CHAOS, 15(1), March, 2005
The truncated forced non-linear Schrödinger (NLS) model is
known to mimic well the forced NLS solutions in the regime at
which only one linearly unstable mode exists. Using a novel
framework in which a hierarchy of bifurcations is constructed, we
analyze this truncated model and provide insights regarding its
global structure and the type of instabilities which appear in it.
In particular, the significant role of the forcing frequency is
revealed and it is shown that a parabolic resonance mechanism of
instability arises in the relevant parameter regime of this model.
Numerical experiments demonstrating the different types of chaotic
motion which appear in the model are provided.
Putting an order in a multi-dimensional chaotic system by classifying all the different types of trajectories and finding their corresponding phase space regions is, in general, a formidable and perhaps even unattainable task. Near integrable Hamiltonian systems are a fascinating playground in this respect as some rough classification may be found. Indeed, we demonstrate here that in some cases their structure may be well described via the construction of a three level hierarchy of bifurcations. The analysis reveals, in a systematic way, what are the typical and singular solutions on a given energy level and how these are altered as the energy level and the parameters are varied. In particular, all the different types of singular unperturbed solutions arising in a given model may be classified. The various types of chaotic trajectories which are produced by the perturbation in the neighborhood of such solutions are shown. The concrete system we analyze is a two-mode truncation of the forced one dimensional non-linear Schrödinger equation, an equation which describes many phenomena in Physics such as the Bose-Einstein condensation. Our analysis explains the phase space structure of this extensively studied reduced model, discloses the significance of the forcing frequency parameter and reveals new types of chaotic solutions in it.