Nonlinear dynamics of market segment evolution: mathematical modeling based on the cusp catastrophe concept

DOI: 10.31673/2412-9070.2025.061208

Abstract

A mathematical model describing the nonlinear evolution of market segments is developed based on the cusp catastrophe (catastrophe of the “cusp” type) from René Thom’s catastrophe theory. The proposed framework conceptualizes market development not as a smooth, continuous trajectory but as a sequence of stable phases separated by abrupt transitions. These transitions emerge from the interaction of three core factors: unsatisfied demand, consumer satisfaction, and the intensity of competition. Their dynamic interplay shapes the structure of the potential function whose minima correspond to stable market states.
The model introduces the canonical potential function of the cusp catastrophe along with its equilibrium condition, discriminant, and bifurcation manifold, which together determine the number and stability of possible market equilibria. It is shown that under specific parameter configurations, the system exhibits bistability, where two stable phases (growth and saturation) coexist alongside an unstable transitional state. This structure explains the possibility of sudden market shifts driven by small variations in external parameters—analogous to market saturation events, competitive shocks, or abrupt changes in consumer behavior.
In addition, the model highlights the role of characteristic diffusion—when key product attributes spread among competitors—in reducing differentiation and increasing competitive pressure. This mechanism gradually pushes the system toward the bifurcation region, where phase transitions become unavoidable. Illustrative examples from the smartphone industry and historical production cycles demonstrate how these nonlinear dynamics manifest in real markets.
The results support the applicability of catastrophe theory as a tool for analyzing market instability and structural transformations. The model offers a compact yet powerful mathematical instrument for identifying critical points, forecasting nonlinear responses, and enhancing strategic decision-making in competitive economic environments.

Keywords: cusp catastrophe; nonlinear dynamics; potential function; market segment; phase transitions; catastrophe theory; diffusion of characteristics.