The bifurcations and dynamics of certain quadratic maps of the plane

Abstract

Discrete models of density-dependent population growth provide simple
examples of dynamical systems which exhibit complicated dynamics. Single
age-class models lead to the study of maps of an interval into itself and
we outline the main results which are known in this case.
Here our main concern is with a two age-class model, due to Maynard Smith,
which takes the form of a two-parameter family of maps of the plane with a
quadratic non-linearity. After a description of the local bifurcations of
general two-parameter families .in the plane we give a linear stability
analysis for the fixed points in our model and analyse their bifurcations.
This local theory is extended by finding numerically the boundaries of the
regions in the parameter space where the map has a periodic orbit of low
period created by resonant Hopf bifurcation. A series of computer-drawn
phase portraits is presented for a one-parameter path through the parameter
plane, showing .the creation of an attracting invariant circle by a Hopf
bifurcation followed by a passage to a more complicated attractor.
We examine a three-parameter family connecting our map to the quadratic
diffeomorphism studied by Henon and conclude with a discussion of a more
realistic model which, however, contains the same complexities associated
with resonant Hopf bifurcation found in the simpler model.

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