COMPUTATIONAL UNFOLDING OF DOUBLE-CUSP MODELS OF OPINION FORMATION

Abstract

In 1975, Isnard and Zeeman proposed a cusp catastrophe model for the polarization of a social group, such as the population of a democratic nation. Ten years later, Kadyrov combined two of these cusps into a model for the opinion dynamics of two "nonsocialist" nations. This is a nongradient dynamical system, more general than the double-cusp catastrophe studied by Callahan and Sashin [1987]. Here, we present a computational study of the nongradient double cusp, in which the degeneracy of Kadyrov's model is unfolded in codimension eight. Also, we develop a discrete-time cusp model, study the corresponding double cusp, establish its equivalence to the continuous-time double cusp, and discuss some potential applications. We find bifurcations for multiple critical-point attractors, periodic attractors, and (for the discrete case) bifurcations to quasiperiodic and chaotic attractors.