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


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 oscillations shooting methods homoclinic chaos 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Byatt-Smith, J. G. B., “Resonant oscillations in shallow water with small mean-square disturbances,”J. Fluid Mech. 193 (1988), 369–390.Google Scholar
  2. 2.
    Guckenheimer, J. and Holmes, P.,Nonlinear Oscillations, Dynamical Systems, andBifurcation of Vector Fields. Springer-Verlag (New York), 1983.Google Scholar
  3. 3.
    Hastings, S. P. and Troy, W. C., “Oscillating solutions of the Falkner-Skan equation for positive β,”J. Differential Equations 71 (1988), 123–144.Google Scholar
  4. 4.
    Ockendon, H., Ockendon, J. R., and Johnson, A. D., “Resonant sloshing in shallow water,”J. Fluid Mech. 167 (1986), 465–479.Google Scholar
  5. 5.
    Palmer, K. J., “Transversal heteroclinic points and Cherry's example of a nonintegrable Hamiltonian system,”J. Differential Equations 65 (1986), 321–360.Google Scholar
  6. 6.
    Wiggins, S., “On the detection and dynamical consequences of orbits homoclinic to hyperbolic periodic orbits and normally hyperbolic invariant tori in a class of ordinary differential equations,”SIAMJ. Appl. Math. 48 (1988), 262–285.Google Scholar

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

Personalised recommendations