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.

Dynamic emergence of domino effects in systems of interacting tipping elements in ecology and climate

“…The dynamics of the tipping elements is modeled with the topological normal form of the cusp bifurcation [51,81], i. e. the most generic polynomial system exhibiting this type of tipping behavior (Fig. 1).…”

“…where a i , b i > 0 and f 0 i corresponds to the uncoupled case. The parameter a i corresponds to the height between the two stable equilibria layers of the cusp [81] and the parameter b i controls the strength of the nonlinearity in the system. The control parameter c i is associated with a tipping parameter the size of which determines whether the system undergoes a critical transition from one stable state to another ( Fig.…”

“…For simplicity, we choose a linear coupling [51,81]. The linear coupling function for subsystem X i then reads…”

“…Tab. 1 provides an overview of special cases of coupling between interacting cusp catastrophes investigated in the literature [51,80,81]. Table 1: Overview of linearly coupled tipping elements studied in the literature.…”

“…With increasing coupling strengths d 21 and d 12 the system behavior changes. The system has only one unstable fixed point and Kadyrov oscillations [81] occur for a wide range of the control parameters in the considered part of the (c 1 , c 2 )-parameter space (upper left of Fig. 4).…”