Nonlinear Dynamics

, Volume 95, Issue 2, pp 1549–1563 | Cite as

Two bifurcation routes to multiple chaotic coexistence in an inertial two-neural system with time delay

  • Shengwei Yao
  • Liwang Ding
  • Zigen SongEmail author
  • Jieqiong Xu
Original Paper


In this article, we establish an inertial two-neural system with time delay and illustrate the stable coexistence of three chaotic attractors that arise via two different bifurcation routes, i.e., the period-doubling and quasi-periodic bifurcations. So, we firstly analyze the system equilibria by nullcline curves. By the pitchfork/saddle-node bifurcation of the trivial/nontrivial equilibria, the system parameter (\(c_{1}\), \(c_{2})\)-plane is divided into the different regions having the different number of equilibrium. Further, the trivial and nontrivial equilibria will lose their stability and bifurcate into periodic orbits as the effect of time delay. The system has the stable coexistence of two periodic orbits near the nontrivial equilibria. For some delayed regions, the system illustrates the stability switching, i.e., the dynamic behaviors lost, retrieved, and lastly lost their stability with increase in delay. Using the Hopf–Hopf bifurcation analysis, we find a quasi-periodic orbit surrounded by the trivial equilibrium. Lastly, based on numerical simulations, such as phase portrait, Poincare section, Lyapunov exponent, and one-dimensional bifurcation diagram, we further investigate the dynamical evolution of the periodic and quasi-periodic orbits. The results show that the neural system presents the multiple stable coexistence with three chaotic attractors by the different bifurcation routes, i.e., the period-doubling and quasi-periodic bifurcations.


Inertial neural system Time delay Chaos route Multiple coexistence Hopf–Hopf bifurcation 



The authors declare that there are no possible conflicts of interest. The authors acknowledge the referees and the editor for suggesting many helpful comments. This research is supported by the National Natural Science of China under Grant Nos. 11672177 and 11602059, Promotion project of basic ability of young and middle-aged teachers in Universities in Guangxi No. 2018KY0521, and Guangxi Natural Science Foundation No. 2018GXNSFBA138023.


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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  • Shengwei Yao
    • 1
    • 2
  • Liwang Ding
    • 3
  • Zigen Song
    • 4
    Email author
  • Jieqiong Xu
    • 5
  1. 1.Guangxi (ASEAN) Research Center of Finance & EconomyGuangxi University of Finance and EconomicsNanningChina
  2. 2.Guangxi Key Laboratory Cultivation Base of Cross-Border E-commerce Intelligent Information ProcessingGuangxi University of Finance and EconomicsNanningChina
  3. 3.School of Information and StatisticsGuangxi University of Finance and EconomicsNanningChina
  4. 4.College of Information TechnologyShanghai Ocean UniversityShanghaiChina
  5. 5.College of Mathematics and Information ScienceGuangxi UniversityNanningChina

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