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Complex dynamics in a permanent-magnet synchronous motor model. (English) Zbl 1129.70329
Summary: This paper characterizes the complex dynamics of the permanent-magnet synchronous motor (PMSM) model with a non-smooth-air-gap, extending the work on the smooth case studied elsewhere. The stability, the number of equilibrium points, and the pitchfork and Hopf bifurcations are analyzed by using bifurcation theory and the center manifold theorem. Numerical simulations not only confirm the theoretical analysis results but also show some more new results including the period-doubling bifurcation, cyclic fold bifurcation, single-scroll and double-scroll chaotic attractors, ribbon-chaotic attractor, as well as intermittent chaos that are different from those reported in the literature before. Moreover, analytical expressions of an approximate stability boundary are given, by computing the local quadratic approximation of the two-dimensional stable manifold at an order-2 saddle point. Combining the existing results with the new results reported in this paper, a fairly complete description of the complex dynamics of the PMSM model is now obtained.

MSC:
70K99 Nonlinear dynamics in mechanics
78A55 Technical applications of optics and electromagnetic theory
Software:
AUTO
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[1] Abed, E.H.; Wang, H.O.; Alexander, J.C., Dynamic bifurcation in a power system model exhibiting voltage collapse, Int. J. bifurcat. chaos, 3, 5, 1169-1176, (1993) · Zbl 0900.34042
[2] Abed, E.H.; Varaiya, P.P., Nonlinear oscillations in power systems, Int. J. electr. power energy syst., 6, 37-43, (1994)
[3] Cahill, D.P.M.; Adkins, B., The permanent magnet synchronous motor, Proc. inst. elec. eng. A, 109, 48, 483-491, (1962)
[4] Chiang, H.D.; Liu, C.W., Chaos in simple power system, IEEE trans. power syst., 8, 4, 1407-1417, (1993)
[5] Doedel EJ, Fairgrient FT, Wang X. Auto 97 continuation and bifurcation software for ordinary differential equations, 1997
[6] Guckenheimer, J.; Holemes, P.J., Nonlinear oscillations dynamical systems and bifurcations of vector fields, (1983), Springer NY
[7] Hsu, I.-D.; Kazavinoff, N.D., Existence and stability of periodic solutions of a third-order nonlinear autonomous systems simulating immune response in animals, Proc. R. soc. edinb. (ser. A), 77, 163-175, (1977) · Zbl 0361.34040
[8] Jing, Z.; Xu, D.S.; Chang, Y.; Chen, L.N., Bifurcations, chaos, and system collapse in a three-node power system, Electr. power energy syst., 25, 443-461, (2003)
[9] Kwatny, H.G.; Pasrija, A.K.; Bahar, L.Y., Static bifurcation in electric power network: loss of steady-state stability and voltage collapse, IEEE trans. circ. syst., 33, 981-991, (1986) · Zbl 0621.94024
[10] Lee, B.; Ajjarapu, V., Period-doubling route to chaos in an electrical power system, Proc. IEEE, 140, 6, 490-496, (1993)
[11] Li, Z.; Park, J.B., Bifurcations and chaos in a permanent-magnet synchronous motor, IEEE trans. CAS (I), 49, 3, 601-611, (2002)
[12] Nayfeh, A.H.; Harb, A.M.; Chin, C.M., Bifurcations in a power system model, Int. J. bifurcat. chaos, 6, 3, 497-512, (1996) · Zbl 0874.34036
[13] Rajesh, K.G.; Padiyar, K.R., Bifurcation analysis of a three node power system with detailed models, Electr. power energy syst., 21, 375-393, (1999)
[14] Shen, J.; Jing, Z., A new detecting method for conditions of existence of Hopf bifurcation, Acta math. appl. sinica, 11, 1, 79-93, (1995) · Zbl 0834.34043
[15] Tan, C.W.; Varghese, M.; Varaiya, P.; Wu, F.F., Bifurcation, chaos, and voltage collapse in power systems, Proc. IEEE, 183, 11, 1484-1496, (1995)
[16] Venkatasubramanian, V.; Ji, W., Numerical approximation of (n−1)-dimensional stable manifolds in large systems such as the power system, Automatica, 33, 10, 1877-1883, (1997) · Zbl 0908.93007
[17] Wiggins, S., Introduction to applied nonlinear dynamical systems and chaos, (1990), Springer Berlink · Zbl 0701.58001
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