizasyon şeması tanıtılmaktadır. Senkronizasyon sürecinin kararlılığını değerlendirmek için ilgili hata dinamikleri kullanılmakta ve kontrol parametrelerinin yakınsama koşulları ve sistem özdeğerleri üzerindeki etkisi analiz edilmektedir. Önerilen yöntemin sadece teorik bir çalışmanın ötesine geçtiğini göstermek için bir sinyal maskeleme ve kurtarma uygulaması incelenmektedir. Kontrolörün açılmasından önce ve sonra sistemin davranışı kontrol edilmektedir. Senkronizasyondan sonra, maskelenmiş form gönderilen sinyali doğru bir şekilde yeniden oluşturmak için kullanılabilir. Farklı giriş sinyallerinin kullanılması da yöntemin ne kadar tutarlı olduğunu test etmektedir. Her durum benzer şekilde işlediğinden, yöntem birçok farklı durumda hala iyi çalışmaktadır. Bu sonuçlar topluca, aktif kontrole dayalı senkronizasyon çerçevesinin Lorenz hiperkaotik sistemi için uygulanabilir ve güvenilir bir çözüm sunduğunu göstermektedir.

Anahtar Kelimeler

• Kaos
• Hiperkaotik sistemler
• Aktif senkronizasyon
• Sinyal maskeleme

Etik Beyan

Hazırlanan makale için etik kurul onayı gerekmemektedir. Hazırlanan makalede herhangi bir kişi/kurumla çıkar çatışması bulunmamaktadır.

Kaynakça

1. S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, 3rd ed. New York, NY, USA: Chapman and Hall/CRC, 2024.
2. G. Kolumban, M. P. Kennedy, and L. O. Chua, “The role of synchronization in digital communications using chaos. II. Chaotic modulation and chaotic synchronization,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 45, no. 11, pp. 1129–1140, Nov. 1998.
3. E. N. Lorenz, “Deterministic Nonperiodic Flow,” in Universality in Chaos, 2nd ed. New York, NY, USA: Routledge, 1989.
4. S. Banerjee and J. Kurths, “Chaos and Cryptography: A new dimension in secure communications,” Eur. Phys. J. Spec. Top., vol. 223, no. 8, pp. 1441–1445, Jun. 2014.
5. L. Jin, J. Mei, and L. Li, “Chaos control of parametric driven Duffing oscillators,” Appl. Phys. Lett., vol. 104, no. 13, p. 134101, Apr. 2014.
6. O. E. Rössler, “An equation for continuous chaos,” Phys. Lett. A, vol. 57, no. 5, pp. 397–398, Jul. 1976.
7. G. Chen, “Yet another chaotic attractor,” Int. J. Bifurc. Chaos, vol. 2, no. 3, pp. 705–708, 1993.
8. C. Liu, T. Liu, L. Liu, and K. Liu, “A new chaotic attractor,” Chaos Solitons Fractals, vol. 22, no. 5, pp. 1031–1038, Dec. 2004.

9. A. Xiong, X. Zhao, J. Han, and G. Liu, “Application of the Chaos Theory in the Analysis of EMG on Patients with Facial Paralysis,” in Robot Intelligence Technology and Applications 2: Results from the 2nd International Conference on Robot Intelligence Technology and Applications, J.-H. Kim, E. T. Matson, H. Myung, P. Xu, and F. Karray, Eds. Cham, Switzerland: Springer, 2014, pp. 805–819.
10. Z. Huang, W. Dong, H. Duan, and H. Li, “Similarity Measure Between Patient Traces for Clinical Pathway Analysis: Problem, Method, and Applications,” IEEE J. Biomed. Health Inform., vol. 18, no. 1, pp. 4–14, Jan. 2014.
11. Z. Kang, J. Sun, L. Ma, Y. Qi, and S. Jian, “Multimode Synchronization of Chaotic Semiconductor Ring Laser and its Potential in Chaos Communication,” IEEE J. Quantum Electron., vol. 50, no. 3, pp. 148–157, Mar. 2014.
12. M. L. Barakat, A. S. Mansingka, A. G. Radwan, and K. N. Salama, “Hardware stream cipher with controllable chaos generator for colour image encryption,” IET Image Process., vol. 8, no. 1, pp. 33–43, 2014.
13. A. Anees, A. M. Siddiqui, and F. Ahmed, “Chaotic substitution for highly autocorrelated data in encryption algorithm,” Commun. Nonlinear Sci. Numer. Simul., vol. 19, no. 9, pp. 3106–3118, Sep. 2014.
14. Y. Ji et al., “Microwave-Photonic Sensor for Remote Water-Level Monitoring Based on Chaotic Laser,” Int. J. Bifurc. Chaos, vol. 24, no. 3, p. 1450032, Mar. 2014.
15. M. Awais, M. A. Khan, and Z. Bashir, “Exploring the stochastic patterns of hyperchaotic Lorenz systems with variable fractional order and radial basis function networks,” Clust. Comput., vol. 27, no. 7, pp. 9031–9064, Oct. 2024.
16. O. E. Rossler, “An equation for hyperchaos,” Phys. Lett. A, vol. 71, no. 2, pp. 155–157, Apr. 1979.
17. L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett., vol. 64, no. 8, pp. 821–824, Feb. 1990.
18. A. Khan and M. Shahzad, “Synchronization of circular restricted three body problem with Lorenz hyper chaotic system using a robust adaptive sliding mode controller,” Complexity, vol. 18, no. 6, pp. 58–64, 2013.
19. H.-K. Chen, “Global chaos synchronization of new chaotic systems via nonlinear control,” Chaos Solitons Fractals, vol. 23, no. 4, pp. 1245–1251, Feb. 2005.
20. H.-H. Chen, “Global synchronization of chaotic systems via linear balanced feedback control,” Appl. Math. Comput., vol. 186, no. 1, pp. 923–931, Mar. 2007.
21. P. He, C.-G. Jing, T. Fan, and C.-Z. Chen, “Robust decentralized adaptive synchronization of general complex networks with coupling delayed and uncertainties,” Complexity, vol. 19, no. 3, pp. 10–26, 2014.
22. R. Rakkiyappan and N. Sakthivel, “Cluster synchronization for T–S fuzzy complex networks using pinning control with probabilistic time-varying delays,” Complexity, vol. 21, no. 1, pp. 59–77, 2015.
23. I. Ahmad, A. B. Saaban, A. B. Ibrahim, and M. Shahzad, “Global Chaos Identical and Nonidentical Synchronization of a New 3-D Chaotic Systems Using Linear Active Control,” Asian J. Appl. Sci., vol. 2, no. 1.
24. M. T. Akter, A. Tarammim, and S. Hussen, “Chaos control and synchronization of modified Lorenz system using active control and backstepping scheme,” Waves Random Complex Media, pp. 1–20, May 2023.
25. A. Durdu and Y. Uyaroğlu, “Comparison of synchronization of chaotic Burke-Shaw attractor with active control and integer-order and fractional-order P-C method,” Chaos Solitons Fractals, vol. 164, p. 112646, Nov. 2022.
26. A. Tarammim and M. T. Akter, “Shimizu–Morioka’s chaos synchronization: An efficacy analysis of active control and backstepping methods,” Front. Appl. Math. Stat., vol. 9, Apr. 2023.
27. Y. Zhou, Y. Guo, W. Zhang, and P. Zhao, “Novel Hyperchaotic System with Linear Memristor: Active Synchronization and an Image Encryption Application,” Nonlinear Dyn., vol. 113, no. 22, pp. 31669–31689, Nov. 2025.
28. M. Kopp and I. Samuilik, “A New 6D Two-wing Hyperchaotic System: Dynamical Analysis, Circuit Design, and Synchronization,” Chaos Theory Appl., vol. 6, no. 4, pp. 273–283, Nov. 2024.
29. E. Ozpolat, V. Celik, and A. Gulten, “A Novel Four-Dimensional Hyperchaotic System: Design, Dynamic Analysis, Synchronization, and Image Encryption,” IEEE Access, vol. 12, pp. 126063–126073, 2024.
30. A. S. Lekshmi, V. Balakumar, and D. Cafagna, “On the dynamics and synchronization of a new fractional-order hyperchaotic system,” Int. J. Dyn. Control, vol. 14, no. 1, p. 16, Dec. 2025.
31. X. Wang and M. Wang, “A hyperchaos generated from Lorenz system,” Physica A, vol. 387, no. 14, pp. 3751–3758, 2008.
32. H. D. I. Abarbanel, R. Brown, and M. B. Kennel, “Lyapunov Exponents in Chaotic Systems: Their Importance and Their Evaluation Using Observed Data,” Int. J. Mod. Phys. B, vol. 5, pp. 1347–1375, Jan. 1991.
33. A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, “Determining Lyapunov exponents from a time series,” Physica D, vol. 16, no. 3, pp. 285–317, Jul. 1985.
34. M. Baia, F. Bagnoli, T. Matteuzzi, and A. Pikovsky, “Synchronization of branching chain of dynamical systems,” Physica D, vol. 477, p. 134664, Jul. 2025.
35. F. Setoudeh, M. M. Dezhdar, and M. Najafi, “Nonlinear analysis and chaos synchronization of a memristive-based chaotic system using adaptive control technique in noisy environments,” Chaos Solitons Fractals, vol. 164, p. 112710, 2022.
36. H. Cheng, H. Li, Q. Dai, and J. Yang, “A deep reinforcement learning method to control chaos synchronization between two identical chaotic systems,” Chaos Solitons Fractals, vol. 174, p. 113809, 2023.
37. L. O. Chua, M. Itoh, L. Kocarev, and K. Eckert, “Chaos Synchronization in Chua’s Circuit,” J. Circuits Syst. Comput., vol. 3, no. 1, pp. 93–108, Mar. 1993.
38. M. T. Yassen, “Chaos synchronization between two different chaotic systems using active control,” Chaos Solitons Fractals, vol. 23, no. 1, pp. 131–140, Jan. 2005.