Study of a class of subharmonic motions using a non-smooth temporal transformation (NSTT)

Valery N. Pilipchuk, Alexander F. Vakakis, M. A.F. Azeez

Research output: Contribution to journalArticlepeer-review


A nonlinear boundary value problem (NBVP) formulation for computing strongly nonlinear subharmonic orbits of a class of harmonically forced conservative systems is presented. The formulation is based on a non-smooth temporal transformation (NSTT), to replace the independent temporal variable of the problem with two new piecewise-linear periodic temporal variables. As a result, the problem of computing the subharmonic motions is reduced to a set of NBVPs with homogeneous boundary conditions. Some numerical and analytical solutions of the reduced NBVPs are given, and the asymptotic behavior in the limit of large period of the derived analytical approximations is discussed. In particular, the formulation shows a definite connection between vibro-impact oscillations considered in mechanics and strongly nonlinear oscillations close to a heteroclinic orbit. An interesting feature of a family of subharmonic solutions analyzed in this work is that in the limit of large periods it degenerates to the perturbed stable and unstable manifolds of an unstable periodic orbit of the system.

Original languageEnglish (US)
Pages (from-to)145-164
Number of pages20
JournalPhysica D: Nonlinear Phenomena
Issue number1-2
StatePublished - 1997

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics


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