×

zbMATH — the first resource for mathematics

Resonant parametric perturbations of the Hopf bifurcation. (English) Zbl 0588.34031
The effects of periodic parametric perturbations on a system undergoing Hopf bifurcation are studied in detail. Of primary interest is the case of resonance between the Hopf bifurcation frequency and the perturbation frequency. The method of alternative problems is used to obtain the small nonlinear periodic solutions. It is shown that for a certain range of parameters, the Hopf solution is modified to give rise to jump response as well as isolated solutions. For some parameter combinations, stable solutions can get unstable and may bifurcate into aperiodic or amplitude modulated motions.

MSC:
34C25 Periodic solutions to ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Marsden, J.E; McCracken, M, The Hopf bifurcation and its applications, (1976), Springer-Verlag New York · Zbl 0346.58007
[2] Rosenblat, S; Cohen, D.S, Periodically perturbed bifurcation. I. simple bifurcation, Stud. appl. math., 63, 1-23, (1980) · Zbl 0442.34040
[3] Rosenblat, S; Cohen, D.S, Periodically perturbed bifurcation. II. Hopf bifurcation, Stud. appl. math., 64, 143-175, (1981) · Zbl 0482.34037
[4] Bajaj, A.K, Interactions between self and parametrically excited motions in articulated tubes, J. appl. mech., 51, 423-429, (1984) · Zbl 0543.73066
[5] Kath, W.L, Resonance in periodically perturbed Hopf bifurcation, Stud. appl. math., 65, 95-112, (1981) · Zbl 0487.34041
[6] Smith, H.L, Nonresonant periodic perturbation of the Hopf bifurcation, Appl. anal., 12, 173-195, (1981) · Zbl 0482.58024
[7] Sinay, L.R; Reiss, E.L, Perturbed panel flutter: A simple model, Aiaa j., 19, 1476-1483, (1981) · Zbl 0468.73060
[8] Strumolo, G.S, Perturbed bifurcation theory for Poiseuille annular flow, J. fluid mech., 130, 59-72, (1983) · Zbl 0514.76056
[9] Nayfeh, A.H; Mook, D.T, Nonlinear oscillations, (1979), Wiley-Interscience New York
[10] Bajaj, A.K, Bifurcating periodic solutions in rotationally symmetric systems, SIAM J. appl. math., 42, 1078-1098, (1982) · Zbl 0506.34041
[11] Hale, J.K, Ordinary differential equations, (1969), Wiley-Interscience New York · Zbl 0186.40901
[12] Chow, S.N; Hale, J.K, Methods of bifurcation theory, (1982), Springer-Verlag New York
[13] Tezak, E.G; Nayfeh, A.H; Mook, D.T, Parametrically excited non-linear multidegree-of-freedom systems with repeated natural frequencies, J. sound vibration, 85, 459-472, (1982) · Zbl 0506.73053
[14] Bajaj, A.K, Primary parametric resonance in a self-excited oscillator, (), 541-544
[15] {\scA. K. Bajaj}, Bifurcations in a parametrically excited nonlinear oscillator, Internat. J. Nonlinear Mech., submitted. · Zbl 0607.70026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.