Understanding the global solutions of the capillary-gravity wave problem
Abstract We attempt to unravel the maze of solutions which occur in the capillary-gravity wave problem. This is done by following two-parameter paths of turning and bifurcation points which originate at the trivial solution at points of mode interaction. Important changes in the solution structure occur at points on these paths which correspond to higher order singularities. From the numerical results obtained, the bifurcation diagrams are sketched for various values of the surface tension parameter using local information on the changes which occur at the higher order singularities and the mode interactions. This leads to the discovery of more paths of turning points which do not connect to the trivial solution. Cascades of period doubling bifurcations are also considered.
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