Title: A class of quadratic system with six limit cycles

Authors: Du Chaoxiong; Liu Yirong; Wang Qinlong

Addresses: Department of Mathematics, Hunan Shaoyang University, Shaoyang, Hunan 422000, China ' School of Mathematics, Central South University, Changsha, Hunan 410075, China ' Department of Mathematics, Hezhou University, Hezhou, Guangxi 542800, China

Abstract: In this paper, the Hopf bifurcation for a class of three-dimensional quadratic system is investigated by making use of singular values methods. Studied system lies in symmetrical vector field with regard to planar y = x and it has two symmetrical singular points (1,2,1) and (2,1,1). We give the expressions of the first three focal values of the singular point (1,2,1) and show that each one of the two singular points (1,2,1) and (2,1,1) of the investigated system can become a fine focus of third order at the same time. Moreover, we obtain that each one of the two singular points (1,2,1) and (2,1,1) of the investigated system can bifurcate three small limit cycles under a certain coefficient's disturbed condition. In sum, six limit cycles can bifurcate from system (1). In terms of Hopf bifurcation of a three-dimensional quadratic system, our results are new and interesting.

Keywords: 3-D quadratic systems; Hopf bifurcation; symmetrical vector field; singular values; limit cycles.

DOI: 10.1504/IJDSDE.2015.071000

International Journal of Dynamical Systems and Differential Equations, 2015 Vol.5 No.3, pp.175 - 190

Received: 07 Nov 2013
Accepted: 28 Nov 2014
Published online: 05 Aug 2015 *

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