Nonlinear Dynamics

, Volume 85, Issue 3, pp 1635–1650 | Cite as

Bifurcation analysis and circuit realization for multiple-delayed Wang–Chen system with hidden chaotic attractors

  • Zhouchao Wei
  • Viet-Thanh Pham
  • Tomasz Kapitaniak
  • Zhen Wang
Original Paper


In this paper, we investigate the effect of multiple delays on the 3-D simple chaotic system with hidden chaotic attractors coexisting with one stable equilibrium by Wang and Chen. In order to investigate complex dynamics of hidden chaotic attractors with multiple delays, we choose \(\tau _1\) and \(\tau _2\) as bifurcating parameters and consider the stability of equilibrium and the existence of Hopf bifurcations. Some explicit formulas for determining the direction of bifurcations and the stability of bifurcating periodic solutions are obtained by using normal form theory and center manifold theory. The numerical simulations are performed to support the correctness and effectiveness of the analytical results. The effect of multiple delays can be applied to the chaotic system only with one stable node-foci for the purpose of control and anti-control of hidden chaos by delayed switchover. Furthermore, to reach deep and clear understanding of the dynamics of such hidden attractors, circuit implementation of the multiple time-delay system is analyzed using the PSpice.


Hidden attractor Periodic solution Hopf bifurcation Multiple delays Stable node-foci 



We would like to express our gratitude to the referee for his or her valuable comments and suggestions that led to a truly significant improvement of the manuscript. This work was supported by the Polish National Science Centre, MAESTRO Programme—Project (No. 2013/08/A/ST8/00/780), the Natural Science Foundation of China (No. 11401543), the China Scholarship Council (CSC) (No. 201506415023) Beijing Postdoctoral Research Foundation (No. 2015ZZ17), the China Postdoctoral Science Foundation funded project (Nos. 2014M560028 and 2015T80029), the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) (No. CUGL150419), the Natural Science Foundation of Hubei Province (No. 2014CFB897), the Government of Chaoyang District Postdoctoral Research Foundation (No. 2015ZZ-7) 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 2016

Authors and Affiliations

  • Zhouchao Wei
    • 1
    • 2
    • 3
    • 4
  • Viet-Thanh Pham
    • 5
  • Tomasz Kapitaniak
    • 4
  • Zhen Wang
    • 6
  1. 1.School of Mathematics and PhysicsChina University of GeosciencesWuhanChina
  2. 2.College of Mechanical EngineeringBeijing University of TechnologyBeijingChina
  3. 3.Mathematical InstituteUniversity of OxfordOxfordEngland, UK
  4. 4.Division of DynamicsTechnical University of LodzLodzPoland
  5. 5.School of Electronics and TelecommunicationsHanoi University of Science and TechnologyHanoiVietnam
  6. 6.Department of Applied SciencesXijing UniversityXi’anChina

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