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Active control with delay of catastrophic motion and horseshoes chaos in a single well Duffing oscillator. (English) Zbl 1071.37061
Summary: The control of escape and Melnikov chaos of an harmonically excited particle from a catastrophic (unbounded) single well \(\phi^4\) potential is considered. In the linear limit, the range of the control gain parameter leading to good control is obtained and the effect of time delays on the control force is taken into account. The approximate critical external forcing amplitudes for catastrophe and chaos are obtained by using the energy and Melnikov methods. The control efficiency is found by analysing the behaviour of the external critical forcing amplitude of the controlled system as compared to that of the uncontrolled system.

37N35 Dynamical systems in control
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
93C10 Nonlinear systems in control theory
70Q05 Control of mechanical systems
70K55 Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics
37N05 Dynamical systems in classical and celestial mechanics
Full Text: DOI
[1] Virgin, N.L.; Plaut, R.H.; Cheng, C.C., Prediction of escape from a potential well under harmonic excitation, Int. J. nonlinear mech, 27, 357-365, (1992)
[2] Thompson, J.M.T., Chaotic phenomena triggering the escape from a potential well, Proc. soc. lond. A, 421, 195-225, (1989) · Zbl 0674.70035
[3] Zhang, L.; Yang, C.Y.; Chajes, M.J.; Cheng, A.H.D., Stability of active-tendon structural control with time delay, J. eng. mech. div. ASCE, 119, 1017-1024, (1993)
[4] Cheng, A.H.D.; Yang, C.Y.; Hackl, K.; Chajes, M.J., Stability bifurcation and chaos of non linear structures with control-II, Int. J. nonlinear mech, 28, 549-565, (1993) · Zbl 0789.70019
[5] Nana Nbendjo, B.R.; Tchoukuegno, R.; Woafo, P., Active control with delay of vibration and chaos in a double-well Duffing oscillator, Chaos, solitons & fractals, 18, 345-353, (2003) · Zbl 1057.37081
[6] Melnikov, V.K., On the stability of the center for some periodic perturbations, Trans. Moscow math. soc, 12, 1-57, (1963)
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