## Bifurcation of multiple limit cycles for a rotor-active magnetic bearings system with time-varying stiffness.(English)Zbl 1147.37349

Summary: The bifurcations of multiple limit cycles for a rotor-active magnetic bearings (AMB) system with the time-varying stiffness are considered in this paper. The governing nonlinear equation of motion is established for the rotor-AMB system with single-degree-of-freedom and parametric excitation. Using the method of multiple scales, the governing nonlinear equation of motion is first transformed to the averaged equation, which is in the form of a $$Z_2$$-symmetric perturbed polynomial Hamiltonian system of degree 5. Then, the bifurcation theory of planar dynamical system and the method of detection function are utilized to analyze the bifurcations of multiple limit cycles of the averaged equation. Four groups of parametric controlling conditions are given to obtain the configurations of compound eyes. It is found that there exist respectively at least 17, 19, 21 and 22 limit cycles in the rotor-AMB system with the time-varying stiffness under the different controlling conditions.

### MSC:

 37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics) 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) 37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems 34C23 Bifurcation theory for ordinary differential equations 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
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