Skip to main content
SpringerLink
Log in

Visualization of Four Normal Size Limit Cycles in Two-Dimensional Polynomial Quadratic System

  • Original Research
  • Published:
Differential Equations and Dynamical Systems Aims and scope Submit manuscript

Abstract

This paper is devoted to analytical and numerical investigation of limit cycles in two-dimensional polynomial quadratic systems. The appearance of modern computers permits one to use a numerical simulation of complicated nonlinear dynamical systems and to obtain new information on a structure of their trajectories. However the possibilities of naive approach, based on the construction of trajectories by numerical integration of the considered differential equations, turns out to be very limited. In the paper the effective analytical-numerical methods for investigation of limit cycles in two-dimensional polynomial quadratic system are discussed. Estimations of domains of parameters, corresponding to existence of different configurations of large limit cycles, are obtained and visualization of four large limit cycles in quadratic system is presented.

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

Similar content being viewed by others

References

  1. Perko L.: Differential Equations and Dynamical Systems, p. 553. Springer, New York (2001)

    Google Scholar 

  2. Li J.: Hilbert’s 16th problem and bifurcations of planar polynomial vector field. Int. J. Bifurc. Chaos 13(1), 47–106 (2003)

    Article  MATH  Google Scholar 

  3. Dumortier F., Llibre J., Artes J.: Qualitative Theory of Planar Differential Systems, p. 298. Springer-Verlag, Berlin (2006)

    Google Scholar 

  4. Christopher C., Li C.: Limit Cycles of Differential Equations, p. 171. Birkhauser, Basel (2007)

    Google Scholar 

  5. Bragin V.O., Vagaitsev V.I., Kuznetsov N.V., Leonov G.A.: Algorithms for finding hidden oscillations in nonlinear systems. The Aizerman and Kalman conjectures and Chua’s circuits. J. Comput. Syst. Sci. Int. 50(4), 511–543 (2011)

    Article  MathSciNet  Google Scholar 

  6. Leonov G.A., Kuznetsov N.V., Vagaitsev V.I.: Localization of hidden Chua’s attractors. Phys. Lett. A 375(23), 2230–2233 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Arnol’d, V.I.: Experimental Mathematics. Fazis, Moscow (2005) [in Russian]

  8. Leonov G.A.: Effective methods for investigation of limit cycles in dynamical systems. Appl. Math. Mech. 74(1), 37–73 (2010)

    MathSciNet  Google Scholar 

  9. Leonov G.A., Kuznetsova O.A.: Lyapunov quantities and limit cycles of two-dimensional dynamical systems. Analytical methods and symbolic computation. Regul. chaot. Dyn 15(2–3), 354–377 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kuznetsov N.V., Leonov G.A.: Lyapunov quantities, limit cycles and strange behavior of trajectories in two-dimensional quadratic systems. J. Vibroeng. 10(4), 460–467 (2008)

    Google Scholar 

  11. Leonov G.A., Kuznetsov N.V., Kudryashova E.V.: A direct method for calculating Lyapunov quantities of two-dimensional dynamical systems. Proc. Steklov Inst. Math. 272(Supplement 1), S119–S127 (2011)

    Article  Google Scholar 

  12. Leonov G.A., Kuznetsov N.V.: Computation of the first Lyapunov quantity for the second-order dynamical system. IFAC Proc. Vol. (IFACPapersOnline) 3((PART 1), 87–89 (2007)

    Article  Google Scholar 

  13. Leonov G.A., Kuznetsov N.V., Kudryashova E.V.: Cycles of two-dimensional systems: Computer calculations, proofs, and experiments. Vestnik St. Petersburg Univ. Math. 41(3), 216–250 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Shi S.L.: A concrete example of the existence of four limit cycles for plane quadratic systems. Sci. Sin. 23, 153–158 (1980)

    MATH  Google Scholar 

  15. Artes J.C., Llibre J.: Quadratic vector fields with a weak focus of third order. Publ. Math. 41, 7–39 (1997)

    MathSciNet  MATH  Google Scholar 

  16. Leonov G.A., Kuznetsov N.V.: Limit cycles of quadratic systems with a perturbed weak focus of order 3 and a saddle equilibrium at infinity. Doklady Math. 82(2), 693–696 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Leonov G.A.: Four limit cycles in quadratic two-dimensional systems with a perturbed first-order weak focus. Doklady Math. 81(2), 248–250 (2010)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Applied Cybernetics Chair, St. Petersburg State University, Universitetsky pr. 28, St. Petersburg, 198504, Russia

    N. V. Kuznetsov, O. A. Kuznetsova & G. A. Leonov

  2. Department of Mathematical Information Technology, University of Jyväskylä, P.O. Box 35, Agora, 40014, Finland

    N. V. Kuznetsov, O. A. Kuznetsova & G. A. Leonov

Authors
  1. N. V. Kuznetsov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. O. A. Kuznetsova
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. G. A. Leonov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. V. Kuznetsov.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kuznetsov, N.V., Kuznetsova, O.A. & Leonov, G.A. Visualization of Four Normal Size Limit Cycles in Two-Dimensional Polynomial Quadratic System. Differ Equ Dyn Syst 21, 29–34 (2013). https://doi.org/10.1007/s12591-012-0118-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12591-012-0118-6

Keywords

Search

Navigation