Centers and isochronous centers of a class of quasi-analytic switching systems



Abstract In this paper, we study the integrability and linearization of a class of quadratic quasi-analytic switching systems. We improve an existing method to compute the focus values and periodic constants of quasi-analytic switching systems. In particular, with our method, we demonstrate that the dynamical behaviors of quasi-analytic switching systems are more complex than those of continuous quasi-analytic systems, by showing the existence of six and seven limit cycles in the neighborhood of the origin and infinity, respectively, in a quadratic quasi-analytic switching system. Moreover, explicit conditions are obtained for classifying the centers and isochronous centers of the system.


quasi-analytic switching systems Lyapunov constant limit cycle center isochronous center 


34C07 34C23 


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This work was supported by National Natural Science Foundation of China (Grant Nos. 11371373 and 11601212), Applied Mathematics Enhancement Program of Linyi University and the Natural Science and Engineering Research Council of Canada (Grant No. R2686A02).


  1. 1.
    Amelikin B, Lukashivich H, Sadovski A. Nonlinear Oscillations in Second Order Systems (in Russian). Minsk: BI Press, 1982Google Scholar
  2. 2.
    Bautin N N. On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type. Mat Sb, 1952, 30: 181—196; Amer Math Soc Transl, 1954, 100: 1—19Google Scholar
  3. 3.
    Chen X, Du Z. Limit cycles bifurcate from centers of discontinuous quadratic systems. Comput Math Appl, 2010, 59: 3836–3848MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Du Z, Zhang W. Melnikov method for homoclinic bifurcation in nonlinear impact oscillators. Comput Math Appl, 2005, 50: 445–458MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dulac H. Determination et integration d’une certaine classe d’equations differentielle ayant par point singulier un centre. Bull Sci Math, 1908, 32: 230–252MATHGoogle Scholar
  6. 6.
    Fredriksson M H, Nordmark A B. On normal form calculations in impact oscillators. Proc R Soc Lond Ser A Math Phys Eng Sci, 2000, 456: 315–329MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Freire E. The focus-center-limit cycle bifurcation in symmetric 3D piecewise linear systems. SIAM J Appl Math, 2005, 65: 1933–1951MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gasull A, Torregrosa J. Center-focus problem for discontinuous planar differential equations. Internat J Bifur Chaos, 2003, 13: 1755–1765MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hauer B, Engelhardt A P, Taubner T. Quasi-analytical model for scattering infrared near-field microscopy on layered systems. Optics Express, 2012, 12: 13173–13188Google Scholar
  10. 10.
    Kapteyn W. On the midpoints of integral curves of differential equations of the first degree (in Dutch). Nederl Akad Wetensch Verslag Afd Natuurk Konikl Nederland, 1911, 2: 1446–1457Google Scholar
  11. 11.
    Kapteyn W. New investigations on the midpoints of integrals of differential equations of the first degree (in Dutch). Nederl Akad Wetensch Verslag Afd Natuurk Konikl Nederland, 1912, 20: 1354–1365Google Scholar
  12. 12.
    Kukucka P. Melnikov method for discontinuous planar systems. Nonlinear Anal, 2007, 66: 2698–2719MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Leine R I, Nijmeijer H. Dynamics and Bifurcations of Nonsmooth Mechanical Systems. Lecture Notes in Applied and Computational Mechanics, vol. 18. Berlin: Springer-Verlag, 2004Google Scholar
  14. 14.
    Li F, Yu P, Tian Y, et al. Center and isochronous center conditions for switching systems associated with elementary singular points. Commun Nonlinear Sci Numer Simul, 2015, 28: 81–97MathSciNetCrossRefGoogle Scholar
  15. 15.
    Liu Y. The generalized focal values and bifurcations of limit circles for quasi-quadratic system (in Chinese). Acta Math Sinica Chin Ser, 2002, 45: 671–682MathSciNetGoogle Scholar
  16. 16.
    Liu Y, Li J. Center and isochronous center problems for quasi analytic systems. Acta Math Sin Engl Ser, 2012, 24: 1569–1582MathSciNetGoogle Scholar
  17. 17.
    Liu Y, Li J, Huang W. Singular Point Values, Center Problem and Bifurcations of Limit Circles of Two Dimensional Differential Autonomous Systems. Beijing: Science Press, 2009Google Scholar
  18. 18.
    Llibre J, Lopes B D, de Moraes J R. Limit cycles for a class of continuous and discontinuous cubic polynomial differential systems. Qual Theory Dyn Syst, 2014, 13: 129–148MathSciNetCrossRefGoogle Scholar
  19. 19.
    Llibre J, Mereu A C. Limit cycles for discontinuous quadratic differential systems with two zones. J Math Anal Appl, 2014, 413: 763–775MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Llibre J, Valls C. Classification of the centers, their cyclicity and isochronicity for a class of polynomial differential systems generalizing the linear systems with cubic homogeneous nonlinearities. J Differential Equations, 2009, 246: 2192–2204MathSciNetCrossRefGoogle Scholar
  21. 21.
    Llibre J, Valls C. Classification of the centers and isochronous centers for a class of quartic-like systems. Nonlinear Anal, 2009, 71: 3119–3128MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Llibre J, Valls C. Classification of the centers, their cyclicity and isochronicity for the generalized quadratic polynomial differential systems. J Math Anal Appl, 2009, 357: 427–437MathSciNetCrossRefGoogle Scholar
  23. 23.
    Llibre J, Valls C. Classification of the centers, of their cyclicity and isochronicity for two classes of generalized quintic polynomial differential systems. NoDEA Nonlinear Differential Equations Appl, 2009, 16: 657–679MathSciNetCrossRefGoogle Scholar
  24. 24.
    Lloyd N G, Pearson J M. Bifurcation of limit cycles and integrability of planar dynamical systems in complex form. J Phys A, 1999, 32: 1973–1984MathSciNetCrossRefGoogle Scholar
  25. 25.
    Loud W S. Behavior of the period of solutions of certain plane autonomous systems near centers. Contrib Differential Equations, 1964, 3: 323–336MathSciNetGoogle Scholar
  26. 26.
    Musolino A, Rizzo R, Tripodi E. A quasi-analytical model for remote field eddy current inspection. Prog Electromagnet Res, 2012, 26: 237–249CrossRefGoogle Scholar
  27. 27.
    Oerlemans J. A quasi-analytical ice-sheet model for climate studies. Nonlinear Process Geophys, 1999, 10: 441–452Google Scholar
  28. 28.
    Pavlovskaia E, Wiercigroch M. Low-dimensional maps for piecewise smooth oscillators. J Sound Vibration, 2007, 305: 750–771MathSciNetCrossRefGoogle Scholar
  29. 29.
    Pleshkan I. A new method for investigating the isochronicity of a system of two differential equations. Differ Equ, 1969, 5: 796–802Google Scholar
  30. 30.
    Sabatini M. Characterizing isochronous centers by Lie brackets. Differential Equations Dynam Systems, 1997, 5: 91–99MathSciNetMATHGoogle Scholar
  31. 31.
    Tian Y, Yu P. Center conditions in a switching Bautin system. J Differential Equations, 2015, 259: 1203–1226MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Vulpe N I, Sibirskii K S. Centro-affine invariant conditions for the existence of a center of a differential system with cubic nonlinearities (in Russian). Dokl Akad Nauk, 1988, 301: 1297–1301; translation in Soviet Math Dokl, 1989, 38: 198–201Google Scholar
  33. 33.
    Xiao P. Critical point quantities and integrability conditions for complex planar resonant polynomial differential systems. PhD Thesis. Changsha: Central South University, 2005Google Scholar
  34. 34.
    Yu P, Han M. Four limit cycles from perturbing quadratic integrable systems by quadratic polynomials. Internat J Bifur Chaos, 2015, 22: 1250254MathSciNetGoogle Scholar
  35. 35.
    Zoladek H. Quadratic systems with center and their perturbations. J Differential Equations, 1994, 109: 223–273MathSciNetCrossRefGoogle Scholar
  36. 36.
    Zou Y, Kupper T, Beyn W J. Generalized Hopf Bifurcation for planar Filippov systems continuous at the origin. J Nonlinear Sci, 2006, 16: 159–177MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLinyi UniversityLinyiChina
  2. 2.Department of Applied MathematicsWestern UniversityLondon, OntarioCanada
  3. 3.School of Mathematics and StatisticsCentral South UniversityChangshaChina

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