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Characterization and bifurcation diagram of the family of quadratic differential systems with an invariant ellipse in terms of invariant polynomials

Abstract

Consider the class QS of all non-degenerate planar quadratic systems and its subclass QSE of all its systems possessing an invariant ellipse. This is an interesting family because on one side it is defined by an algebraic geometric property and on the other, it is a family where limit cycles occur. Note that each quadratic differential system can be identified with a point of \({{\mathbb {R}}}^{12}\) through its coefficients. In this paper we provide necessary and sufficient conditions for a system in QS to have at least one invariant ellipse. We give the global “bifurcation” diagram of the family QS which indicates where an ellipse is present or absent and in case it is present, the diagram indicates if the ellipse is or it is not a limit cycle. The diagram is expressed in terms of affine invariant polynomials and it is done in the 12-dimensional space of parameters. This diagram is also an algorithm for determining for any quadratic system if it possesses an invariant ellipse and whether or not this ellipse is a limit cycle.

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Acknowledgements

The first author is partially supported by FAPESP Grants “Projeto Temático” 2014/00304-2 and 2017/20854-5. The work of the third and the fourth authors was partially supported by the NSERC Grant RN000355.

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Correspondence to Regilene Oliveira.

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Oliveira, R., Rezende, A.C., Schlomiuk, D. et al. Characterization and bifurcation diagram of the family of quadratic differential systems with an invariant ellipse in terms of invariant polynomials. Rev Mat Complut 35, 361–413 (2022). https://doi.org/10.1007/s13163-021-00398-8

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  • DOI: https://doi.org/10.1007/s13163-021-00398-8

Keywords

  • Quadratic vector fields
  • Affine invariant polynomials
  • Invariant algebraic curve
  • Invariant ellipse
  • Limit cycle

Mathematics Subject Classification

  • 58K30
  • 34A26
  • 34C05
  • 34C40