Skip to main content
SpringerLink
Log in

Dynamics, Circuitry Implementation and Control of an Autonomous Helmholtz Jerk Oscillator

  • Published:
Journal of Control, Automation and Electrical Systems Aims and scope Submit manuscript

Abstract

In this paper, an autonomous three-dimensional Helmholtz-type oscillator is designed based on conversion of an autonomous Helmholtz two-dimensional oscillator to a jerk oscillator. For a suitable choice of the parameters, the proposed autonomous Helmholtz jerk oscillator can generate Hopf bifurcation, bistable period-2 limit cycles, two types of one-scroll chaotic attractors and coexistence between period-3 limit cycle and one-scroll chaotic attractors. Using a weak modulation of a parameter of the proposed Helmholtz jerk oscillator, it is possible to destroy the coexisting attractors found and transform the proposed Helmholtz jerk oscillator to period-3 oscillations. Moreover using experiments and OrCAD-PSpice software, circuit implementation of the proposed autonomous Helmholtz jerk oscillator is realized in order to check the one-scroll chaotic attractors and the coexisting attractors found during the numerical simulations. Numerical and experimental/OrCAD-PSpice results have a good qualitative agreement. Finally, by adding two new parameters in the proposed autonomous Helmholtz jerk oscillator, it is possible to control the amplitude of the attractor and the largest Lyapunov exponent.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  • Argyris, J., & Andreadis, I. (2000). On the influence of noise on the coexistence of chaotic attractors. Chaos, Solitons & Fractals, 11, 941–946.

    Article  MathSciNet  MATH  Google Scholar 

  • Baer, T. (1986). Large-amplitude fluctuations due to longitudinal mode coupling in diode-pumped intracavity-doubled Nd:YAG lasers. The Journal of the Optical Society of America B, 3, 1175–1180.

    Article  Google Scholar 

  • Bao, B., Bao, H., Wang, N., Chen, M., & Xu, Q. (2017). Hidden extreme multistability in memristive hyperchaotic system. Chaos, Solitons & Fractals, 94, 102–111.

    Article  MathSciNet  MATH  Google Scholar 

  • Bao, B., Jiang, T., Xu, Q., Chen, M., Wu, H., & Hu, Y. (2016). Coexisting infinitely many attractors in active band-pass filter-based memristive circuit. Nonlinear Dynamics, 86, 1711–1723.

    Article  Google Scholar 

  • Benitez, S., Acho, L., & Guerra, R. J. R. (2006). Chaotification of the Van der Pol system using jerk architecture. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 89, 1088–1091.

    Article  Google Scholar 

  • Carbajal-Gomez, V., Tlelo-Cuautle, E., Fernandez, F., de la Fraga, L. G., & Sanchez-Lopez, C. (2014). Maximizing Lyapunov exponents in a chaotic oscillator by applying differential evolution. International Journal of Nonlinear Sciences and Numerical Simulation, 15, 11–17.

    Article  MathSciNet  MATH  Google Scholar 

  • de la Fraga, L. G., & Tlelo-Cuautle, E. (2014). Optimizing the maximum Lyapunov exponent and phase space portraits in multi-scroll chaotic oscillators. Nonlinear Dynamics, 76, 1503–1515.

    Article  Google Scholar 

  • de la Fraga, L. G., Tlelo-Cuautle, E., Carbajal-Gomez, V., & Munoz-Pacheco, J. (2012). On maximizing positive Lyapunov exponents in a chaotic oscillator with heuristics. Revistamexicana de fisica, 58, 274–281.

    Google Scholar 

  • del Río, E., Rodriguez Lozano, A., & Velarde, M. G. (1992). A prototype Helmholtz–Thompson nonlinear oscillator. AIP Review of Scientific Instruments, 63, 4208–4212.

    Article  Google Scholar 

  • Goswami, B. K., & Pisarchik, A. N. (2008). Controlling multistability by small periodic perturbation. International Journal of Bifurcation and Chaos, 18, 1645–1673.

    Article  MathSciNet  Google Scholar 

  • Gottlieb, H. P. W. (1996). What is the simplest Jerk function that gives chaos? American Journal of Physics, 64, 525–529.

    Article  Google Scholar 

  • Gottwald, J. A., Virgin, L. N., & Dowell, E. H. (1995). Routes to escape from an energy well. Journal of Sound and Vibration, 187, 133–144.

    Article  MathSciNet  MATH  Google Scholar 

  • Helmholtz, H. L. F. (1954). On the sensations of tone. As a physiological basis for the theory of music. New York: Dover Reprints.

    Google Scholar 

  • Jafari, S., Pham, V. T., & Kapitaniak, T. (2016). Multiscroll chaotic sea obtained from a simple 3D system without equilibrium. International Journal of Bifurcation and Chaos, 26, 1650031.

    Article  MathSciNet  MATH  Google Scholar 

  • Kang, I. S., & Leal, L. G. (1990). Bubble dynamics in time-periodic straining flows. Journal of Fluid Mechanics, 218, 41–69.

    Article  MathSciNet  MATH  Google Scholar 

  • Kengne, J., Njitacke, Z. T., & Fotsin, H. B. (2016). Dynamical analysis of a simple autonomous jerk system with multiple attractors. Nonlinear Dynamics, 83, 751–765.

    Article  MathSciNet  Google Scholar 

  • Kingni, S. T., Jafari, S., Simo, H., & Woafo, P. (2014). Three-dimensional chaotic autonomous system with only one stable equilibrium: Analysis, circuit design, parameter estimation, control, synchronization and its fractional-order form. The European Physical Journal Plus, 129, 76–91.

    Article  Google Scholar 

  • Li, C., & Sprott, J. (2013). Amplitude control approach for chaotic signals. Nonlinear Dynamics, 73, 1335–1341.

    Article  MathSciNet  MATH  Google Scholar 

  • Li, C., & Sprott, J. (2014). Finding coexisting attractors using amplitude control. Nonlinear Dynamics, 78, 2059–2064.

    Article  MathSciNet  Google Scholar 

  • Li, C., Sprott, J. C., Yuan, Z., & Li, H. (2015). Constructing chaotic systems with total amplitude control. International Journal of Bifurcation and Chaos, 25, 1530025.

    Article  MathSciNet  MATH  Google Scholar 

  • Louodop, P., Kountchou, M., Fotsin, H. B., & Bowong, S. (2014). Practical finite-time synchronization of jerk systems: Theory and experiment. Nonlinear Dynamics, 78, 597–607.

    Article  MathSciNet  MATH  Google Scholar 

  • Malasoma, J. M. (2000). What is the simplest dissipative chaotic jerk equation which is parity invariant? Physics Letters A, 264, 383–389.

    Article  MathSciNet  MATH  Google Scholar 

  • Mortu, S., Nofiele, B., & Marquié, P. (2007). On the use of multistability for image processing. Physics Letters A, 367, 192–198.

    Article  MATH  Google Scholar 

  • Munmuangsaen, B., Sprott, J. C., Thio, W. J. C., Buscarino, A., & Fortuna, L. (2015). A simple chaotic flow with a continuously adjustable attractor dimension. International Journal of Bifurcation and Chaos, 25, 1530036.

    Article  MathSciNet  MATH  Google Scholar 

  • Pham, V. T., Vaidyanathan, S., Volos, C., Jafari, S., & Kingni, S. T. (2016). A no-equilibrium hyperchaotic system with a cubic nonlinear term. Optik International Journal for Light and Electron Optics, 127, 3259–3265.

    Article  Google Scholar 

  • Pisarchik, A. N., & Feudel, U. (2014). Control of multistability. Physics Reports, 540(4), 167–218.

    Article  MathSciNet  MATH  Google Scholar 

  • Sprott, J. C. (1997a). Simplest dissipative chaotic flows. Physics Letters A, 228, 271–274.

    Article  MathSciNet  MATH  Google Scholar 

  • Sprott, J. C. (1997b). Some simple chaotic jerk functions. American Journal of Physics, 65, 537–543.

    Article  Google Scholar 

  • Sprott, J. C. (2000a). A new class of chaotic circuit. Physics Letters, 266, 19–23.

    Article  Google Scholar 

  • Sprott, J. C. (2000b). Simple chaotic systems and circuits. American Journal of Physics, 68, 758–763.

    Article  Google Scholar 

  • Sprott, J. C. (2011). A new chaotic Jerk circuit. IEEE Transactions on Circuits and Systems II: Express Briefs, 58, 240–243.

    Article  Google Scholar 

  • Spyrou, K. J., Cotton, B., & Cotton, Gurd. (2002). Analytical expressions of capsize boundary for a ship with roll bias in beam waves. Journal of Ship Research, 46, 167–174.

    Google Scholar 

  • Tamba, V. K., Fotsin, H. B., Kengne, J., Kapche Tagne, F., & Talla, P. K. (2015). Coupled inductors-based chaotic Colpitts oscillators: Mathematical modelling and synchronization issues. The European Physical Journal Plus, 130, 137–155.

    Article  Google Scholar 

  • Tamba, V. K., Kingni, S. T., Kuiate, G. F., Fotsin, H. B., & Talla, P. K. (2018a). Coexistence of attractors in autonomous Van der Pol–Duffing jerk oscillator: Analysis, chaos control and synchronisation in its fractional-order form. Pramana—Journal of Physics, 91, 1–12.

    Article  Google Scholar 

  • Tamba, V. K., Kuiate, G. F., Kingni, S. T., & Talla, P. K. (2018b). An autonomous Helmholtz like-jerk oscillator: Analysis, electronic circuit realization and synchronization issues. Nonlinear Dynamical Systems with Self-Excited and Hidden Attractors Studies in Systems, Decision and Control, 133, 203–227.

    MathSciNet  Google Scholar 

  • Thompson, J. M. T. (1989). Chaotic phenomena triggering the escape from a potential well. Proceedings of the Royal Society of London Series A, 421, 195–225.

    Article  MathSciNet  MATH  Google Scholar 

  • Thompson, J. M. T. (1997). Designing against capsize in beam seas: Recent advances and new insights. Applied Mechanics Reviews, 50, 307–325.

    Article  Google Scholar 

Download references

Acknowledgements

S.T.K. wishes to thank Dr. Viet-Thanh Pham (Faculty of Electrical & Electronics Engineering, Ton Duc Thang University, Vietnam) for interesting discussions.

Author information

Authors and Affiliations

  1. Institut de Mathématiques et de Sciences Physiques, Université d’Abomey-Calavi, BP: 613, Porto Novo, Benin

    Cyrille Ainamon & Jean Bio Chabi Orou

  2. Department of Mechanical, Petroleum and Gas Engineering, Faculty of Mines and Petroleum Industries, University of Maroua, P.O. Box 46, Maroua, Cameroon

    Sifeu Takougang Kingni

  3. Department of Telecommunication and Network Engineering, IUT-Fotso Victor of Bandjoun, University of Dschang, P. O. Box: 134, Bandjoun, Cameroon

    Victor Kamdoum Tamba

  4. Laboratory of Modelling and Simulation in Engineering, Biomimetics and Prototypes (LaMSEBP) and TWAS Research Unit, Department of Physics, Faculty of Science, University of Yaoundé I, Po. Box 812, Yaoundé, Cameroon

    Paul Woafo

Authors
  1. Cyrille Ainamon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sifeu Takougang Kingni
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Victor Kamdoum Tamba
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jean Bio Chabi Orou
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Paul Woafo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Victor Kamdoum Tamba.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ainamon, C., Kingni, S.T., Tamba, V.K. et al. Dynamics, Circuitry Implementation and Control of an Autonomous Helmholtz Jerk Oscillator. J Control Autom Electr Syst 30, 501–511 (2019). https://doi.org/10.1007/s40313-019-00463-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40313-019-00463-0

Keywords

Search

Navigation