Hopf Bifurcation Analysis Of A Class Of Finance Systems

In this paper we have studied a class of delayed fnance system. By linearizing thesystem at the positive equilibrium and analyzing the associated characteristic equation,the asymptotic stability the posirive equilibrium is investigated and Hopf bifurcation aredemonstrated. Furthermore, the direction of Hopf bifurcation and the stability of thebifurcating periodic solutions are determined by the normal form theory and the centermanifold theorem for functional diferential equations. Finally, some numerical simulationsare carried out for illustrating the theoretical results. The main contents are as follows:First, a fnance system with multiply delays is considered. By applying the theoremof Hopf bifurcation and regarding the delay τ1, τ2, τ3as the bifurcation parameter, thestability of positive equilibrium and existence of Hopf bifurcation are demonstrated. Thenthe direction of the Hopf bifurcation and stability of the bifurcating periodic solutionis considered. Finally, numerical simulations are given. Moreover, the efect multi-timedelays is considered as an important factor, which will be more close to really nature.Comparing to the case of single time delay or the case without delay, research multi-timedelays will need to more complicated and technique mass. Moreover, all the three statevariables are infuenced by time delays, multi-time delays can better impact the economicproblems.Second, a delayed fnance system is considered. By applying the theorem of Hopfbifurcation and regarding the delay τ as the bifurcation parameter, the stability of positiveequilibrium and existence of Hopf bifurcation are demonstrated. Then the direction of theHopf bifurcation and stability of the bifurcating periodic solution is considered. Finally,numerical simulations are given.