Studying basin bifurcations in nonlinear triopoly games by using 3D visualization

We consider three-dimensional discrete dynamical system, obtained by the iteration of a noninvertible map of , which simulates the time evolution of an oligopoly game with three competing firms. The model is characterized by the presence of several coexisting stable equilibria, each with its own basin of attraction. In this paper we face the question of the delimitation of the basins and the detection of the global bifurcations that cause the creation of non-connected basins. This requires a study of the global properties of the 3dimensional noninvertible map by the method of critical sets, based on the determination of the contact bifurcations through a systematic computer-assisted study. This requires the visualization of surfaces (the critical surfaces and the basins’ boundaries) which sometimes are nested one inside the other. Enhanced graphical methods, based on two-level volume rendering, are employed in order to modulate the opacity of outer objects so that the contacts between the basins’ boundaries and critical surfaces can be visualized. This is obtained through the realization of ad-hoc routines, which allow interactive 3D visualization.

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