Skip to main content
SpringerLink
Log in

Analysis of atypical orbits in one-dimensional piecewise-linear discontinuous maps

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this paper, boundary regions of 1-D linear piecewise-smooth discontinuous maps are examined analytically. It is shown that, under certain parameter conditions, maps exhibit atypical orbits like a continuum of periodic orbits and quasi-periodic orbits. Further, we have derived the conditions under which such phenomenon occurs. The paper also illustrates that there exists a specific parameter region where as the parameter is varied, there is a transition from stable to unstable periodic orbits. Moreover, we have derived an expression for the value of parameter at which this transition from stable to unstable periodic orbits occurs. Additionally, the dynamics concerning this value of parameter is also given.

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

Similar content being viewed by others

Data Availability

Not applicable.

References

  1. Tse, C.K.: Flip bifurcation and chaos in three-state boost switching regulators. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 41(1), 16–23 (1994)

    Article  Google Scholar 

  2. Avrutin, V., Zhusubaliyev, Z.T.: Piecewise-linear map for studying border collision phenomena in dc/ac converters. International Journal of Bifurcation and Chaos 30(07), 2030015 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  3. Zhang, R., Wu, A., Wang, Z., Cang, S.: Chaotic and subharmonic oscillations in a dc-dc boost converter with pwm voltage-current hybrid controller and parallel mr load. Nonlinear Dyn. 99(2), 1321–1339 (2020)

    Article  Google Scholar 

  4. Avrutin, V., Zhusubaliyev, Z.T., Suissa, D., El Aroudi, A.: Non-observable chaos in piecewise smooth systems. Nonlinear Dynamics 99(3), 2031–2048 (2020)

    Article  Google Scholar 

  5. Souhail, W., Khammari, H., Mimouni, M.F.: Bifurcation surfaces and multi-stability analysis of state feedback control of pmsm. International Journal of Dynamics and Control 7(1), 276–294 (2019)

    Article  MathSciNet  Google Scholar 

  6. Carvalho, T., Cristiano, R., Rodrigues, D.S., Tonon, D.J.: Global analysis of a piecewise smooth epidemiological model of covid-19. Nonlinear Dyn. 105(4), 3763–3773 (2021)

    Article  Google Scholar 

  7. Zhusubaliyev, Z., Churilov, A., Medvedev, A.: Bifurcation phenomena in an impulsive model of non-basal testosterone regulation. Chaos An Interdiscip. J. Nonlinear Sci. (AIP) 22(1), 013121 (2012)

    Article  MathSciNet  Google Scholar 

  8. Hu, H.: Experimental observation of transition from chaotic bursting to chaotic spiking in a neural pacemaker. Chaos: An Interdiscip. J. Nonlinear Sci. (AIP) 23(2), 023126 (2013)

    Article  Google Scholar 

  9. Jeffrey, M., Dankowicz, H.: Discontinuity-induced bifurcation cascades in flows and maps with application to models of the yeast cell cycle. Physica D 271, 32–47 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bischi, G., Gardini, L., Tramontana, F.: Bifurcation curves in discontinuous map. Discr. Contin. Dyna. Syst. Series B 13(2), 249–267 (2010)

    MathSciNet  MATH  Google Scholar 

  11. Tramontana, F., Westerhoff, F., Gardini, L.: On the complicated price dynamics of a simple one-dimensional discontinuous financial market model with heterogeneous interacting traders. J. Econ. Behav. Organiz. 74(3), 187–205 (2010)

    Article  Google Scholar 

  12. Agliari, A., Rillosi, F., Vachadze, G.: Credit market imperfection, financial market globalization, and catastrophic transition. Mathematics and Computers in Simulation 108, 41–62 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Panchuk, A., Sushko, I., Westerhoff, F.: A financial market model with two discontinuities: Bifurcation structures in the chaotic domain. Chaos An Interdiscip. J. Nonlinear Sci. 28(5), 055908 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  14. Toniol Cardin, P.: Relaxation oscillation in planar discontinuous piecewise smooth fast–slow systems. Chaos An Interdiscip. J. Nonlinear Sci. 32(1), 013104 (2022)

    Article  MathSciNet  Google Scholar 

  15. Gardini, L., Sushko, I., Matsuyama, K.: 2d discontinuous piecewise linear map: Emergence of fashion cycles. Chaos An Interdiscip. J. Nonlinear Sci. 28(5), 055917 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  16. Avrutin, V., Zhusubaliyev, Z.T.: Nested closed invariant curves in piecewise smooth maps. International Journal of Bifurcation and Chaos 29(07), 1930017 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  17. Patra, M., Gupta, S., Banerjee, S.: Local and global bifurcations in 3d piecewise smooth discontinuous maps. Chaos An Interdiscip. J. Nonlinear Sci. 31(1), 013126 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  18. Guo, B., Liu, Y.: Three-dimensional map for a piecewise-linear capsule system with bidirectional drifts. Physica D: Nonlinear Phenomena 399, 95–107 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  19. di Bernardo, M., Budd, C., Champneys, A.: Normal form maps for grazing bifurcations in n-dimensional piecewise-smooth dynamical systems. Physica D: Nonlinear Phenomena 160(3), 222–254 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Augusto Cardoso Pereira F, Colli E, Carlos Sartorelli J (2012): Period adding cascades: experiment and modeling in air bubbling. Chaos An Interdiscip. J. Nonlinear Sci. 22(1), 013135

    Article  MathSciNet  Google Scholar 

  21. Biswas, D., Seth, S., Bor, M.: A study of the dynamics of a new piecewise smooth map. International Journal of Bifurcation and Chaos 30(01), 2050018 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhusubaliyev, Z.T., Mosekilde, E.: Multistability and hidden attractors in a multilevel dc/dc converter. Mathematics and Computers in Simulation 109, 32–45 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Avrutin, V., Jeffrey, M.R.: Remarks on bifurcations of hidden orbits in discontinuous maps (2020). Submitted to Nonlinearity

  24. Jeffrey, M.R., Webber, S.: The hidden unstable orbits of maps with gaps. Proceedings of the Royal Society A 476(2234), 20190473 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  25. Avrutin, V., Panchuk, A., Sushko, I.: Border collision bifurcations of chaotic attractors in one-dimensional maps with multiple discontinuities. Proceedings of the Royal Society A 477(2253), 20210432 (2021)

    Article  MathSciNet  Google Scholar 

  26. Avrutin, V., Gardini, L., Sushko, I., Tramontana, F.: Continuous and discontinuous piecewise-smooth one-dimensional maps: invariant sets and bifurcation structures. World Scientific, USA (2019)

    Book  MATH  Google Scholar 

  27. Simpson, D., Meiss, J.: Aspects of bifurcation theory for piecewise-smooth, continuous systems. Physica D: Nonlinear Phenomena 241(22), 1861–1868 (2012)

    Article  MathSciNet  Google Scholar 

  28. Jain, P., Banerjee, S.: Border collision bifurcation in one-dimensional discontinuous maps. International Journal of Bifurcation and Chaos 13(11), 3341–3351 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  29. Metri, R.A., Mounica, M., Rajpathak, B.A.: Characterization of 1d linear piecewise-smooth discontinuous map. In: 2020 IEEE First International Conference on Smart Technologies for Power, Energy and Control (STPEC), pp. 1–6, IEEE (2020)

  30. Sha, J., Xu, J., Bao, B., Yan, T.: Effects of circuit parameters on dynamics of current-mode-pulse-train-controlled buck converter. IEEE Transactions on Industrial Electronics 61(3), 1562–1573 (2013)

    Article  Google Scholar 

  31. Avrutin, V., Clüver, M., Mahout, V., Fournier-Prunaret, D.: Bandcount adding structure and collapse of chaotic attractors in a piecewise linear bimodal map. Physica D: Nonlinear Phenomena 309, 37–56 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. Avrutin, V., Sushko, I.: A gallery of bifurcation scenarios in piecewise smooth 1d maps. In: Global analysis of dynamic models in economics and finance, pp. 369–395. Springer (2013)

  33. Rajpathak, B., Pillai, H., Bandyopadhyay, S.: Analysis of stable periodic orbits in the 1-d linear piecewise-smooth discontinuous map. Chaos 22(3), 0331261–0331269 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  34. Rajpathak, B., Pillai, H.K., Bandyopadhyay, S.: Analysis of unstable periodic orbits and chaotic orbits in the one-dimensional linear piecewise-smooth discontinuous map. Chaos 25(10), 1031011–10310112 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. Qin, Z., Yang, J., Banerjee, S., Jiang, G.: Border collision bifurcations in a generalized piecewise-linear power map. Discrete and Continuous Dynamical Systems Series B 16(2), 547–567 (2011)

    MathSciNet  MATH  Google Scholar 

  36. Iñiguez, A., Ruiz Leal, B.: Atractores en funciones lineales crecientes por parte en la recta real. Revista Digital Novasinergia 4(2), 48–61 (2021)

    Google Scholar 

  37. Alligood, K., Sauer, T., Yorke, J.: Chaos: An Introduction to Dynamical Systems. Textbooks in Mathematical Sciences. New York, Springer (1997)

    Book  MATH  Google Scholar 

  38. Avrutin, V., Schanz, M.: On multi-parametric bifurcation in a scalar piecewise-linear map. Nonlinearity 19(3), 531–552 (2006)

  39. Gardini, L., Avrutin, V., Sushko, I.: Codimension-2 border collision, bifurcations in one-dimensional, discontinuous piecewise smooth maps. International Journal of Bifurcation and Chaos 24(02), 1450024 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

The authors declare that no funds, grants or other support were received during the preparation of this manuscript. The authors have no relevant financial or non-financial interests to disclose.

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, VNIT, Nagpur, Maharashtra, 440010, India

    Rajanikant Metri & Bhooshan Rajpathak

  2. Department of Electrical Engineering, IIT Bombay, Mumbai, 400076, India

    Harish Pillai

Authors
  1. Rajanikant Metri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bhooshan Rajpathak
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Harish Pillai
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed to the study conception, methods and analysis. Material preparation, writing original draft, editing and supervision were performed by RAM, Dr. BAR and Dr. HP.

Corresponding author

Correspondence to Bhooshan Rajpathak.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Metri, R., Rajpathak, B. & Pillai, H. Analysis of atypical orbits in one-dimensional piecewise-linear discontinuous maps. Nonlinear Dyn 111, 9395–9408 (2023). https://doi.org/10.1007/s11071-023-08333-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-023-08333-w

Keywords

Mathematics Subject Classification

Search

Navigation