Limit cycles for two families of cubic systems

M. J. Álvarez, A. Gasull, R. Prohens

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Abstract

In this paper we study the number of limit cycles of two families of cubic systems introduced in previous papers to model real phenomena. The first one is motivated by a model of star formation histories in giant spiral galaxies and the second one comes from a model of Volterra type. To prove our results we develop a new criterion on the non-existence of periodic orbits and we extend a well-known criterion on the uniqueness of limit cycles due to Kuang and Freedman. Both results allow to reduce the problem to the control of the sign of certain functions that are treated by algebraic tools. Moreover, in both cases, we prove that when the limit cycles exist they are non-algebraic. © 2012 Elsevier Ltd. All rights reserved.
Original languageEnglish
Pages (from-to)6402-6417
JournalNonlinear Analysis, Theory, Methods and Applications
Volume75
Issue number18
DOIs
Publication statusPublished - 1 Dec 2012

Keywords

  • Bifurcation
  • Cubic system
  • Kolmogorov system
  • Limit cycle

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    Álvarez, M. J., Gasull, A., & Prohens, R. (2012). Limit cycles for two families of cubic systems. Nonlinear Analysis, Theory, Methods and Applications, 75(18), 6402-6417. https://doi.org/10.1016/j.na.2012.07.012