Homoclinic crossings and pattern selection

The correlations between a homoclinic orbit (HMO) and coherent patterns in the nonlinear Schr\"odinger model are discussed. In a Hamiltonian situation, two independent patterns are revealed to exist: One corresponds to a motion within an HMO, and the other outside of an HMO. The study further illustrates, when the Hamiltonian constraint is released, the significance of an HMO on the patterns dynamics by presenting the irregular HMO crossings and the resulting chaotic selections between the patterns.