Bifurcation structure and asymmetric sequences of localized structures

A mathematical description of the bifurcation sequence of a general class
of localised structures (LS) was recently given in Refs. [1,2]. Here, we
analyse in detail the structure of the phase space on a specific nonlinear
optical system. We find that the stable and unstable manifolds of homogeneous and pattern solutions present a much higher level of complexity than previously assumed, including the existence of additional fronts and localised solutions.

We also analyse the properties of asymmetric LS clusters (non-reversible
trajectories). Such states are not, in general, homoclinic trajectories of a
reversible dynamical system. We show, however, that asymmetric states can be
moving solutions of the partial differential equation. The extra degree of freedom associated with their motion then allows such solutions to persist
over a finite parameter range. The level of complexity is therefore increased with respect to that present in the ordinary differential equations describing the stationary solutions.

[1] P.D. Woods and A.R. Champneys, Physica D 129, 147 (1999).

[2] P. Coullet, C. Riera and C. Tresser, Phys. Rev. Lett. 84, 3069 (2000); Chaos 14, 193 (2004).