The asymptotic perturbation method is used to analyze the nonlinear vibrations and chaotic dynamics of a rotor-active magnetic bearing (AMB) system with 16-pole legs and the time-varying stiffness. Based on the expressions of the electromagnetic force resultants, the influences of some parameters, such as the cross-sectional area Aα of one electromagnet and the number N of windings in each electromagnet coil, on the electromagnetic force resultants are considered for the rotor-AMB system with 16-pole legs. Based on the Newton law, the governing equation of motion for the rotor-AMB system with 16-pole legs is obtained and expressed as a two-degree-of-freedom system with the parametric excitation and the quadratic and cubic nonlinearities. According to the asymptotic perturbation method, the four-dimensional averaged equation of the rotor-AMB system is derived under the case of 1 : 1 internal resonance and 1 : 2 subharmonic resonances. Then, the frequency-response curves are employed to study the steady-state solutions of the modal amplitudes. From the analysis of the frequency responses, both the hardening-type nonlinearity and the softening-type nonlinearity are observed in the rotor-AMB system. Based on the numerical solutions of the averaged equation, the changed procedure of the nonlinear dynamic behaviors of the rotor-AMB system with the control parameter is described by the bifurcation diagram. From the numerical simulations, the periodic, quasiperiodic, and chaotic motions are observed in the rotor-active magnetic bearing (AMB) system with 16-pole legs, the time-varying stiffness, and the quadratic and cubic nonlinearities.
This paper introduces a new four-dimensional hyperchaotic financial system on the basis of an established three-dimensional nonlinear financial system and a dynamic model by adding a controller term to consider the effect of control on the system. In terms of the proposed financial system, the sufficient conditions for nonexistence of chaotic and hyperchaotic behaviors are derived theoretically. Then, the solutions of equilibria are obtained. For each equilibrium, its stability and existence of Hopf bifurcation are validated. Based on corresponding first Lyapunov coefficient of each equilibrium, the analytical proof of the existence of periodic solutions is given. The ultimate bound and positively invariant set for the financial system are obtained and estimated. There exists a stable periodic solution obtained near the unstable equilibrium point. Finally, the dynamic behaviors of the new system are explored from theoretical analysis by using the bifurcation diagrams and phase portraits. Moreover, the hyperchaotic financial system has been simulated using a specially designed electronic circuit and viewed on an oscilloscope, thereby confirming the results of the numerical integrations and its real contribution to engineering.
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