Dynamic Analysis of a Snap Oscillator Based on a Unique Diode Nonlinearity Effect, Offset Boosting Control and Sliding Mode Control Design for Global Chaos Synchronization

  • S. F. Takougang TchindaEmail author
  • G. Mpame
  • A. C. Nzeukou Takougang
  • V. Kamdoum Tamba


This paper focuses on the behavior of a specific class of oscillator (snap) under a unique diode effect by checking the complex dynamic of the proposed oscillator. From linear and nonlinear analysis methods, the numerical integration of the system such as, fixed-point and stability analysis, bifurcation diagrams, Kaplan–Yorke dimension, Lyapunov exponent spectrum, frequency spectra, Poincaré section and cross section of basins of attraction reveals that oscillator is chaotic and the chaotic robustness of the system depends on parameters changing. To the best knowledge of the authors, the system is the simplest in its category but the study revealed some interesting phenomena: asymmetric coexisting attractors, antimonotonicity phenomenon and even period-doubling bifurcation. These phenomena confer oscillating the quality of multistable or resistance to attacks in engineering application of data encryption. Furthermore, an offset boosting operation of a variable is used to control attractors and a robust sliding mode control for control engineering application of the system is designed to achieve global chaos synchronization based on Lyapunov stability theory. An appropriate electronic circuit or analog simulator is designed; PSpice simulations demonstrate feasible the proposed snap.


Snap oscillator Asymmetric coexisting attractors Offset boosting Circuit realization Sliding mode control Chaos synchronization 



  1. Akgul, A., Calgan, H., Koyuncu, I., Pehlivan, I., & Istanbullu, A. (2016). Chaos-based engineering applications with a 3D chaotic system without equilibrium points. Nonlinear Dynamics, 84, 481–495.MathSciNetGoogle Scholar
  2. Akgul, A., & Pehlivan, I. (2016). A new three dimensional chaotic system without equilibrium points, its dynamical analysis. Technical Gazette, 23, 209–214.Google Scholar
  3. Chen, Q. G., & Chen, G. R. (2008). A chaotic system with one saddle and two stable node-foci. International Journal of Bifurcation and Chaos, 18, 1393–1414.MathSciNetzbMATHGoogle Scholar
  4. Chen, G. R., & Ueta, T. (1999). Yet another chaotic attractor. International Journal of Bifurcation and Chaos, 9, 1465–1466.MathSciNetzbMATHGoogle Scholar
  5. Chunbiao, Li, Xiong, Wang, & Chen, Guanrong. (2017). Diagnosing multistability by offset boosting. Nonlinear Dynamics, 90, 1335–1341.MathSciNetGoogle Scholar
  6. Dalkiran, F. Y., & Sprott, J. C. (2016). Simple chaotic hyperjerk system. International Journal of Bifurcation and Chaos, 26, 1650189.MathSciNetGoogle Scholar
  7. Dawson, S. P., Grebogi, C., Yorke, J. A., Kan, I., & Kocak, H. (1992). Antimonotonicity: Inevitable reversals of period-doubling cascades. Physics Letters A, 162, 249–254.MathSciNetGoogle Scholar
  8. Dudkowski, D., Jafari, S., Kapitaniak, T., Kuznetsov, N., Leonov, G. A., & Prasad, A. (2016). Hidden attractors in dynamical systems. Physics Reports, 637, 1–50.MathSciNetzbMATHGoogle Scholar
  9. Elsonbaty, A. R., & El-Sayed, A. M. A. (2016). Further nonlinear dynamical analysis of simple jerk system with multiple attractors. Nonlinear Dynamics, 87, 1169–1186.zbMATHGoogle Scholar
  10. Hanias, M. P., Giannaris, G., & Spyridakis, A. R. (2006). Time series analysis in chaotic diode resonator circuit. Chaos, Solitons and Fractals, 27, 569–573.zbMATHGoogle Scholar
  11. Jay, P. S., & Roy, B. K. (2017). Multistability and hidden chaotic attractors in a new simple 4D chaotic system with chaotic 2-torus behavior. Int: International Journal of Dynamics and Control. Scholar
  12. Jinkun, L., & Xinhua, W. (2012). Advanced sliding mode control for mechanical systems. New York: Springer.zbMATHGoogle Scholar
  13. Kengne, J., Njitacke, Z. T., & Fotsin, H. (2016a). Dynamical analysis of a simple autonomous jerk system with multiple attractors. Nonlinear Dynamics, 83, 751–765.MathSciNetGoogle Scholar
  14. Kengne, J., Njitacke, Z. T., Negou, A. N., Tsostop, M. F., & Fotsin, H. B. (2016b). Coexistence of multiple attractors and crisis route to chaos in a novel chaotic jerk circuit. International Journal of Bifurcation and Chaos, 26(1650), 081. Scholar
  15. Kiers, K., Schmidt, D., & Sprott, J. C. (2004). Precision measurement of a simple chaotic circuit. American Journal of Physics, 72, 503–509.Google Scholar
  16. Klouverakis, K. E., & Sprott, J. C. (2006). Chaotic hyperjerk systems. Chaos, Solitons and Fractals, 28, 739–746.MathSciNetzbMATHGoogle Scholar
  17. Kocarev, L., Halle, K., Eckert, K., & Chua, L. (1993). Experimental observation of antimonotonicity in Chua’s circuit. International Journal of Bifurcation and Chaos, 3(3), 1051–1055.zbMATHGoogle Scholar
  18. Koyuncu, I., Ozcerit, A. T., & Pehlivan, I. (2013). An analog circuit design and FPGA-based implementation of the Burke-Shaw chaotic system. Optoelectronics and Advanced Materials-Rapid Communications, 7, 635–638.Google Scholar
  19. Kyprianidis, I., Stouboulos, I., Haralabidis, P., & Bountis, T. (2000). Antimonotonicity and chaotic dynamics in a fourth-order autonomous nonlinear electric circuit. International Journal of Bifurcation and Chaos, 10, 1903–1915.Google Scholar
  20. Leonov, G. A., & Kuznetsov, N. V. (2013). Hidden attractors in dynamical systems: from hidden oscillation in Hilbert-Kolmogorov, Aizerman and Kalman problems to hidden chaotic attractor in Chua circuits. International Journal of Bifurcation and Chaos, 23(1330), 002.MathSciNetzbMATHGoogle Scholar
  21. Leonov, G. A., Kuznetsov, N. V., Kuznetsova, O. A., Seldedzhi, S. M., & Vagaitsev, V. I. (2011a). Hidden oscillations in dynamical systems. Transactions on Control Systems, 6, 54–67.Google Scholar
  22. Leonov, G. A., Kuznetsov, N. V., & Vagaitsev, V. I. (2011b). Localization of hidden Chua’s attractors. Physics Letters A, 375, 2230–2233.MathSciNetzbMATHGoogle Scholar
  23. Leonov, G. A., Kuznetsov, N. V., & Vagaitsev, V. I. (2012). Hidden attractor in smooth Chua system. Physica D, 241, 1482–1486.MathSciNetzbMATHGoogle Scholar
  24. Letellier, C., & Gilmore, R. (2007). Symmetry groups for 3D dynamical systems. Journal of Physics A Mathematical and Theoretical, 40, 5597–5620.MathSciNetzbMATHGoogle Scholar
  25. Leutcho, G. D., Kengne, J., & Kamdjeu, Kengne L. (2018). Dynamical analysis of a novel autonomous 4-D hyperjerk circuit with hyperbolic sine nonlinearity: Chaos, antimonotonicity and a plethora of coexisting attractors. Chaos, Solitons and Fractals, 107, 67–87.MathSciNetzbMATHGoogle Scholar
  26. Linz, S. J. (2008). On hyperjerk systems. Chaos, Solitons and Fractals, 37, 741–747.MathSciNetzbMATHGoogle Scholar
  27. Lü, J., & Chen, G. R. (2002). A new chaotic attractor coined. International Journal of Bifurcation and Chaos, 12, 659–661.MathSciNetzbMATHGoogle Scholar
  28. Munmuangsaen, B. (2011). Elementary chaotic snap flows. Chaos, Solitons and Fractals, 44, 995–1003.Google Scholar
  29. Parlitz, U., & Lauterborn, W. (1985). Superstructure in the bifurcation set of the Duffing equation \( \ddot{x} + d \ddot{x} + \)x + x3 = f cos (ω t). Physics Letters A, 107, 351–355.MathSciNetGoogle Scholar
  30. Pehlivan, I., & Uyaroglu, Y. (2010). A new chaotic attractor from general Lorenz system family and its electronic experimental implementation. Turkish Journal of Electrical Engineering and Computer Sciences, 18, 171–184.Google Scholar
  31. Pham, V. T., Volos, C., Jafari, S., & Kapitaniak, T. (2017). Coexistence of hidden chaotic attractors in a novel no-equilibrium system. Nonlinear Dynamics, 87, 2001–2010.Google Scholar
  32. Rössler, O. (1976). An equation for continuous chaos. Physics Letters A, 57, 397–398.zbMATHGoogle Scholar
  33. Shaw, R. (1981). Strange attractor, chaotic behavior and information flow. Z. Naturforsch. A, 36, 60–112.MathSciNetGoogle Scholar
  34. Slotine, J., & Li, W. (1991). Applied nonlinear control. New Jersey: Prentice Hall.zbMATHGoogle Scholar
  35. Sprott, J. C. (1997). Some simple chaotic jerk functions. American Journal of Physics, 65, 537–543.Google Scholar
  36. Sprott, J. C. (2010). Elegant Chaos: Algebraically simple flow. Singapore: World Scientific Publishing.zbMATHGoogle Scholar
  37. Sprott, J. C. (2011a). A new chaotic jerk circuit. Transactions on Circuits and Systems II Express Briefs, 58, 240–243.Google Scholar
  38. Sprott, J. C. (2011b). A proposed standard for the publication of new chaotic systems. International Journal of Bifurcation and Chaos, 21(9), 2391–2394.Google Scholar
  39. Utkin, V. I. (1977). Variable structure systems with sliding modes. IEEE Transactions on Automatic Control, 22(2), 212–222.MathSciNetzbMATHGoogle Scholar
  40. Vaidyanathan, S., Akgul, A., Kaçar, S., & çavusoğlu, U. (2018). A new 4-D chaotic hyperjerk system, its synchronization, circuit design and applications in RNG, image encryption and chaos-based steganography. The European Physical Journal Plus, 133, 46.Google Scholar
  41. Vaidyanathan, S., Sampath, S., & Azar, A. T. (2015a). Global chaos synchronization of identical chaotic systems via novel sliding mode control method and its application to Zhu system. International Journal of Modelling, Identification and Control, 23, 92–100.Google Scholar
  42. Vaidyanathan, S., Volos, C., Pham, V. T., & Madhavan, K. (2015b). Analysis, adaptive control and synchronization of a novel 4-D hyperchaotic hyperjerk system and its SPICE implementation. Archives of Control Sciences, 25, 135–158.MathSciNetGoogle Scholar
  43. Van der Schrier, G., & Maas, L. R. M. (2000). The diffusionless Lorenz equations; Shil’nikov bifurcations and reduction to an explicit map. Physica D, 141, 19–36.MathSciNetzbMATHGoogle Scholar
  44. Wolf, A., Swift, J. B., Swinney, H. L., & Vastano, J. A. (1985). Determining Lyapunov exponents from time series. Physica D Nonlinear Phenomena, 16, 285–317.MathSciNetzbMATHGoogle Scholar
  45. Yang, Q. G., Wei, Z. C., & Chen, G. R. (2010). An unusual 3D autonomous quadratic chaotic system with two stable nodefoci. International Journal of Bifurcation and Chaos, 20, 1061–1083.MathSciNetzbMATHGoogle Scholar
  46. Yu, S., Lü, J., Leung, H., & Chen, G. (2005). Design and implementation of n-scroll chaotic attractors from a general Jerk circuit. IEEE Transactions on Circuits and Systems I Regular Papers, 52, 1459–1476.MathSciNetzbMATHGoogle Scholar
  47. Zeraoulia, E., & Sprott, J. C. (2013). Transformation of 4-D dynamical systems to hyperjerk form. Palestine Journal of Mathematics, 2, 38–45.MathSciNetzbMATHGoogle Scholar
  48. Zhusubaliyev, Z. T., & Mosekilde, E. (2015). Multistability and hidden attractors in a multilevel DC/DC converter. Mathematics and Computers in Simulation, 109, 32–45.MathSciNetGoogle Scholar
  49. Zhusubaliyev, Z. T., Mosekilde, E., Churilov, A. N., & Medvedev, A. (2015). Multistability and hidden attractors in an impulsive Goodwin oscillator with time delay. The European Physical Journal Special Topics, 224, 1519–1539.Google Scholar

Copyright information

© Brazilian Society for Automatics--SBA 2019

Authors and Affiliations

  1. 1.UR de Matière Condensée, d’Electronique et de Traitement du Signal(LAMACETS), Department of PhysicsUniversity of DschangDschangCameroun
  2. 2.UR d’Automatique et Informatique Appliquée (LAIA), Department of Electrical Engineering, IUT-FV BandjounUniversity of DschangDschangCameroun
  3. 3.Department of Telecommunication and Network Engineering, IUT-FVBandjoun, University of DschangDschangCameroun

Personalised recommendations