Limiting behavior and complex dynamics of all solitary waves in the two-component Dullin-Gottwald-Holm equation

All possible exotic and smooth solitary wave solutions to the two-component Dullin-Gottwald-Holm equation are investigated. We classify this equation in specified regions of the parametric space. Moreover, we give the limiting relations of all different solitary waves as the parameters trend to some special values. All solitary waves suffer from external perturbations, and these solutions turn to the chaotic state easily. In view of the variation of the control coefficient, the smooth solitary wave is the easiest one to be controlled into a stable state and the cusped solitary wave is the most difficult to be controlled under the same controller condition.