Some global bifurcations related to the appearance of closed invariant curves (original) (raw)

2005, Mathematics and Computers in Simulation

In this paper, we consider a two-dimensional map (a duopoly game) in which the fixed point is destabilized via a subcritical Neimark-Hopf (N-H) bifurcation. Our aim is to investigate, via numerical examples, some global bifurcations associated with the appearance of repelling closed invariant curves involved in the Neimark-Hopf bifurcations. We shall see that the mechanism is not unique, and that it may be related to homoclinic connections of a saddle cycle, that is to a closed invariant curve formed by the merging of a branch of the stable set of the saddle with a branch of the unstable set of the same saddle. This will be shown by analyzing the bifurcations arising inside a periodicity tongue, i.e., a region of the parameter space in which an attracting cycle exists. A. Agliari et al. / Mathematics and Computers in Simulation 68 (2005) 201-219 suppliers, oligopoly, many suppliers, polypoly, to perfect competition. In the last case, each supplier is so small that it cannot in any way influence market price. In the opposite case, the monopolist deliberately limits supply so as to be able to charge a high market price to the end of obtaining a maximum monopoly profit. The cases of duopoly and oligopoly are the most complicated, because each competitor has to take account not only of consumer demand, as reflected in the demand function, but also of the expected retaliations of the competitors. Oligopoly theory is one of the oldest branches of mathematical economics, created in 1838 by the mathematician Augustin Cournot .