TY - JOUR
T1 - Global dynamics of parametrically excited nonlinear reversible systems with nonsemisimple 1:1 resonance
AU - Malhotra, N.
AU - Namachchivaya, N. Sri
N1 - Funding Information:
This research was partially supported by Air Force Office of Scientific Research (AFOSR) through Grant 93-0063 and the National Science Foundation (NSF) through Grant MSS 90-57437 PYI.
PY - 1995/12/15
Y1 - 1995/12/15
N2 - In this paper, we analytically investigate the global dynamics associated with the nonlinear reversible systems that exhibit Hopf bifurcation in the presence of one-to-one nonsemisimple internal resonance. The effect of periodic parametric excitations is examined on such systems near the principal subharmonic resonance in presence of dissipation. The nonlinear and nonautonomous system is simplified considerably by reducing it to the corresponding four-dimensional normal form. The normal form associated with the reversible systems is obtained as a special case from the general normal form equations obtained in [N. Sri Namachchivaya, M.M. Doyle, W.F. Langford and N. Evans, Normal form for generalized hopf bifurcation with non-semisimple 1:1 resonance, Z. Angew. Math. Phys. (ZAMP) 45 (1994) 312-335]. Under small perturbations arising from parametric excitations and nonreversible dissipation, two mechanisms are identified in such systems that may lead to chaotic dynamics. Explicit restrictions on the system parameters are obtained for both of these mechanisms which lead to this complex behavior. Finally, the results are demonstrated through a two-degree-of-freedom model of a thin rectangular beam vibrating under the action of a pulsating follower force.
AB - In this paper, we analytically investigate the global dynamics associated with the nonlinear reversible systems that exhibit Hopf bifurcation in the presence of one-to-one nonsemisimple internal resonance. The effect of periodic parametric excitations is examined on such systems near the principal subharmonic resonance in presence of dissipation. The nonlinear and nonautonomous system is simplified considerably by reducing it to the corresponding four-dimensional normal form. The normal form associated with the reversible systems is obtained as a special case from the general normal form equations obtained in [N. Sri Namachchivaya, M.M. Doyle, W.F. Langford and N. Evans, Normal form for generalized hopf bifurcation with non-semisimple 1:1 resonance, Z. Angew. Math. Phys. (ZAMP) 45 (1994) 312-335]. Under small perturbations arising from parametric excitations and nonreversible dissipation, two mechanisms are identified in such systems that may lead to chaotic dynamics. Explicit restrictions on the system parameters are obtained for both of these mechanisms which lead to this complex behavior. Finally, the results are demonstrated through a two-degree-of-freedom model of a thin rectangular beam vibrating under the action of a pulsating follower force.
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U2 - 10.1016/0167-2789(95)00214-6
DO - 10.1016/0167-2789(95)00214-6
M3 - Article
AN - SCOPUS:17144373280
SN - 0167-2789
VL - 89
SP - 43
EP - 70
JO - Physica D: Nonlinear Phenomena
JF - Physica D: Nonlinear Phenomena
IS - 1-2
ER -