Nonlinear Dynamics

, Volume 82, Issue 1–2, pp 131–141 | Cite as

Study of hidden attractors, multiple limit cycles from Hopf bifurcation and boundedness of motion in the generalized hyperchaotic Rabinovich system

  • Zhouchao Wei
  • Pei Yu
  • Wei Zhang
  • Minghui Yao
Original Paper


Based on Rabinovich system, a 4D Rabinovich system is generalized to study hidden attractors, multiple limit cycles and boundedness of motion. In the sense of coexisting attractors, the remarkable finding is that the proposed system has hidden hyperchaotic attractors around a unique stable equilibrium. To understand the complex dynamics of the system, some basic properties, such as Lyapunov exponents, and the way of producing hidden hyperchaos are analyzed with numerical simulation. Moreover, it is proved that there exist four small-amplitude limit cycles bifurcating from the unique equilibrium via Hopf bifurcation. Finally, boundedness of motion of the hyperchaotic attractors is rigorously proved.


Rabinovich system Hidden attractor Hopf bifurcation Boundedness of motion 



The authors acknowledge the referees and the editor for carefully reading this manuscript and suggesting many helpful comments. This work was supported by the Natural Science Foundation of China (11401543, 11290152, 11072008, 41230637), the Natural Science Foundation of Hubei Province (No. 2014CFB897), the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) (No. CUGL150419), China Postdoctoral Science Foundation funded project (No. 2014M560028), Beijing Postdoctoral Research Foundation (2015ZZ), the Natural Science and Engineering Research Council of Canada (No. R2686A02) and the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHRIHLB).


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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.School of Mathematics and PhysicsChina University of GeosciencesWuhanPeople’s Republic of China
  2. 2.College of Mechanical EngineeringBeijing University of TechnologyBeijingPeople’s Republic of China
  3. 3.Department of Applied MathematicsWestern UniversityLondonCanada

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