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Chaos in sociobiology

Published online by Cambridge University Press:  17 April 2009

J.R. Christie ,
K. Gopalsamy  and
Jibin Li
J.R. Christie
Affiliation:
School of Information Science and Technology, The Flinders University of South Australia, GPO Box 2100, Adelaide SA 5001, Australia
K. Gopalsamy
Affiliation:
School of Information Science and Technology, The Flinders University of South Australia, GPO Box 2100, Adelaide SA 5001, Australia
Jibin Li
Affiliation:
Institute of Appplied Mathematics of Yunnan Province, Department of Mathematics, Kunming Institute of Technology, Yunnan 650093, People's Republic of China
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Abstract

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It is shown that the dynamical game theoretic mating behaviour of males and females can be modelled by a planar system of autonomous ordinary differential equations. This system occurs in modelling “the battle of the sexes” in evolutionary biology. 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 heteroclinic cycle and hence to possess chaotic behaviour in the sense of Smale. Finally, subharmonic Melnikov theory is used to obtain a criterion for the existence of subharmonic periodic orbits of the perturbed system.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 51 , Issue 3 , June 1995 , pp. 439 - 451
Copyright
Copyright © Australian Mathematical Society 1995

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