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.

本文利用几何奇异摄动理论（GSPT）对具有参数和外部余弦激励的非线性形状记忆合金振荡器中反映多时标动力学的对称跳跃现象进行了分析研究）。应用梅尔尼科夫方法，在数学模型中研究了存在周期性扰动的同宿和异宿混沌的存在条件。然后，将激励项作为作用在快速子系统的分叉结构上的慢变量，并给出了该子系统的关键流形的显式代数表达式，以研究整个系统中的快慢运动。研究揭示了由不同分支上的交替稳态所形成的弛豫振荡的动力学特征。另外，系统的参量激励幅度和刚度值的范围在奇异性和成形振荡模式中起着重要作用。我们通过数值确定参数值在活动的快速过渡期间如何影响快速子系统的流形。使用快速慢速分析，可以对参数空间的区域提供正确的分析预测，其中在快速运动和慢速运动的周期之间交替出现的周期性轨道。还进行了数值模拟以说明我们研究的有效性。