Abstract
In the field of marketing, business and finance, the rise and fall and the associated bifurcations can be better interpreted with the aid of differential equations and the dynamical systems. The differential equations, when Incorporated with the delayed dynamics can interpret the financial delays in more realistic manner. The parametric approximation of such differential equations can be achieved with the aid of robust optimization tool, that is termed as the genetic algorithm. In this manuscript, the concept of delay is invoked with the aid of specific factors linked with the raise and fall of investment frequency in the field of fashion industry and the corresponding stability is modeled with the aid of detailed dynamical analysis. The stability and instability criteria of delay differential equations, governing the dynamics of the investment trends is reported in this manuscript. The bifurcation and their relative criteria of occurrence is discussed. The Hopf bifurcation is important since with the aid of the bifurcation analysis, one can obtain a region of instability, and can thus invest in a safer manner by avoiding such circumstances.