Skip to main content

Limit Cycles for Piecewise Linear Differential Systems via Poincaré–Miranda Theorem

  • Conference paper
  • First Online:
Extended Abstracts Spring 2018

Part of the book series: Trends in Mathematics ((RPCRMB,volume 11))

  • 458 Accesses

Abstract

In Gasull and Mañosa (Periodic orbits of discrete and continuous dynamical systems via Poincaré–Miranda theorem, Preprint 2018 [2]), we develop an effective procedure to prove the existence, determine the number, and locate periodic orbits of dynamical systems of both discrete and continuous nature. It is based on the use of the Poincaré–Miranda theorem. This note presents one of the results obtained in that paper: a new example of piecewise linear differential system with three limit cycles.

The authors are supported by Ministry of Economy, Industry and Competitiveness, State Research Agency of the Spanish Government through grants MTM2016-77278-P (MINECO/AEI/FEDER, UE, first author) and DPI2016-77407-P (MINECO/AEI/FEDER, UE, second author). The first author is also supported by the grant 2017-SGR-1617 from AGAUR, Generalitat de Catalunya. The second author acknowledges the group’s research recognition 2017-SGR-388 from AGAUR, Generalitat de Catalunya.

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

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Similar content being viewed by others

References

  1. E. Freire, E. Ponce, F. Torres, The discontinuous matching of two planar linear foci can have three nested crossing limit cycles. Publ. Mat. 221–253 (2014)

    Article  MathSciNet  Google Scholar 

  2. A. Gasull, V. Mañosa, Periodic orbits of discrete and continuous dynamical systems via Poincaré–Miranda theorem. Discrete Contin. Dyn. Syst. Ser. B (submitted). See arXiv:1809.06208 [math.DS]

  3. S.M. Huan, X.S. Yang, On the number of limit cycles in general planar piecewise linear systems. Discrete Contin. Dyn. Syst. A 32, 2147–2164 (2012)

    Article  MathSciNet  Google Scholar 

  4. W. Kulpa, The Poincaré-Miranda theorem. Amer. Math. Month. 104, 545–550 (1997)

    MATH  Google Scholar 

  5. J. Llibre, E. Ponce, Three nested limit cycles in discontinuous piecewise linear differential systems with two zones. Dynam. Contin. Discrete Impuls. Syst. 19, 325–335 (2012)

    MathSciNet  MATH  Google Scholar 

  6. C. Miranda, Una osservazione su una teorema di Brouwer. Boll. Unione Mat. Ital. 3, 5–7 (1940)

    MATH  Google Scholar 

  7. H. Poincaré, Sur certaines solutions particulieres du probléme des trois corps. C.R. Acad. Sci. Paris 97, 251–252; and Bull. Astronomique 1(1884), 63–74 (1883)

    Google Scholar 

  8. H. Poincaré, Sur les courbes définies par une équation différentielle IV. J. Math. Pures Appl. 85, 151–217 (1886)

    MATH  Google Scholar 

  9. J. Stoer, R. Bulirsch, Introduction to Numerical Analysis (Springer, New York, 2002)

    Book  Google Scholar 

  10. M.N. Vrahatis, A short proof and a generalization of Miranda’s existence theorem. Proc. AMS 107, 701–703 (1989)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Armengol Gasull .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Gasull, A., Mañosa, V. (2019). Limit Cycles for Piecewise Linear Differential Systems via Poincaré–Miranda Theorem. In: Korobeinikov, A., Caubergh, M., Lázaro, T., Sardanyés, J. (eds) Extended Abstracts Spring 2018. Trends in Mathematics(), vol 11. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-25261-8_8

Download citation

Publish with us

Policies and ethics