Chaos theory leads to new understanding of when the next fashion trend will hit

Whether it’s the clothing decisions of teenagers, tastes in art, or political thought, human behavior is powerfully driven by fashion cycles. As one idea falls out of favor within groups of people, another one is often rediscovered and takes its place.

Based on the hypothesis that a delicate balance between the two fundamentally irreconcilable desires of human beings — the desire to look or act the same with others and the desire to look or act differently from others — are driving persistent fashion cycles, an international research team looked into a causal mechanism. They investigated the underlying factors that lead to fashion cycles emerging through the lens of the bifurcation theory in nonlinear dynamical systems.

The work, published in Chaos, reformulated the so-called continuous-time Matsuyama fashion cycle model and generated a 2-D discontinuous piecewise linear map to analyze how fashion cycles bifurcate. Aside from its appeal for applied uses, the new map looks to shed light on the bifurcation theory of this lesser understood class of maps.

In bifurcation theory, bifurcations are sudden qualitative changes in the dynamics of a system under smooth variation of its parameters. The team discovered a new kind of bifurcation structure in the parameter space of the map, where periodicity regions of attracting cycles are organized in different incrementing structures.

The team found that if the time delay in the discrete time formulation approaches zero, the number of incrementing structures goes to infinity and the dynamics of the discontinuous map converges to that of the continuous-time model. They hope the model will stoke further interest in bridging theory of nonlinear dynamics with social sciences and also look to apply the theory to different theoretical and applied models.

Source: “2D discontinuous piecewise linear map: Emergence of fashion cycles,” by L. Gardini, I. Sushko, and K. Matsuyama, Chaos (2018). The article can be accessed at https://doi.org/10.1063/1.5018588 .