On the limit cycles of a quartic model for Evolutionary Stable Strategies

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 (4x2−1)(4y2−1)=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 Berlinskiĭ’s type result for this family of vector fields.

Keywords

Berlinskiĭ’s theorem, Center-focus, Cyclicity, Evolutionary Stable Strategies, Limit cycles, Evolutionary stable strategies, Invariant algebraic curves, Limit-cycle, Number of centers, Polynomial vector field, Quartic polynomial, Vector fields