Hopf bifurcations in an extended Lorenz system
详细信息 查看全文
- 作者：Zhiming Zhou ; Gheorghe Tigan ; Zhiheng Yu
- 关键词：extended Lorenz system ; Hopf bifurcation ; center manifold ; Lyapunov constant ; degeneration
- 刊名：Advances in Difference Equations
- 出版年：2017
- 出版时间：December 2017
- 年：2017
- 卷：2017
- 期：1
- 全文大小：1427KB
- 刊物主题：Difference and Functional Equations; Mathematics, general; Analysis; Functional Analysis; Ordinary Differential Equations; Partial Differential Equations;
- 出版者：Springer International Publishing
- ISSN：1687-1847
- 卷排序：2017
文摘
In this work, we study Hopf bifurcations in the extended Lorenz system, \(\dot{x}=y\), \(\dot{y}=mx-ny-mxz-px^{3}\), \(\dot{z}=-az+bx^{2}\), with five parameters \(m,n,p,a,b\in \mathbb{R}\). For some values of the parameters, this system can be transformed to the classical Lorenz system. In this paper, we give conditions for occurrence of Hopf bifurcation at the equilibrium points. We find totally three limit cycles, each of them located around one of the three equilibria of the system. Numerical simulations illustrate the validity of these conditions and the existence of limit cycles.