Resonance surfaces for forced oscillators. (English) Zbl 0822.34035

McGehee, Richard P.; Peckham, Bruce B.

Exp. Math. 3, No. 3, 221-244 (1994).

Authors’ abstract: “The study of resonances in systems such as periodically forced oscillators has traditionally focused on understanding the regions in the parameter plane where these resonances occur. Resonance regions can also be viewed as projections to the parameter plane of resonance surfaces in the four-dimensional Cartesian product of the state space with the parameter space.
This paper reports on a computer study of resonance surfaces for a particular family and illustrates some advantages of viewing resonance regions in this light”.

Reviewer: H.Rüßmann (Mainz)

MSC:

34C15	Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C23	Bifurcation theory for ordinary differential equations
34C25	Periodic solutions to ordinary differential equations
70K40	Forced motions for nonlinear problems in mechanics
65C20	Probabilistic models, generic numerical methods in probability and statistics

Keywords:

resonance regions; periodically forced oscillators; computer study of resonance surfaces

References:

[1]	Arnol’d V. I., Geometric Methods in the Theory of Ordinary Differential Equations (1982)
[2]	DOI: 10.1007/BF01213607 · Zbl 0499.70034 · doi:10.1007/BF01213607
[3]	Aronson D., Phys. Rev. 33 pp 2190– (1986) · doi:10.1103/PhysRevA.33.2190
[4]	Bogdanov R. I., Trudy Seminara Imeni I. G. Petrovskogo 2 pp 37– (1976)
[5]	DOI: 10.1007/BF01388549 · Zbl 0578.58031 · doi:10.1007/BF01388549
[6]	DOI: 10.1142/S0218127491000075 · Zbl 0754.70023 · doi:10.1142/S0218127491000075
[7]	Guckenheimer J., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (1983) · Zbl 0515.34001 · doi:10.1007/978-1-4612-1140-2
[8]	DOI: 10.1137/0515083 · Zbl 0554.58040 · doi:10.1137/0515083
[9]	DOI: 10.1143/PTP.61.54 · doi:10.1143/PTP.61.54
[10]	DOI: 10.1016/0009-2509(86)85237-X · doi:10.1016/0009-2509(86)85237-X
[11]	McGehee R. P., ”Resonance City” (1992)
[12]	McGehee, R. P. and Peckham, B. B. ”Determining the Global Topology of Resonance Surfaces for Periodically Forced Oscillator Families”. Proceedings of the Workshop on Normal Forms and Homoclinic Chaos. Providence, RI: Fields Institute Communications Series, Amer. Math. Soc. [McGehee and Peckham 1995] · Zbl 0860.70019
[13]	Peckham B., Ph.D. Thesis, in: ”The closing of resonance horns for periodically forced oscillators” (1988)
[14]	DOI: 10.1088/0951-7715/3/2/002 · Zbl 0704.58035 · doi:10.1088/0951-7715/3/2/002
[15]	DOI: 10.1137/0522099 · Zbl 0744.58059 · doi:10.1137/0522099
[16]	Phillips M., Notices of the Amer. Math. Soc. 40 pp 985– (1993)
[17]	DOI: 10.1016/0375-9601(88)91045-6 · doi:10.1016/0375-9601(88)91045-6
[18]	Takens F., Coram. Math. Inst. Rijksuniversiteit Utrecht 3 pp 1– (1974)
[19]	DOI: 10.1063/1.457235 · doi:10.1063/1.457235