Study For The Multi-scale Effects Of A Kind Of Nonlinear Financial System

The financial system is a complex non-linear system related to many factors such as investment and savings.With the development of society,the financial system is becoming more and more complex,which prompts the majority of scholars to continuously explore the nonlinear phenomenon in the financial system,reveal the operation law of the financial system.Based on fast and slow analysis method,stability theory,bifurcation theory and numerical simulation,this thesis studies the nonlinear financial model coupled at two scales,discovers various cluster oscillation phenomena in the financial system,reveals the mechanism of production,and provides a theoretical basis for government and decision-making departments.Work done and main conclusions are as follows:Considering that there may be small disturbances with a long period in unit investment costs,the nonlinear financial system is a fast and slow coupling system.Based on the stability and bifurcation theory of differential equations,the critical parameter values for the occurrence of Hopf bifurcation are calculated from the conditions under which Hopf bifurcation occurs.It is also found that the pitchfork bifurcation behavior of autonomous systems leads to randomness in the symmetric focus attractors accessed by non-autonomous systems.When the amplitude range gradually expands,the type of attractor involved in the autonomous system changes,and the non-autonomous system presents a transition from periodic oscillation to singular non-chaotic cluster oscillation to chaotic cluster oscillation.When the investment needs in the financial system is disturbed by the small disturbances with a long period,the whole system becomes a fast-slow coupling non-autonomous system.The stability of the autonomous system is obtained by the Routh-Hurwitz criterion.Based on the bifurcation theory,it is found that the system has pitchfork bifurcation and Hopf bifurcation.And there is a bifurcation form where the period to chaos.Using the fast and slow analysis method,it is found that with the change of bifurcation parameters,the system will have periodic cluster oscillation,symmetric singular non-chaotic cluster oscillation,symmetric critical cluster oscillation and chaotic cluster oscillation.Finally,the influence of disturbance amplitude on cluster oscillation is discussed.Considering the internal relationship between investment demand and price index with small disturbances,and the periodic disturbance term is added to the system,forming a class of fast and slow coupling systems.Firstly,according to the characteristic equation of the equilibrium point of the autonomous system and the conditions for the occurrence of Hopf bifurcation,the existence of the Hopf bifurcation point is proved and the critical parameter value of the Hopf bifurcation occurs is calculated.Secondly,based on the Lyapunov index,the type of Hopf bifurcation is judged.Finally,the single-Hopf cluster oscillation phenomenon in the non-autonomous system is discovered through numerical simulation.According to the slow-fast analysis method,it reveals the mechanism of cluster oscillation.