Kayma Kipli Kontrol Yöntemiyle Newton-Leipnik Sisteminin Dinamik Analizi ve Sağlam Kaos Bastırılması

Abstract

Bu çalışma, doğrusal olmayan dinamik sistemler çerçevesinde Newton-Leipnik kaotik sisteminin matematiksel analizini ve sağlam kontrolünü sunmaktadır. Sistem önce parametrik biçimde formüle edilir ve ardından kaotik özelliklerini ve parametreye bağlı davranışını belirlemek için zaman tepkileri, faz portreleri, Lyapunov üsleri ve çatallanma analizi yoluyla incelenir. Sonuçlar, kontrolsüz sistemin sınırlı ancak periyodik olmayan bir hareket, garip bir çekici yapı ve başlangıç ​​koşullarına belirgin bir duyarlılık sergilediğini, yani tamamen gelişmiş bir kaotik rejimi gösterdiğini doğrulamaktadır. Dinamik analizden sonra, kaotik salınımları bastırmak ve tüm durum değişkenlerini denge noktasına getirmek için bir kayma modu kontrol stratejisi tasarlanmıştır. Önerilen kontrolör, hem nominal hem de bozulmuş çalışma koşulları altında değerlendirilmiştir. Simülasyon sonuçları, kontrol yasasının sistemi hızla stabilize ettiğini, kaotik davranışı etkili bir şekilde ortadan kaldırdığını ve sınırlı kontrol çabasını koruduğunu göstermektedir. Sağlamlığı daha da değerlendirmek için, kapalı döngü tepkisi dürtüsel ve salınımlı bozulmalar altında test edilmiştir. Her iki durumda da, kontrolör kararlılığı korur ve bozulmadan sonra sistemi istenen çalışma bölgesine hızla geri döndürür. Elde edilen bulgular, kayma modlu kontrolün Newton-Leipnik kaotik sisteminin stabilizasyonu için güvenilir ve sağlam bir çözüm sağladığını ve önerilen çerçevenin, sağlam kaos bastırma ve doğrusal olmayan kontrol tasarımı üzerine gelecekteki çalışmalar için yararlı bir temel sunduğunu göstermektedir.

Keywords

• Kaos analizi
• Kayma kipli kontrol
• Newton-Leipnik sistemi

Dynamical Analysis and Robust Chaos Suppression of the Newton-Leipnik System via Sliding Mode Control

Abstract

This study presents the mathematical analysis and robust control of the Newton-Leipnik chaotic system within a nonlinear dynamical systems framework. The system is first formulated in parametric form and then examined through time responses, phase portraits, Lyapunov exponents, and bifurcation analysis in order to identify its chaotic characteristics and parameter dependent behavior. The results confirm that the uncontrolled system exhibits bounded but aperiodic motion, a strange attractor structure, and clear sensitivity to initial conditions, indicating a fully developed chaotic regime. Following the dynamical analysis, a sliding mode control strategy is designed to suppress chaotic oscillations and regulate all state variables to the equilibrium point. The proposed controller is evaluated under both nominal and disturbed operating conditions. Simulation results show that the control law rapidly stabilizes the system, effectively eliminates chaotic behavior, and maintains bounded control effort. To further assess robustness, the closed loop response is tested under impulsive and oscillatory disturbances. In both cases, the controller preserves stability and quickly restores the system to the desired operating region after perturbation. The findings demonstrate that sliding mode control provides a reliable and robust solution for the stabilization of the Newton-Leipnik chaotic system and that the proposed framework offers a useful basis for future studies on robust chaos suppression and nonlinear control design.

Keywords

• Chaos analysis
• Sliding mode control
• Newton-Leipnik system

References

1. [1] G. Boeing, “Visual analysis of nonlinear dynamical systems: chaos, fractals, self-similarity and the limits of prediction,” Systems, vol. 4, no. 4, p. 37, 2016.
2. [2] P. B. Persson and C. D. Wagner, “General principles of chaotic dynamics,” Cardiovascular Research, vol. 31, no. 3, pp. 332–341, 1996.
3. [3] X. Zeng, R. A. Pielke, and R. Eykholt, “Chaos theory and its applications to the atmosphere,” Bulletin of the American Meteorological Society, vol. 74, no. 4, pp. 631–644, 1993.
4. [4] R. I. Sujith and S. A. Pawar, “An introduction to dynamical systems theory,” in Thermoacoustic Instability: A Complex Systems Perspective. Cham, Switzerland: Springer, 2021, pp. 31–85.
5. [5] C. Letellier et al., “Some elements for a history of the dynamical systems theory,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 31, no. 5, 2021.
6. [6] N. A. Magnitskii, “Bifurcation theory of dynamical chaos,” Chaos Theory, vol. 11, pp. 197–215, 2018.
7. [7] A. Robledo, “Generalized statistical mechanics at the onset of chaos,” Entropy, vol. 15, no. 12, pp. 5178–5222, 2013.
8. [8] F. Baldovin and A. Robledo, “Universal renormalization-group dynamics at the onset of chaos in logistic maps and nonextensive statistical mechanics,” Physical Review E, vol. 66, no. 4, p. 045104, 2002.

9. [9] B. Luque, L. Lacasa, F. J. Ballesteros, and A. Robledo, “Feigenbaum graphs: a complex network perspective of chaos,” PLoS ONE, vol. 6, no. 9, p. e22411, 2011.
10. [10] A. D. Morozov, Invariant Sets for Windows, vol. 37. Singapore: World Scientific, 1999.
11. [11] S. P. Kuznetsov, “A variety of critical phenomena associated with the golden mean quasiperiodicity,” Izvestiya VUZ. Applied Nonlinear Dynamics, vol. 10, no. 3, p. 22, 2002.
12. [12] S. Banerjee and G. C. Verghese, Nonlinear Phenomena in Power Electronics. Piscataway, NJ, USA: IEEE Press, 1999.
13. [13] A. Uthamacumaran, “A review of dynamical systems approaches for the detection of chaotic attractors in cancer networks,” Patterns, vol. 2, no. 4, 2021.
14. [14] V. E. Orel, S. I. Korovin, J. Molnár, and V. B. Orel, “Non-linear dynamics theory and malignant melanoma,” Experimental Oncology, vol. 41, no. 4, pp. 291–299, 2019.
15. [15] M. S. Demirsoy, M. S. Sarıkaya, S. Kurnaz, Y. H. El Naser, M. A. Hazar, S. Coşkun, et al., “Enhancing stirring performance through chaotic position control in a magnetic stirrer,” Journal of the Brazilian Society of Mechanical Sciences and Engineering, vol. 47, no. 11, p. 580, 2025.
16. [16] Y. H. El Naser, B. Demirsoy, K. Erin, and M. S. Demirsoy, “Chaotic speed control of a DC motor using the Sprott-A system for robotic end-effector applications,” Black Sea Journal of Engineering and Science, vol. 8, no. 5, pp. 1406–1414, 2025.
17. [17] M. S. Sarıkaya, M. S. Demirsoy, and M. Ç. Kutlu, “Design of a chaotic speed-controlled mixing device and efficiency analysis in biogas,” Black Sea Journal of Engineering and Science, vol. 8, no. 3, pp. 672–679, 2025.
18. [18] H. Gao, H. Zhang, and L. Xiong, “Sliding mode control of Lorenz chaotic system of fluid thermal convection between infinite plates based on LMI,” Advances in Mechanical Engineering, vol. 15, no. 1, 2023.
19. [19] S. Vaidyanathan and S. Pakiriswamy, “A 3-D novel conservative chaotic system and its generalized projective synchronization via adaptive control,” Journal of Engineering Science & Technology Review, vol. 8, no. 2, 2015.
20. [20] M. Shafiq, I. Ahmad, O. A. Almatroud, and M. M. Al-Sawalha, “Robust attitude control of the three-dimensional unknown chaotic satellite system,” Transactions of the Institute of Measurement and Control, vol. 44, no. 7, pp. 1484–1504, 2022.
21. [21] X. Lu, “A financial chaotic system control method based on intermittent controller,” Mathematical Problems in Engineering, vol. 2020, Article ID 5810707, 2020.
22. [22] H. Tian, M. Zhao, J. Liu, Q. Wang, X. Yu, and Z. Wang, “Dynamic analysis and sliding mode synchronization control of chaotic systems with conditional symmetric fractional-order memristors,” Fractal and Fractional, vol. 8, no. 6, p. 307, 2024.
23. [23] L. Xiong, Y. J. Lu, Y. F. Zhang, X. G. Zhang, and P. Gupta, “Design and hardware implementation of a new chaotic secure communication technique,” PLoS ONE, vol. 11, no. 8, p. e0158348, 2016.
24. [24] J. Petrzela, “Chaos in analog electronic circuits: Comprehensive review, solved problems, open topics and small example,” Mathematics, vol. 10, no. 21, p. 4108, 2022.
25. [25] J. Zheng, Q. Zhang, Q. Xu, F. Xu, and V. Shi, “Synchronization of a supply chain model with four chaotic attractors,” Discrete Dynamics in Nature and Society, vol. 2022, Article ID 6390456, 2022.
26. [26] A. Biswas, S. S. Chaurasia, P. Parmananda, and S. Sinha, “Asymmetry induced suppression of chaos,” Scientific Reports, vol. 10, no. 1, p. 15582, 2020.
27. [27] H. P. Ren, C. Bai, K. Tian, and C. Grebogi, “Dynamics of delay induced composite multi-scroll attractor and its application in encryption,” International Journal of Non-Linear Mechanics, vol. 94, pp. 334–342, 2017.
28. [28] Y. Ye and J. He, “Constructing a new multi-scroll chaotic system and its circuit design,” Mathematics, vol. 12, no. 13, p. 1931, 2024.
29. [29] W. Gao and J. C. Hung, “Variable structure control of nonlinear systems: A new approach,” IEEE Transactions on Industrial Electronics, vol. 40, no. 1, pp. 45–55, 1993.
30. [30] M. S. Sarıkaya, O. Demirel, S. Kaçar, and A. Derdiyok, “Modelling and chaotic based parameter optimization of sliding mode controller,” Journal of Mathematical Sciences and Modelling, vol. 8, no. 2, pp. 42–55, 2025.