Journal of Nonlinear Science

, Volume 1, Issue 2, pp 225–245

On the periodic solutions of a forced second-order equation

  • S. P. Hastings
  • J. B. McLeod

DOI: 10.1007/BF01209067

Cite this article as:
Hastings, S.P. & McLeod, J.B. J Nonlinear Sci (1991) 1: 225. doi:10.1007/BF01209067


We study the equations for two-dimensional irrotational motion induced in a rectangular tank of water which is forced to oscillate horizontally in a periodic fashion (“shallow water sloshing”). The problem reduces to a second-order non-autonomous ordinary differential equation. We rigorously show the existence of many solutions obtained previously by numerical computations, and in addition we show that there are many other bounded solutions. We use simple shooting to obtain irregular solutions of the kind obtained by the methods of dynamical systems. In addition, we obtain a kind of “kneading” theory whereby we can order the initial conditions according to the pattern of “spikes” exhibited by the solutions.

Key words

nonlinear oscillationsshooting methodshomoclinic chaos

Copyright information

© Springer-Verlag New York Inc 1991

Authors and Affiliations

  • S. P. Hastings
    • 1
  • J. B. McLeod
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of PittsburghPittsburghUSA