Dynamics at Infinity, Degenerate Hopf and Zero-Hopf Bifurcation for Kingni-Jafari System with Hidden Attractors

Zhouchao Wei
School of Mathematics and Physics, China University of Geosciences,
Wuhan 530074, P. R. China
Division of Dynamics, Technical University of Lodz, Lodz,
Stefanowskiego 1/15, 90-924, Poland
College of Mechanical Engineering, Beijing University of Technology,
Beijing 100124, P. R. China
Mathematical Institute, University of Oxford,
Oxford, 0X2 6GG, England

Irene Moroz
Mathematical Institute, University of Oxford,
Oxford, 0X2 6GG, England

Zhen Wang
Department of Applied Sciences,
Xijing University, Xi'an 710123, P. R. China

Julien Clinton Sprott
Department of Physics, University of Wisconsin
Madison, WI 53706, USA

Tomasz Kapitaniak
Division of Dynamics, Technical University of Lodz,
Lodz, Stefanowskiego 1/15 90-924, Poland

Received February 15, 2016

ABSTRACT

To understand the complex dynamics of Kingni-Jafari system with hidden attractors, the first objective of this paper is to study the global dynamics, and give a complete description of the dynamics of Kigni-Jafari system at infinity by using the Poincare compactification of a polynomial vector field in R³. The second objective of this paper is to prove the existence of periodic solutions in the Kigni-Jarari system by classical Hopf bifurcation and degenerate Hopf bifurcation. Moreover, based on averaging theory, a small amplitude periodic solution that bifurcates from a zero-Hopf equilibrium was derived in the Kingni-Jafari system. The theoretical analysis and simulations demonstrate the rich dynamics of the system.

Ref: Z. Wei, I. Moroz, Z. Wang, J. C. Sprott, and T. Kapitaniak, International Journal of Bifurcation and Chaos 26 1650125 (2016)