Abstract
We discuss in this paper the bifurcation, stability and chaos of the non-linear Duffing oscillator with a PID controller. Hopf bifurcation can occur and we show that there is a global stable fixed point. The PID controller works well in some fields of the parameter space, but in other fields of the parameter space, or if the reference input is not equal to zero, chaos is common for hard spring type system and so is fractal basin boundary for soft spring system. The Melnikov method is used to obtain the criterion of fractal basin boundary.
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References
Ottino, J. M., Kinematics of Mixing: Stretching, Chaos and Transport, Cambridge University Press, Cambridge, 1989.
Ott, E., Grebogi, C., and Yorke, J. A., ‘Controlling chaos’, Physical Review Letters 64, 1990, 1196–1199.
Romeiras, F. J., Grebogi, C., Ott, E., and Dayawansa, W. P., ‘Controlling chaotic dynamical systems’, Physica D 58, 1992, 165–192.
Pyragas, K., ‘Continuous control of chaos by self-controlling feedback’, Physics Letters A 170, 1992, 421–428.
Guemez, J. and Matias, M. A., ‘Control of chaos in unidimensional maps’, Physics Letters A 181, 1993, 29–32.
Lima, R. and Pettini, M., ‘Suppression of chaos by resonant parametric perturbations’, Physical Review A 41, 1990, 726–733.
Kapitaniak, T., ‘Controlling chaotic oscillator without feedback’, Chaos, Solitons & Fractals 2, 1992, 519–530.
Nayfeh, A.H. and Sanchez, N. E., ‘Bifurcation in a forced softening Duffing oscillator’, International Journal of Non-Linear Mechanics 24, 1989, 483–497.
Holmes, P., ‘Bifurcation and chaos in a simple feedback system’, in Proceedings of the 22th IEEE Conference on Decision and Control, San Antonio, Texas, U.S.A., 1983, pp. 365–370.
Cook, P. A., ‘Simple feedback systems with chaotic behavior’, System Control Letters 6, 1985, 223–227.
Cook, P. A., ‘Chaotic behavior in feedback systems’, in Proceedings of the 25th IEEE Conference on Decision and Control, Athens, Greece, 1986, pp. 1151–1154.
Carr, J., Application of Manifold Theory, Springer-Verlag, New York, Heidelberg, Berlin, 1981.
Gukenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields, Springer-Verlag, New York, 1983.
Zhang, J., Bifurcation and Geometrical Theory of Ordinary Differential Equations, Peking University, Beijing, 1987 (in Chinese).
Wiggins, S., Global Bifurcation and Chaos, Springer-Verlag, New York, 1988.
Lurie, A. I. and Postnikov, V. N., ‘On the theory of stability of control systems’, Prikladnaya Matematika i Mekhanika 8, 1944, 246–248.
Huang, L., Theory of Stability, Peking University, Beijing, 1992 (in Chinese).
Ueda, Y., ‘Randomly transitional phenomena in the system governed by Duffing's equation’, Journal of Statistical Physics 20, 1979, 181–196.
Yang, C. Y., Cheng, A. H. D., and Roy, R. V., ‘Chaotic and stochastic dynamics for a nonlinear structural system with hysteresis and degration’, Journal of Probabilistic Engineering Mechanics 6, 1991, 193–203.
Cheng, A. H. D., Yang, C. Y., Hackl, K., and Chajes, M. J., ‘Stability, bifurcation and chaos of non-linear structures with control – II. Non-autonomous case’, International Journal of Non-Linear Mechanics 28, 1993, 549–565.
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Cui, F., Chew, C.H., Xu, J. et al. Bifurcation and Chaos in the Duffing Oscillator with a PID Controller. Nonlinear Dynamics 12, 251–262 (1997). https://doi.org/10.1023/A:1008204332684
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DOI: https://doi.org/10.1023/A:1008204332684