Nonlinear Dynamics

, Volume 94, Issue 4, pp 2511–2525 | Cite as

A repeated yielding model under periodic perturbation

  • Yiwen Tao
  • Xueping Li
  • Jingli Ren
Original Paper


The Ananthakrishna model, seeking to explain the phenomenon of repeated yielding of materials, is studied with or without periodic perturbation. For the unforced model, Hopf bifurcation, degenerate Hopf bifurcation and saddle-node bifurcation are detected. For the periodically forced model, two elementary periodic mechanisms are analyzed corresponding to five bifurcation cases of the unforced one. Rich dynamical behaviors arise, including stable and unstable periodic solutions of different periods, quasi-periodic solutions, chaos through torus destruction or cascade of period doublings. Moreover, even small change of a parameter can lead to bifurcation of different periodic solutions. Finally, according to the forced Ananthakrishna model, four types of stress–time curves are simulated, which can well interpret various experimental phenomena of repeated yielding.


Ananthakrishna model Bifurcation Attractor Periodically forced Repeated yielding 



This research is supported by the National Natural Science Foundation of China (11771407), the Innovative Research Team of Science and Technology in Henan Province (17IRTSTHN007) and the National Key R&D Program of China (2017YFB0702500).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest concerning the publication of this manuscript.


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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsZhengzhou UniversityZhengzhouPeople’s Republic of China

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