Date of Award

Summer 2016

Document Type


Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)


Mathematical Sciences

First Advisor

Roy Goodman

Second Advisor

Denis L. Blackmore

Third Advisor

Richard O. Moore

Fourth Advisor

Casey Diekman

Fifth Advisor

Marian Gidea


The phase space of the three-mode discrete NLS in the nonlinear regime with periodic boundary conditions is investigated by reducing the degree of freedom from three down to two. The families of standing waves are enumerated and normal forms are used to describe several families of relative periodic orbits whose topologies change due to Hamiltonian Hopf bifurcations and transcritical bifurcations. The Hamiltonian Hopf bifurcation occurs when eigenvalues on the imaginary axis collide and split and has two types: elliptic and hyperbolic. These two types arise in the DNLS problem, and the families of periodic orbits are discussed as a conserved quantity N is changed. The stability of each standing wave solution is discussed both numerically and analytically to describe how the dynamics change under perturbation of the parameter N.

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Mathematics Commons