A Note on a Generalized Dynamical System Occurs in Modelling ‘The Battle of the Sexes’: Chaos in Sociobiology

Nikolay Kyurkchiev1,2), Anton Iliev1), Vesselin Kyurkchiev1),
Angel Golev1), Todorka Terzieva1), Asen Rahnev1)
1)University of Plovdiv „Paisii Hilendarski“ (Bulgaria)
2)Institute of Mathematics and Informatics,
Bulgarian Academy of Sciences (Bulgaria)

https://doi.org/10.53656/math2025-1-2-ano

Abstract. In this paper, we propose a new system that occurs in modelling “the battle of the sexes” in evolutionary biology (Hofbauer & Sigmund 1988). The existence of a heteroclinic cycle and a continuous family of periodic orbits of the system is established; then the dynamical characteristics of a time-periodic perturbation of the system are investigated. By using the well-known Melnikov’s method, a sufficient condition is obtained for the perturbed system to have a transverse hetero-clinic cycle and hence to possess chaotic behaviour in the sense of Smale. One possible application that Melnikov functions may find in the modelling and synthesis of radiating antenna patterns is considered. We demonstrate some modules for investigating the dynamics of the proposed model. This will be included as an integral part of a planned much more general Web-based application for scientific computing.
The proposed new extended model contains many free parameters (the coefficients ai, i = 1, 2, . . . ,N), which makes it attractive for use in the fields of biological applications, chemistry, sociology, lifetime analysis, reaction
kinetics, biostatistics, population dynamic, medical research, games theory etc. Finally, a special case of subharmonic solutions is discussed.

Keywords: new system occurs in modelling “the battle of the sexes”; hetero-clinic orbit; Melnikov function; subharmonic Melnikov function; chaotic behavior

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