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Relaxation and mixed mode oscillations in a shape memory alloy oscillator driven by parametric and external excitations

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  • Yu, Yue
  • Zhang, Cong
  • Chen, Zhenyu
  • Lim, C.W.

Abstract

This paper performs analytical investigations on symmetric jump phenomena reflecting multi-timescale dynamics in a nonlinear shape memory alloy oscillator with parametric and external cosinoidal excitations by means of geometrical singular perturbation theory (GSPT). The conditions concerning the existence of homoclinic and heteroclinic chaos subjected to periodic perturbations are studied in mathematical models by applying Melnikov method. Then we treat the excitation item as the slow variable acting on the bifurcation structure of the fast subsystem and present explicit algebraic expressions of the critical manifold of this subsystem for the investigation of fast-slow motions in the whole system. The study reveals the dynamical features of the relaxation oscillations formed by the alternative stable states on the different branches. In addition, the range of parametric excitation amplitude and stiffness values of the system plays an important role in the singularities and shaping the oscillation patterns. We determine numerically how the parameter values affect the manifold of the fast subsystem during the active fast transitions. Using the fast-slow analysis, one can provide the correct analytical predictions of the region of parameter space where the periodic orbits that alternate between epochs of fast and slow motions occur. Numerical simulations are also carried out to illustrate the validity of our study.

Suggested Citation

  • Yu, Yue & Zhang, Cong & Chen, Zhenyu & Lim, C.W., 2020. "Relaxation and mixed mode oscillations in a shape memory alloy oscillator driven by parametric and external excitations," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
    • Handle: RePEc:eee:chsofr:v:140:y:2020:i:c:s0960077920305415
    DOI: 10.1016/j.chaos.2020.110145
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    References listed on IDEAS

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    1. Siewe, M. Siewe & Kakmeni, F.M. Moukam & Tchawoua, C. & Woafo, P., 2005. "Bifurcations and chaos in the triple-well Φ6-Van der Pol oscillator driven by external and parametric excitations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 357(3), pages 383-396.
    2. Siewe, M. Siewe & Cao, Hongjun & Sanjuán, Miguel A.F., 2009. "Effect of nonlinear dissipation on the basin boundaries of a driven two-well Rayleigh–Duffing oscillator," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1092-1099.
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    Cited by:

    1. Song, Jian & Liu, Shenquan & Wen, Qixiang, 2022. "Geometric analysis of the spontaneous electrical activity in anterior pituitary corticotrophs," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
    2. Bu, Min & Li, Jing & Guo, Rong & Ma, Jinzhong, 2023. "Dynamical responses of a Gaussian colored noise-driven shape memory alloy oscillator with a periodic force," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    3. Zhang, Xiaofang & Li, Hongqing & Jiang, Wenan & Chen, Liqun & Bi, Qinsheng, 2022. "Exploiting multiple-frequency bursting of a shape memory oscillator," Chaos, Solitons & Fractals, Elsevier, vol. 158(C).
    4. Danjin Zhang & Youhua Qian, 2023. "Bursting Oscillations in General Coupled Systems: A Review," Mathematics, MDPI, vol. 11(7), pages 1-16, April.
    5. Zhang, Chun & Ma, Xindong & Bi, Qinsheng, 2022. "Complex mixed-mode oscillations based on a modified Rayleigh-Duffing oscillator driven by low-frequency excitations," Chaos, Solitons & Fractals, Elsevier, vol. 160(C).

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