Hopf Birurcation of Lorenz-like System about Parameter h

Gaoxiang Yang

doi:10.5539/mas.v4n1p91

Abstract

The article mainly researched the Hopf bifurcation of Lorenz-like system about the coefficient of the quadratic term. When the quadratic term changes, the solution to the Lorenz-like system will become the local periodic solution. Further the stability of this periodic solution and the bifurcation direction of this periodic solution were discussed, and found when the quadratic term comes through a threshold , the direction of hopf bifurcation and stability were given, and the result as follows. If , when , the direction of bifurcation is , when , the direction of bifurcation is ; (b) if , we have the contrary result. That is when , the direction of bifurcation is ; when , the direction of bifurcation is . If , the bifurcation solution is asymptotically stable; if , the bifurcation solution is not asymptotically stable. Finally employing the matlab compute the numerical periodic solution, the results fit the theory well.

Highlights

Nonlinear science plays an important role in science research including biology, chemistry, fluid dynamics, optics and so on, but bifurcation theory is an important part in nonlinear science and many progresses have been made in the past many years
Bifurcation takes very significant role in the nonlinear system because bifurcation of the nonlinear system can lead to the chaotic behavior and induced the much more difficulty behaviors
The Lorenz-like system is very important like the Lorenz system which was found by the Lorenz in 1963 and many dynamical behaviors of this system have been discussed in the recent years

Summary

Introduction

Nonlinear science plays an important role in science research including biology, chemistry, fluid dynamics, optics and so on, but bifurcation theory is an important part in nonlinear science and many progresses have been made in the past many years. The Lorenz-like system is very important like the Lorenz system which was found by the Lorenz in 1963 and many dynamical behaviors of this system have been discussed in the recent years. Such as the bifurcation behavior about the one order term, chaos and so on. This paper mainly employed the very important method of central manifold theory and theory of normal form to conduct a detailed discussion of Hopf bifurcation of the Lorenz-like system. The results of numerical simulation fit the theoretical analysis well

Main results

Numerical stimulation