### QUALITATIVE ANALYSIS OF THE NONLINEAR ADVERTISING DIFFUSION MODEL

В. В. Атамась, В. С. Денисенко, В. О. Денисенко

#### Abstract

In this paper a qualitative analysis of the nonlinear dynamic advertising diffusion model have been carried out. Using the first Lyapunov method, the  asymptotic stability of the equilibrium position of the system of equations of perturbed motion (stability by linear approximation) has been investigated and sufficient asymptotic stability conditions in terms of algebraic inequalities has been obtained based on Routh-Hurwitz conditions. The domain of asymptotic stability has been constructed. To investigate stability of equilibrium position in a critical case we first transform our initial nonlinear system of differential equations to a normal form of Poincare. Then using the second Lyapunov method (quadratic positive definite Lyapunov function), we establish the asymptotic stability of a singular point in the critical case. It is shown that the boundary of the asymptotic stability domain is safe and there is a soft loss of stability. The existence of a stable limit cycle (Hopf bifurcation) is proved. Basing on the Poincare-Bendixon theorem, the existence of the limit cycle (stable periodic solution) has been established. Using the  Poincare normal form, the parameters of self-oscillations and the formula for limit cycle and its period  have been approximately  determined.

#### Keywords

diffusion of innovation model; Lyapunov asymptotic stability; limit cycle; Hopf bifurcation

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