Dai Qinrui
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China.
Chaos. 2022 Aug;32(8):083128. doi: 10.1063/5.0104854.
A modified high-temperature superconducting maglev model is studied in this paper, mainly considering the influence of time delay on the dynamic properties of the system. For the original model without time delay, there are periodic equilibrium points. We investigate its stability and Hopf bifurcation and study the bifurcation properties by using the center manifold theorem and the normal form theory. For the delayed model, we mainly study the co-dimension two bifurcations (Bautin and Hopf-Hopf bifurcations) of the system. Specifically, we prove the existence of Bautin bifurcation and calculate the normal form of Hopf-Hopf bifurcation through the bifurcation theory of functional differential equations. Finally, we numerically simulate the abundant dynamic phenomena of the system. The two-parameter bifurcation diagram in the delayed model is given directly. Based on this, some nontrivial phenomena of the system, such as periodic coexistence and multistability, are well presented. Compared with the original ordinary differential equation system, the introduction of time delay makes the system appear chaotic behavior, and with the increase in delay, the variation law between displacement and velocity becomes more complex, which provides further insights into the dynamics of the high-temperature superconducting maglev model.
本文研究了一种改进的高温超导磁悬浮模型，主要考虑了时延对系统动态特性的影响。对于无时延的原模型，存在周期平衡点。我们研究其稳定性和霍普夫分岔，并利用中心流形定理和范式理论研究分岔特性。对于时延模型，我们主要研究系统的二维余维分岔（鲍廷分岔和霍普夫-霍普夫分岔）。具体来说，我们通过泛函微分方程的分岔理论证明了鲍廷分岔的存在性，并计算了霍普夫-霍普夫分岔的范式。最后，我们对系统丰富的动态现象进行了数值模拟。直接给出了时延模型中的双参数分岔图。在此基础上，很好地呈现了系统的一些非平凡现象，如周期共存和多重稳定性。与原常微分方程系统相比，时延的引入使系统出现混沌行为，并且随着时延的增加，位移和速度之间的变化规律变得更加复杂，这为高温超导磁悬浮模型的动力学提供了进一步的见解。
