IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v148y2021ics0960077921003325.html
   My bibliography  Save this article

Multistability, quasiperiodicity and chaos in a self-oscillating ring dynamical system with three degrees of freedom based on the van der Pol generator

Author

Listed:
  • Astakhov, Sergey
  • Astakhov, Oleg
  • Fadeeva, Natalia
  • Astakhov, Vladimir

Abstract

In this paper, we consider a self-oscillating ring system consisting of a nonlinear amplifier and a chain of three linear oscillators. The equations for this system are derived and the results of numerical simulations are presented. The inherent regimes of the self-oscillating system are studied versus the control parameters, and the results of bifurcation analysis are described. It was found that the addition of linear oscillators to the feedback loop of the classical van der Pol generator leads to the appearance of quasiperiodic and chaotic regimes, as well as the formation of multistability. The bifurcation mechanism of the multistability formation determined by two supercritical Andronov - Hopf bifurcations of the steady state and the subcritical Neimark - Sacker bifurcation of the saddle limit cycle is revealed. Transitions to chaos are investigated for each family of multistable states. Transition to chaos is observed through a sequence of doubling bifurcations of torus in a wide range of values of the control parameters.

Suggested Citation

  • Astakhov, Sergey & Astakhov, Oleg & Fadeeva, Natalia & Astakhov, Vladimir, 2021. "Multistability, quasiperiodicity and chaos in a self-oscillating ring dynamical system with three degrees of freedom based on the van der Pol generator," Chaos, Solitons & Fractals, Elsevier, vol. 148(C).
    • Handle: RePEc:eee:chsofr:v:148:y:2021:i:c:s0960077921003325
    DOI: 10.1016/j.chaos.2021.110978
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077921003325
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2021.110978?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Marcel Tigges & Tatiana T. Marquez-Lago & Jörg Stelling & Martin Fussenegger, 2009. "A tunable synthetic mammalian oscillator," Nature, Nature, vol. 457(7227), pages 309-312, January.
    2. Jesse Stricker & Scott Cookson & Matthew R. Bennett & William H. Mather & Lev S. Tsimring & Jeff Hasty, 2008. "A fast, robust and tunable synthetic gene oscillator," Nature, Nature, vol. 456(7221), pages 516-519, November.
    3. Kazuki Horikawa & Kana Ishimatsu & Eiichi Yoshimoto & Shigeru Kondo & Hiroyuki Takeda, 2006. "Noise-resistant and synchronized oscillation of the segmentation clock," Nature, Nature, vol. 441(7094), pages 719-723, June.
    4. Clément Sayrin & Igor Dotsenko & Xingxing Zhou & Bruno Peaudecerf & Théo Rybarczyk & Sébastien Gleyzes & Pierre Rouchon & Mazyar Mirrahimi & Hadis Amini & Michel Brune & Jean-Michel Raimond & Serge Ha, 2011. "Real-time quantum feedback prepares and stabilizes photon number states," Nature, Nature, vol. 477(7362), pages 73-77, September.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Huang, Juanjuan & Bi, Qinsheng, 2023. "Multiple modes of bursting phenomena in a vector field of triple Hopf bifurcation with two time scales," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Zhou, Peipei & Cai, Shuiming & Liu, Zengrong & Chen, Luonan & Wang, Ruiqi, 2013. "Coupling switches and oscillators as a means to shape cellular signals in biomolecular systems," Chaos, Solitons & Fractals, Elsevier, vol. 50(C), pages 115-126.
    2. Lucia Marucci & David A W Barton & Irene Cantone & Maria Aurelia Ricci & Maria Pia Cosma & Stefania Santini & Diego di Bernardo & Mario di Bernardo, 2009. "How to Turn a Genetic Circuit into a Synthetic Tunable Oscillator, or a Bistable Switch," PLOS ONE, Public Library of Science, vol. 4(12), pages 1-10, December.
    3. Miles Miller & Marc Hafner & Eduardo Sontag & Noah Davidsohn & Sairam Subramanian & Priscilla E M Purnick & Douglas Lauffenburger & Ron Weiss, 2012. "Modular Design of Artificial Tissue Homeostasis: Robust Control through Synthetic Cellular Heterogeneity," PLOS Computational Biology, Public Library of Science, vol. 8(7), pages 1-18, July.
    4. Sara Hooshangi & William E Bentley, 2011. "LsrR Quorum Sensing “Switch” Is Revealed by a Bottom-Up Approach," PLOS Computational Biology, Public Library of Science, vol. 7(9), pages 1-11, September.
    5. Deschamps, Julien, 2015. "Continuous-time limit of repeated interactions for a system in a confining potential," Stochastic Processes and their Applications, Elsevier, vol. 125(1), pages 327-342.
    6. Shunsuke Kawasaki & Hiroki Ono & Moe Hirosawa & Takeru Kuwabara & Shunsuke Sumi & Suji Lee & Knut Woltjen & Hirohide Saito, 2023. "Programmable mammalian translational modulators by CRISPR-associated proteins," Nature Communications, Nature, vol. 14(1), pages 1-17, December.
    7. Mayu Sugiyama & Takashi Saitou & Hiroshi Kurokawa & Asako Sakaue-Sawano & Takeshi Imamura & Atsushi Miyawaki & Tadahiro Iimura, 2014. "Live Imaging-Based Model Selection Reveals Periodic Regulation of the Stochastic G1/S Phase Transition in Vertebrate Axial Development," PLOS Computational Biology, Public Library of Science, vol. 10(12), pages 1-16, December.
    8. Zhdanov, Vladimir P., 2012. "Periodic perturbation of genetic oscillations," Chaos, Solitons & Fractals, Elsevier, vol. 45(5), pages 577-587.
    9. H. Dehne & A. Reitenbach & A. R. Bausch, 2021. "Reversible and spatiotemporal control of colloidal structure formation," Nature Communications, Nature, vol. 12(1), pages 1-9, December.
    10. T. Brown & E. Doucet & D. Ristè & G. Ribeill & K. Cicak & J. Aumentado & R. Simmonds & L. Govia & A. Kamal & L. Ranzani, 2022. "Trade off-free entanglement stabilization in a superconducting qutrit-qubit system," Nature Communications, Nature, vol. 13(1), pages 1-7, December.
    11. Kaslik, Eva & Rădulescu, Ileana Rodica, 2022. "Stability and bifurcations in fractional-order gene regulatory networks," Applied Mathematics and Computation, Elsevier, vol. 421(C).
    12. Evgeni V Nikolaev & Eduardo D Sontag, 2016. "Quorum-Sensing Synchronization of Synthetic Toggle Switches: A Design Based on Monotone Dynamical Systems Theory," PLOS Computational Biology, Public Library of Science, vol. 12(4), pages 1-33, April.
    13. Lukas Aufinger & Johann Brenner & Friedrich C. Simmel, 2022. "Complex dynamics in a synchronized cell-free genetic clock," Nature Communications, Nature, vol. 13(1), pages 1-9, December.
    14. Singh, Vijai & Chaudhary, Dharmendra Kumar & Mani, Indra & Dhar, Pawan Kumar, 2016. "Recent advances and challenges of the use of cyanobacteria towards the production of biofuels," Renewable and Sustainable Energy Reviews, Elsevier, vol. 60(C), pages 1-10.
    15. Lucas Henrion & Juan Andres Martinez & Vincent Vandenbroucke & Mathéo Delvenne & Samuel Telek & Andrew Zicler & Alexander Grünberger & Frank Delvigne, 2023. "Fitness cost associated with cell phenotypic switching drives population diversification dynamics and controllability," Nature Communications, Nature, vol. 14(1), pages 1-13, December.
    16. Samanthe M Lyons & Wenlong Xu & June Medford & Ashok Prasad, 2014. "Loads Bias Genetic and Signaling Switches in Synthetic and Natural Systems," PLOS Computational Biology, Public Library of Science, vol. 10(3), pages 1-16, March.
    17. Alan Veliz-Cuba & Andrew J Hirning & Adam A Atanas & Faiza Hussain & Flavia Vancia & Krešimir Josić & Matthew R Bennett, 2015. "Sources of Variability in a Synthetic Gene Oscillator," PLOS Computational Biology, Public Library of Science, vol. 11(12), pages 1-23, December.
    18. Betz, Ulrich A.K. & Arora, Loukik & Assal, Reem A. & Azevedo, Hatylas & Baldwin, Jeremy & Becker, Michael S. & Bostock, Stefan & Cheng, Vinton & Egle, Tobias & Ferrari, Nicola & Schneider-Futschik, El, 2023. "Game changers in science and technology - now and beyond," Technological Forecasting and Social Change, Elsevier, vol. 193(C).
    19. Yuqian Tang & Debin Qin & Zhexian Tian & Wenxi Chen & Yuanxi Ma & Jilong Wang & Jianguo Yang & Dalai Yan & Ray Dixon & Yi-Ping Wang, 2023. "Diurnal switches in diazotrophic lifestyle increase nitrogen contribution to cereals," Nature Communications, Nature, vol. 14(1), pages 1-16, December.
    20. Chelsea Y. Hu & Richard M. Murray, 2022. "Layered feedback control overcomes performance trade-off in synthetic biomolecular networks," Nature Communications, Nature, vol. 13(1), pages 1-13, December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:148:y:2021:i:c:s0960077921003325. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.