Two-parameter bifurcations analysis of a delayed high-temperature superconducting maglev model with guidance force.

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.