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On the limit cycles of a quartic model for evolutionary stable strategies

Armengol Gasull, Luiz F. S. Gouveia, Paulo Santana

TL;DR

This work investigates centers and limit cycles for planar quartic vector fields with the invariant curve $(4x^2-1)(4y^2-1)=0$ inside the invariant square $Δ$. Focusing on the subfamily $X ∈ X_2$, it proves at most two centers in $Δ$ and constructs dynamic scenarios with five nested limit cycles as well as two disjoint nests of two, complemented by Berlinskii-type results for this family. The authors deploy Lyapunov quantities, reversibility, Darboux integrability, and resultant methods to derive center conditions and bifurcation results, including several explicit examples of multi-cycle bifurcations from the origin and from two centers. They also connect these dynamical findings to two-player Evolutionary Stable Strategy models, highlighting how oscillatory stable states (limit cycles) model oscillating strategies in ESS dynamics. The results advance understanding of how quartic planar systems can realize rich center/limit-cycle configurations and broaden Berlinskiı-type insights beyond quadratic cases.

Abstract

This paper studies the number of centers and limit cycles of the family of planar quartic polynomial vector fields that has the invariant algebraic curve $(4x^2-1)(4y^2-1)=0.$ The main interest for this type of vector fields comes from their appearance in some mathematical models in Game Theory composed by two players. In particular, we find examples with five nested limit cycles surrounding the same singularity, as well as examples with four limit cycles formed by two disjoint nests, each one of them with two limit cycles. We also prove a Berlinski\u ı's type result for this family of vector fields.

On the limit cycles of a quartic model for evolutionary stable strategies

TL;DR

This work investigates centers and limit cycles for planar quartic vector fields with the invariant curve inside the invariant square . Focusing on the subfamily , it proves at most two centers in and constructs dynamic scenarios with five nested limit cycles as well as two disjoint nests of two, complemented by Berlinskii-type results for this family. The authors deploy Lyapunov quantities, reversibility, Darboux integrability, and resultant methods to derive center conditions and bifurcation results, including several explicit examples of multi-cycle bifurcations from the origin and from two centers. They also connect these dynamical findings to two-player Evolutionary Stable Strategy models, highlighting how oscillatory stable states (limit cycles) model oscillating strategies in ESS dynamics. The results advance understanding of how quartic planar systems can realize rich center/limit-cycle configurations and broaden Berlinskiı-type insights beyond quadratic cases.

Abstract

This paper studies the number of centers and limit cycles of the family of planar quartic polynomial vector fields that has the invariant algebraic curve The main interest for this type of vector fields comes from their appearance in some mathematical models in Game Theory composed by two players. In particular, we find examples with five nested limit cycles surrounding the same singularity, as well as examples with four limit cycles formed by two disjoint nests, each one of them with two limit cycles. We also prove a Berlinski\u ı's type result for this family of vector fields.

Paper Structure

This paper contains 13 sections, 20 theorems, 96 equations, 2 figures.

Table of Contents

  1. Main results
  2. Vector fields in $\mathfrak{X}_d$ and Game Theory
  3. Preliminaries
  4. Lyapunov quantities
  5. Bifurcation of limit cycles
  6. Reversibility
  7. Darboux Theory of Integrability
  8. A simple family without limit cycles and centers
  9. Resultants
  10. Finding conditions for a center at the origin
  11. Bifurcation of limit cycles from the origin
  12. Simultaneous bifurcation of limit cycles from two centers
  13. Proof of the main results

Key Result

Theorem 1

Let $X\in\mathfrak{X}_2$ as in 7. Then, the following statements hold.

Figures (2)

  • Figure 1: An illustration of $\ell$.
  • Figure 2: An illustration of the rotation of $X$ along $\infty q$.

Theorems & Definitions (34)

  • Theorem 1
  • Theorem 2: Berlinskiı's Theorem (B)
  • Proposition 1
  • Theorem 3
  • Theorem 4: Poincaré-Lyapunov Theorem
  • Theorem 5: Theorem $3.1$ of GinGouTor2021
  • Remark 1
  • Remark 2
  • Theorem 6: Poincaré reversibility criterion
  • Theorem 7: Theorem $8.7$ of DumLliArt2006
  • ...and 24 more