Time-delay control for stabilization of the Shapovalov mid-size firm model

Abstract

Control and stabilization of irregular and unstable behavior of dynamic systems (including chaotic processes) are interdisciplinary problems of interest to a variety of scientific fields and applications. Using the control methods allows improvements in forecasting the dynamics of unstable economic processes and offers opportunities for governments, central banks, and other policy makers to modify the behaviour of the economic system to achieve its best performance. One effective method for control of chaos and computation of unstable periodic orbits (UPOs) is the unstable delay feedback control (UDFC) approach, suggested by K. Pyragas. This paper proposes the application of the Pyragas' method within framework of economic models. We consider this method through the example of the Shapovalov model, by describing the dynamics of a mid-size firm. The results demonstrate that suppressing chaos is capable in the Shapovalov model, using the UDFC method.

Dynamics of the Shapovalov mid-size firm model

“…Further, we analyze the system (2) and apply the inverse transformation (4) to obtain conditions on the parameters of system (1). To solve the Shapovalov problem, using the standard stability analysis of dynamic systems, we calculate the equilibria of system (1). System (1) always has three equilibria…”

“…It is important to show that system (1) does not have trajectories tending to infinity either for a finite or for an infinite period of time for a correct mathematical description of economic processes in a model and the possibility of studying its limit dynamics. Next, we distinguish the domain of the parameters of system (1) for which all trajectories are bounded and, moreover, which over time fall into a limited closed region called an absorbing set [2]. For the corresponding set of parameters system (1) has a global attractor.…”

“…in which system (2) is globally stable. Using (4), we obtain the statement of Theorem for system (1).…”

“…For the set of parameters considered, we use a MATLAB realization of the adaptive algorithm of finite-time Lyapunov dimension and Lyapunov exponents computation [35] and obtain the maximum of the finite-time local Lyapunov dimensions at the points of grid ( max This estimation is consistent with the hypothesis on the Lyapunov dimension of a self-excited attractor. Here the difficulties of reliable estimation of the Lyapunov exponents and dimension along one randomly chosen trajectory over long-time interval are caused by unstable periodic orbits embedded into attractor and finite precision numerical integration of ODE (see [1]). Note that the existence of multistability with hidden attractors in the model under consideration, as well as for the classical Lorenz system, is an open problem [19,36,42,43,56].…”