Dynamical Analysis For A Model Of Asset Prices With Two Delays

Most empirical researches show that financial market is a complicated nonlinear system. People expect to get information from the financial market to make prediction about the future. It is efficient to study asset price theory by establishing the differential equation model to describe the wide price fluctuation in the financial market. On this basis, using numerical simulation to show the phenomenon called a speculative bubble in the asset market is extensive discussion and attention. In fact, asset prices across financial markets exist correlation. Scholars consider not only the single asset market,but also the situation of coexistence of multiple assets. Some heterogeneous phenomena in financial market can be illuminated through stability analysis and bifurcation analysis of the model.This paper mainly carry out dynamic properties of two-dimensional asset price model with two delays which was established by Dibeh in 2007. Here the time delay is used to describe the chartists estimating the trend of asset price in the future based on the price over a period of past time. First, we analyze the existence of equilibrium point of the model, then we study the stability of equilibrium point and obtain the sufficient conditions of the existence of local Hopf bifurcation when2t=0 and2t 10. For the particular case, the stability of equilibrium point and the sufficient conditions of the existence of local Hopf bifurcation are also investigated when1 2t=t=t. The explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions is derived, using the center manifold theorem and the normal form theory established by Hassard. The global existence of periodic solutions is proved using a global Hopf bifurcation theorem given by Wu and the Bendixson criterion on higher dimensional ordinary differential equations due to Li and Muldowney.Finally, some numerical simulations are given by using Matlab. They show that the price of assets appear periodic fluctuations around the basis value. All the results of numerical simulation agree with the theoretical analysis. Several branches of periodic solutions are observed with the aid of DDE- BIFTOOL. In addition, the global Hopf bifurcation diagrams are given, which illustrated the global Hopf bifurcation results.