Skip to main content
SpringerLink
Log in

An Autonomous Helmholtz Like-Jerk Oscillator: Analysis, Electronic Circuit Realization and Synchronization Issues

  • Chapter
  • First Online:
Nonlinear Dynamical Systems with Self-Excited and Hidden Attractors

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 133))

Abstract

This chapter introduces an autonomous self-exited three-dimensional Helmholtz like oscillator which is built by converting the well know autonomous Helmholtz two-dimensional oscillator to a jerk oscillator. Basic properties of the proposed Helmholtz like-jerk oscillator such as dissipativity, equilibrium points and stability are examined. The dynamics of the proposed jerk oscillator is investigated by using bifurcation diagrams, Lyapunov exponent plots, phase portraits, frequency spectra and cross-sections of the basin of attraction. It is found that the proposed jerk oscillator exhibits some interesting phenomena including Hopf bifurcation, period-doubling bifurcation, reverse period-doubling bifurcation and hysteretic behaviors (responsible of the phenomenon of coexistence of multiple attractors). Moreover, the physical existence of the chaotic behavior and the coexistence of multiple attractors found in the proposed autonomous Helmholtz like-jerk oscillator are verified by some laboratory experimental measurements. A good qualitative agreement is shown between the numerical simulations and the experimental results. In addition, the synchronization of two identical coupled Helmholtz like-jerk oscillators is carried out using an extended backstepping control method. Based on the considered approach, generalized weighted controllers are designed to achieve synchronization in chaotic Helmholtz like-jerk oscillators. Numerical simulations are performed to verify the feasibility of the synchronization method. The approach followed in this chapter shows that by combining both numerical and experimental techniques, one can gain deep insight about the dynamics of chaotic systems exhibiting hysteretic behavior.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Azar AT, Vaidyanathan S (2015) Chaos modeling and control systems design. Springer, Berlin

    Book  MATH  Google Scholar 

  • Balibrea F, Chacon R, Lopez MA (1998) Inhibition of chaotic escape by an additional driven term. Int J Bifurcat Chaos 8:1719–1724

    Article  MathSciNet  MATH  Google Scholar 

  • Banerjee R (2010) Chaos Synchronization and Cryptography for Secure communications. IGI Global, USA

    Google Scholar 

  • Benitez MS, Zuppa LA, Guerra RJR (2006) Chaotification of the Van der Pol system using Jerk architecture. IEICE Trans. Fundam. 89-A, 375–378

    Google Scholar 

  • Blekhman II (1988) Synchronization in science and technology. AMSE Press, New York

    Google Scholar 

  • Brun E, Derighetti B, Meier D, Holzner R, Ravani M (1985) Observation of order and chaos in a nuclear spin-flip laser. J. Opt. Soc. Am. B 2:156–167

    Google Scholar 

  • Buscarino A, Fortuna L, Frasca M (2009) Experimental robust synchronization of hyperchaotic circuits. Physica D 238:1917–1922

    Article  MATH  Google Scholar 

  • Del Río E, Rodriguez Lozano A, Velarde MG (1992) A prototype Helmholtz-Thompson nonlinear oscillator. AIP Rev Sci Instrum 63:4208–4212

    Article  Google Scholar 

  • Dalkiran FY, Sprott JC (2016) Simple chaotic hyperjerk system. Int J Bifurcat Chaos 26:1650189

    Article  MathSciNet  Google Scholar 

  • Foss J, Longtin A, Mensour B, Milton J (1996) Multistability and delayed recurrent loops. Phys Rev Lett 76:708–711

    Article  Google Scholar 

  • Frederickson P, Kaplan JL, Yorke HL et al (1983) The Lyapunov dimension of strange attractor. J Differ Equ 49:185–207

    Article  MATH  Google Scholar 

  • Gottwald JA, Virgin LN, Dowell EH (1995) Routes to escape from an energy well. J Sound Vib 187:133–144

    Article  MathSciNet  MATH  Google Scholar 

  • Hampton A, Zanette HD (1999) Measure synchronization in coupled Hamiltonian systems. Phys Rev Lett 83:2179–2182

    Article  Google Scholar 

  • Hilborn RC (1994) Chaos and nonlinear dynamics: an introduction for scientists and engineers. Oxford University Press, Oxford, UK

    MATH  Google Scholar 

  • Hilborn RC (2001) Chaos and nonlinear dynamics: an introduction for scientists and engineers, 2nd edn. Oxford University Press, Oxford, UK

    MATH  Google Scholar 

  • Junge L, Parlitz U (2000) Synchronization of coupled Ginzburg-Landau equations using local potential. Phys Rev E 61:3736–3742

    Article  Google Scholar 

  • Kamdoum Tamba V, Fotsin HB, Kengne J, Megam Ngouonkadi EB, Talla PK (2016) Emergence of complex dynamical behaviors in improved Colpitts oscillators: antimonotonicity, coexisting attractors, and metastable chaos. Int J Dyn Control. https://doi.org/10.1007/s40435-016-0223-4

  • Kang IS, Leal LG (1990) Bubble dynamics in time-periodic straining flows. J Fluid Mech 218:41–69

    Article  MathSciNet  MATH  Google Scholar 

  • Kengne J, Njitacke ZT, Nguomkam Negou A, Fouodji Tsostop M, Fotsin HB (2016) Coexistence of multiple attractors and crisis route to chaos in a novel chaotic jerk circuit. Int J Bifurcat Chaos 26:1650081

    Article  MATH  Google Scholar 

  • Kengne J, Folifack Signing VR, Chedjou JC, Leutcho GD (2017) Nonlinear behavior of a novel chaotic jerk system: antimonotonicity, crises, and multiple coexisting attractors. Int J Dyn Control 1–18

    Google Scholar 

  • Kengne J, Nguomkam Negou A, Tchiotsop D, Kamdoum Tamba V, Kom GH (2018) On the dynamics of chaotic systems with multiple attractors: a case study. In: Recent advances in nonlinear dynamics and synchronization, studies in systems, decision and control, vol. 109. Springer

    Google Scholar 

  • Koyuncu I, Ozecerit AT, Pehlivan I (2014) Implementation of FPGA-based real time novel chaotic oscillator. Nonlinear Dyn 77:49–59

    Article  MathSciNet  Google Scholar 

  • Krstic M, Kanellakopoulus I, Kokotovic PO (1995) Nonlinear and adaptive control design. Wiley, New York

    Google Scholar 

  • Lakshmanan M, Rajasekhar S (2003) Nonlinear dynamics: integrability, chaos, and patterns. Springer, Berlin

    Book  Google Scholar 

  • Lenci S, Rega G (2001) Optimal control of homoclinic bifurcation in a periodically driven Helmholtz oscillator. In: Proceedings of the ASME design engineering technical conference, Pittsburgh, Pennsylvania, USA

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Luo X, Small M (2007) On a dynamical system with multiple chaotic attractors. Int J Bifurcat Chaos 17(9):3235–3251

    Article  MathSciNet  MATH  Google Scholar 

  • Ma J, Wu X, Chu R et al (2014) Selection of multi-scroll attractors in Jerk circuits and their verification using Pspice. Nonlinear Dyn 76:1951–1962

    Article  Google Scholar 

  • Mascolo S (1997) Backstepping design for controlling Lorenz chaos. In: Proceedings of the 36th IEEE conference on decision and control, San Diego, California, USA, pp 1500–1501

    Google Scholar 

  • Maurer J, Libchaber A (1980) Effect of the Prandtl number on the onset of turbulence in liquid-He-4. J Phys Lett 41:515–518

    Article  Google Scholar 

  • Nagaev RF (2003) Dynamics of synchronizing systems. Springer, Berlin-Heidelberg

    Book  Google Scholar 

  • Njah AN (2010) Tracking control and synchronization of the new hyperchaotic Liu system via backstepping techniques. Nonlinear Dyn 61:1–9

    Article  MathSciNet  MATH  Google Scholar 

  • Olusola OI, Vincent UE, Njah AN, Ali E (2011) Control and synchronization of chaos in biological systems via backstepping design. Int J Nonlinear Sci 11:121–128

    MathSciNet  MATH  Google Scholar 

  • Onma OS, Olusola OI, Njah AN (2014) Control and synchronization of chaotic and hyperchaotic Lorenz systems via extended backstepping techniques. J Nonlinear Dyn, ID 861727

    Google Scholar 

  • Pecora LM, Carrol TL (1990) Synchronization in chaotic systems. Phys Rev Lett 64:821–824

    Article  MathSciNet  MATH  Google Scholar 

  • Pikovsky A, Rosemblum M, Kurths J (2001) Synchronization: a universal concept in nonlinear science. Cambridge University Press, New York

    Book  Google Scholar 

  • Pisarchik AN, Feudel U (2014) Control of multistability. Phys Rep 540(4):167–218

    Article  MathSciNet  MATH  Google Scholar 

  • Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical recipes in Fortran 77. Press, Cambridge U

    MATH  Google Scholar 

  • Rosemblum MG, Pikovsky AS, Kurths J (1997) From phase to lag synchronization in coupled chaotic oscillators. Phys Rev Lett 78:4193–4196

    Article  Google Scholar 

  • Rullkov NF, Sushchik MM, Tsimring LS, Abarbanel HDI (1995) Generalized synchronization of chaos in directionally coupled chaotic systems. Phys Rev E 51:980–994

    Article  Google Scholar 

  • Soliman MS, Thompson JMT (1989) Integrity measures quantifying the erosion of smooth and fractal basins of attraction. J Sound Vib 135:453–475

    Article  MathSciNet  MATH  Google Scholar 

  • Sprott JC (2000a) A new class of chaotic circuit. Phys Lett A 266:19–23

    Article  Google Scholar 

  • Sprott JC (2000b) Simple chaotic systems and circuits. Am J Phys 68:758–763

    Article  Google Scholar 

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

    Google Scholar 

  • Strogatz SH (1994) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Perseus Books, Massachussetts

    MATH  Google Scholar 

  • Thompson JMT, Stewart HB (1986) Nonlinear dynamics and chaos. Wiley, Chichester

    MATH  Google Scholar 

  • Thompson JMT (1989) Chaotic phenomena triggering the escape from a potential well. Proc R Soc Lond A 421:195–225

    Article  MathSciNet  MATH  Google Scholar 

  • Thompson JMT, Rainey RCT, Soliman MS (1990) Ship stability criteria based on chaotic transients from incursive fractals. Philos Trans R Soc Lond A 332:149–167

    Article  MathSciNet  MATH  Google Scholar 

  • Thompson JMT (1997) Designing against capsize in beam seas: recent advances and new insights. Appl Mech Rev 50:307–325

    Article  Google Scholar 

  • Trejo-Guerra R, Tlelo-Cuautle E, Jimenez-Fuentes JM, Sanchez-Lopez C, Munoz-Pacheco JM, Espinosa-Flores-Verdad G, Rocha-Perez JM (2012) Integrated circuit generating 3- and 5- scroll attractors. Commun Nonlinear Sci Numer Simul 17:4328–4335

    Article  MathSciNet  Google Scholar 

  • Vincent UE, Njah AN, Akinlade O, Solarin ART (2005) Synchronization of cross-well chaos in coupled duffing oscillators. Int J Mod Phys B 19:3205–3216

    Article  MATH  Google Scholar 

  • Vaidyanathan S, Pham VT, Volos CK (2016) Adaptive backstepping control, synchronization and circuit simulation of a novel jerk chaotic system with a quartic nonlinearity. In: Advances and applications in chaotic systems, studies in computational intelligence. Springer

    Google Scholar 

  • Volos CK, Kyprianidis IM, Stouboulus INA (2012) Chaotic path planning generator for autonomous mobile robots. Robot Auton Syst 60:651–656

    Article  Google Scholar 

  • Volos CK, Kyprianidis IM, Stouboulus IN (2013a) Image encryption process based onchaotic synchronization phenomena. Signal Process 93:1328–1340

    Article  Google Scholar 

  • Volos CK, Kyprianidis IM, Stouboulus IN (2013b) Experimental investigation on coverage performance of a chaotic autonomous mobile robot. Robot Auton Syst 61:1314–1322

    Article  Google Scholar 

  • Voss HU (2000) Anticipating chaotic synchronization. Phys Rev E 61:5115–5119

    Article  Google Scholar 

  • Wolf A, Swift JB, Swinney HL, Wastano JA (1985) Determining Lyapunov exponents from time series. Phys D 16:285–317

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Victor Kamdoum Tamba

  2. Department of Physics, Higher Teacher Training College, University of Bamenda, P. O. Box 39, Bamenda, Cameroon

    Gaetan Fautso Kuiate

  3. Department of Mechanical and Electrical Engineering, Institute of Mines and Petroleum Industries, University of Maroua, P. O. Box 46, Maroua, Cameroon

    Sifeu Takougang Kingni

  4. Laboratory of Mechanics and Modelling of Physical Systems (L2MPS), Department of Physics, Faculty of Science, University of Dschang, P. O. Box: 67, Dschang, Cameroon

    Pierre Kisito Talla

Authors
  1. Victor Kamdoum Tamba
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gaetan Fautso Kuiate
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sifeu Takougang Kingni
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Pierre Kisito Talla
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Victor Kamdoum Tamba .

Editor information

Editors and Affiliations

  1. School of Electronics and Telecommunications, Hanoi University of Science and Technology, Hanoi, Vietnam

    Viet-Thanh Pham

  2. Research and Development Centre, Vel Tech University, Chennai, Tamil Nadu, India

    Sundarapandian Vaidyanathan

  3. Department of Physics, Aristotle University of Thessaloniki, Thessaloniki, Greece

    Christos Volos

  4. Division of Dynamics, Faculty of Mechanical Engineering, Lodz University of Technology, Łódź, Poland

    Tomasz Kapitaniak

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Tamba, V.K., Kuiate, G.F., Kingni, S.T., Talla, P.K. (2018). An Autonomous Helmholtz Like-Jerk Oscillator: Analysis, Electronic Circuit Realization and Synchronization Issues. In: Pham, VT., Vaidyanathan, S., Volos, C., Kapitaniak, T. (eds) Nonlinear Dynamical Systems with Self-Excited and Hidden Attractors. Studies in Systems, Decision and Control, vol 133. Springer, Cham. https://doi.org/10.1007/978-3-319-71243-7_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-71243-7_9

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-71242-0

  • Online ISBN: 978-3-319-71243-7

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation