Analysis of 1:4 resonance in a monopoly model with memory

Abstract

Resonances generate complicated bifurcation sequences. To design a picture of the bifurcation sequence occurring in the presence of a particular strong resonance, the two-parameters bifurcation of the equilibrium of a monopoly model with gradient adjustment mechanism and log-concave demand function is analyzed. Locally, the equilibrium may be destabilized through a period doubling or a supercritical Neimark–Sacker bifurcation. From a global perspective, it is also shown that the model undergoes strong resonances, such as the resonance 1:4, by using a continuation procedure. One of the interesting issue to tackle for this resonance is the investigation of the heteroclinic bifurcations which occur when pairs of saddles form connections near the 1:4 resonance. Therefore, the global analysis, through the description of the phase portraits and the basins of attractions, illustrates the theoretical features associated with the resonance and display interesting and complex dynamical behaviors, like the emergence of square and clover orbits.