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Global optimal control and system-dependent solutions in the hardening Helmholtz–Duffing oscillator. (English) Zbl 1060.93527
Summary: A method for controlling nonlinear dynamics and chaos based on avoiding homo/heteroclinic bifurcations is applied to the hardening Helmholtz-Duffing oscillator, which is an archetype of a class of asymmetric oscillators having distinct homoclinic bifurcations occurring for different values of the parameters. The Melnikov’s method is applied to analytically detect these bifurcations, and, in the spirit of the control method developed by the authors, the results are used to select the optimal shape of the excitation permitting the maximum shift of the undesired bifurcations in parameter space. The two main novelties with respect to previous authors’ works consist in the occurrence (i) of a more involved control scenario, and (ii) of system-dependent optimal solutions. In particular, the possibility of having global control ”with” or ”without symmetrization”, related to the optimization of the critical amplitudes versus the critical gains, is illustrated. In the former case, two sub-cases are found, namely, ”pursued” and ”achieved” symmetrization, which take care of the possibly simultaneous occurrence of right and left homoclinic bifurcations under optimal excitation.

93C10 Nonlinear systems in control theory
93C15 Control/observation systems governed by ordinary differential equations
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[1] Alaggio R, Benedettini F. The use of experimental tests in the formulation of analytical models for the finite forced dynamics of planar arches. In: Proc. DECT’01 2001 ASME Design Eng Tech Conf, Pittsburgh, Pennsylvania, USA, 9-12 September 2001 (CD ROM DETC2001/VIB21613)
[2] Gjelsvik, A.; Bodner, S.R., The energy criterion and snap buckling of arches, ASCE J. eng. mech. div, October, 89-134, (1962)
[3] Grebogi, C.; Ott, E.; Yorke, J.A., Crises, sudden changes in chaotic attractors, and transient chaos, Physica D, 7, 181-200, (1983) · Zbl 0561.58029
[4] Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, (1983), Springer-Verlag New York · Zbl 0515.34001
[5] Higuchi K, Dowell EH. Effect of constant transverse force on chaotic oscillations of sinusoidally excited buckled beam. In: Schiehlen W, editor. Proc. of IUTAM Symposium, Nonlinear dynamics in engineering systems. Stuttgart, Germany 21-25 August 1989. Berlin: Springer-Verlag; 1989. p. 99-106
[6] Katz, A.; Dowell, E.H., From single well chaos to cross well chaos: a detailed explanation in terms of manifold intersections, Int. J. bif. chaos, 4, 933-941, (1994) · Zbl 0870.58037
[7] Kovacic, G.; Wiggins, S., Orbits homoclinic to resonance, with an application to chaos in a model of the forced and damped sine-Gordon equation, Physica D, 57, 185-225, (1992) · Zbl 0755.35118
[8] Lenci S, Rega G. Global chaos control in a periodically forced oscillator. In: Bajaj AK, Namachchivaya NS, Franchek MA, editors. Proc ASME Int Mech Eng Congress. Nonlinear Dynamics and Control, DE-vol. 91. 17-22 Nov. 1996, Atlanta, GA. p. 111-6
[9] Lenci, S.; Rega, G., A procedure for reducing the chaotic response region in an impact mechanical system, Nonlinear dyn, 15, 391-409, (1998) · Zbl 0914.70017
[10] Lenci, S.; Rega, G., Controlling nonlinear dynamics in a two-well impact system. parts I and II, Int. J. bif. chaos, 8, 2387-2424, (1998) · Zbl 0973.34501
[11] Lenci, S.; Rega, G., Optimal control of nonregular dynamics in a Duffing oscillator, Nonlinear dyn, 33, 71-86, (2003) · Zbl 1038.70019
[12] Lenci, S.; Rega, G., Optimal numerical control of single-well to cross-well chaos transition in mechanical systems, Chaos, solitons & fractals, 15, 173-186, (2003) · Zbl 1058.70027
[13] Lenci, S.; Rega, G., Optimal control of homoclinic bifurcation: theoretical treatment and practical reduction of safe basin erosion in the Helmholtz oscillator, J. vib. control, 9, 281-316, (2003) · Zbl 1156.70314
[14] Lenci S, Rega G. A unified control framework of the nonregular dynamics of mechanical oscillators. J Sound Vibr, in press · Zbl 1236.70050
[15] Mettler, E., Dynamic buckling, ()
[16] Moon, F.C., Chaotic and fractal dynamics. an introduction for applied scientists and engineers, (1992), Wiley New York
[17] Moon, F.C.; Cusumano, J.; Holmes, P.J., Evidence for homoclinic orbits as a precursor to chaos in a magnetic pendulum, Physica D, 24, 383-390, (1987) · Zbl 0607.70027
[18] Nelder, J.A.; Mead, R., A simplex method for function minimization, Comput. J, 7, 308-313, (1964) · Zbl 0229.65053
[19] Pinto, O.C.; Gonçalves, P.B., Non-linear control of buckled beams under step loading, Mech. syst. sign. proc, 14, 967-985, (2000)
[20] Shaw, S.W., The suppression of chaos in periodically forced oscillators, (), 289-296
[21] Szemplinska-Stupnicka, W., Cross-well chaos and escape phenomena in driven oscillators, Nonlinear dyn, 3, 225-243, (1992)
[22] Szemplinska-Stupnicka, W., The analytical predictive criteria for chaos and escape in nonlinear oscillators: a survey, Nonlinear dyn, 7, 129-147, (1995)
[23] Thompson, J.M.T.; Stewart, H.B., Nonlinear dynamics and chaos, (1986), Wiley New York · Zbl 0601.58001
[24] Thompson, J.M.T., Chaotic phenomena triggering the escape from a potential well, Proc. R. soc. lond. A, 421, 195-225, (1989) · Zbl 0674.70035
[25] Thompson, J.M.T.; McRobie, F.A., Indeterminate bifurcations and the global dynamics of driven oscillators, (), 107-128 · Zbl 0794.70017
[26] Tseng, W.-Y.; Dugundji, J., Nonlinear vibrations of a buckled beam under harmonic excitation, ASME J. appl. mech, 38, 467-476, (1971) · Zbl 0218.73059
[27] Wiggins, S., Global bifurcation and chaos: analytical methods, (1988), Springer-Verlag New York, Heidelberg, Berlin
[28] Wiggins, S., Introduction to applied nonlinear dynamical systems and chaos, (1990), Springer-Verlag New York, Heidelberg, Berlin · Zbl 0701.58001
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