FORMALIZATION AND MODELING OF DYNAMIC STABILITY OF BUSINESS STRUCTURES ON THE BASIS OF NONLINEAR DYNAMICS

Abstract

The article substantiates the theoretical and methodological foundations for modeling the dynamic stability of business structures operating within the high-entropy environment of inter-organizational cooperation. The research synthesis is built upon W. R. Ashby's Law of Requisite Variety (LRV) and the mathematical apparatus of non-linear discrete dynamics. The authors propose a fundamental transition from continuous growth models to a discrete recurrence mapping, which enables the formalization of the impact of the management lag ∆t on the system's homeostatic trajectory. A central focus of the study is the concept of "variety engineering," where the parameters of the intrinsic growth rate r(H(R),H(D)) and the carrying capacity of the environment K(H(R),H(D))are redefined as functions of the entropy balance. This approach allows for a quantitative measure of the regulator's variety relative to environmental disturbances. By applying the Euler discretization method to the classical Verhulst logistic equation, the study identifies a specific "numerical chaos" phenomenon inherent in management systems. The stability of the resulting discrete mapping was rigorously tested using the Schur-Cohn criterion. The analysis reveals a fundamental threshold of controllability, identified by the dimensionless parameter ∆t∙r(H(R),H(D)). The research proves that even with a robust repository of management mechanisms, an excessive time lag in decision-making acts as an independent source of entropy, pushing the business structure into a zone of bifurcations and period-doubling. The study establishes that the transition to deterministic chaos occurs when the product of the growth rate and the discretization step exceeds the Feigenbaum point. The scientific novelty of the work lies in the integration of "variety engineering" into iterative management cycles, providing a mathematical basis for predictive business intelligence. The results demonstrate that systemic stability is not a static property but a dynamic equilibrium between the cognitive complexity of the regulator and the temporal resolution of the control loop. These findings provide a framework for developing digital twins and AI-driven management systems capable of maintaining structural integrity in a BANI-world environment by dynamically adjusting the control step to counteract environmental entropy.