BIFURCATION AND CHAOS ANALYSIS IN A DISCRETE-DELAY DYNAMIC MODEL FOR A STOCK MARKET

Abstract

Using a discrete-delay nonlinear dynamic system, we model the time evolution of a stock market price index and net stock of savings in mutual funds. The proposed deterministic model has a unique steady-state, so its time evolution is determined by nonlinear effects acting out of equilibrium. For this model, we find the local stability properties and the local bifurcations conditions, given the parameter space. Specifically, we find that in both versions of the model (with and without delay) a Neimark–Sacker bifurcation can occur. Moreover, we show that the system without delay has a chaotic behavior. Finally, we formulate the associated discrete stochastic model and establish the conditions for asymptotic stability. Several numerical simulations are finally performed for both the deterministic and the stochastic model to justify the theoretical results.