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Bifurcations in a Mathieu Equation With Cubic Nonlinearities: Part II

[+] Author Affiliations
Leslie Ng, Richard Rand

Cornell University, Ithaca, NY

Paper No. IMECE2002-32410, pp. 435-443; 9 pages
doi:10.1115/IMECE2002-32410
From:
  • ASME 2002 International Mechanical Engineering Congress and Exposition
  • Design Engineering
  • New Orleans, Louisiana, USA, November 17–22, 2002
  • Conference Sponsors: Design Engineering Division
  • ISBN: 0-7918-3628-2 | eISBN: 0-7918-1691-5, 0-7918-1692-3, 0-7918-1693-1
  • Copyright © 2002 by ASME

abstract

In a previous paper [6], the authors investigated the dynamics of the equation:

d2xdt2 + (δ + ε cos t)x
  + εAx3 + Bx2dxdt + Cxdxdt2 + Ddxdt3 = 0
. We used the method of averaging in the neighborhood of the 2:1 resonance in the limit of small forcing and small nonlinearity. We found that a degenerate bifurcation point occurs in the resulting slow flow and some of the bifurcations near this point were looked at. In this work we present additional results concerning the bifurcations around this point using analytic techniques and AUTO. An analytic approximation for a heteroclinic bifurcation curve is obtained. Additional results on the bifurcations of periodic orbits in the slow flow are also presented.

Copyright © 2002 by ASME

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