Summary
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Byatt-Smith, J. G. B., “Resonant oscillations in shallow water with small mean-square disturbances,”J. Fluid Mech. 193 (1988), 369–390.
Guckenheimer, J. and Holmes, P.,Nonlinear Oscillations, Dynamical Systems, andBifurcation of Vector Fields. Springer-Verlag (New York), 1983.
Hastings, S. P. and Troy, W. C., “Oscillating solutions of the Falkner-Skan equation for positive β,”J. Differential Equations 71 (1988), 123–144.
Ockendon, H., Ockendon, J. R., and Johnson, A. D., “Resonant sloshing in shallow water,”J. Fluid Mech. 167 (1986), 465–479.
Palmer, K. J., “Transversal heteroclinic points and Cherry's example of a nonintegrable Hamiltonian system,”J. Differential Equations 65 (1986), 321–360.
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.
Author information
Authors and Affiliations
Additional information
Communicated by Stephen Wiggins
Rights and permissions
About this article
Cite this article
Hastings, S.P., McLeod, J.B. On the periodic solutions of a forced second-order equation. J Nonlinear Sci 1, 225–245 (1991). https://doi.org/10.1007/BF01209067
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01209067