Nonlinear Dynamics

, Volume 88, Issue 4, pp 2693–2703 | Cite as

Origin of mixed-mode oscillations through speed escape of attractors in a Rayleigh equation with multiple-frequency excitations

  • Xiujing Han
  • Fubing Xia
  • Chun Zhang
  • Yue Yu
Original Paper


The purpose of this paper is to report a novel route to mixed-mode oscillations (MMOs), i.e., the mechanism which we call speed escape of attractors, based on a Rayleigh equation with multiple-frequency excitations. The fast subsystem exhibits two critical points, which bound the region of a periodic attractor or an equilibrium point attractor and outside which are divergent regions. We show that both the periodic attractor and equilibrium point attractor can rapidly move far away from the original place when the control parameter reaches the critical values. This helps us reveal the novel mechanism to MMOs, and two different types of MMOs, i.e., MMOs of point–point type and MMOs of cycle–cycle type, are thus obtained. Besides, the effects of excitations on the MMOs are explored. We show that the amplitudes and frequencies of excitations may have important influences on MMOs. Our results enrich the possible routes to MMOs as well as the underlying mechanisms of MMOs.


Mixed-mode oscillations Speed escape of attractors Multiple-frequency excitations Fast–slow analysis 



The authors express their gratitude to the associate editor and the reviewers whose comments and suggestions have helped the improvements of this paper. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11572141, 11632008, 11472115, 11502091 and 11402226) and the Training Project for Young Backbone Teacher of Jiangsu University.


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Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Faculty of Civil Engineering and MechanicsJiangsu UniversityZhenjiangPeople’s Republic of China
  2. 2.School of Mathematical ScienceHuaiyin Normal UniversityHuaianPeople’s Republic of China
  3. 3.School of ScienceNantong UniversityNantongPeople’s Republic of China

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