Spatial localisation beyond steady states in the neighbourhood of the
Takens--Bogdanov bifurcation

The coincidence of a pitchfork and Hopf bifurcation at a Takens-Bogdanov (TB)
bifurcation occurs in many physical systems such as double-diffusive
convection, binary convection and magnetoconvection. Analysis of the associated
normal form, in one dimension with periodic boundary condition, shows the
existence of steady patterns, standing waves, modulated waves and travelling
waves, where the values of coefficients of the terms in the normal form
classify all possible different bifurcation scenarios in the neighbourhood of
the TB bifurcation (Dangelmayr & Knobloch, 1987). In this work we develop a new
and simple pattern-forming PDE model, based on the Swift-Hohenberg equation,
adapted to have the TB normal form at onset, which allows us to explore the
dynamics in a wide range of bifurcation scenarios, including in domains much
wider than the lengthscale of the pattern. We identify two bifurcation
scenarios in which coexistence between different types of solutions is
indicated from the analysis of the normal form equation. In these scenarios, we
look for spatially localised solutions by examining pattern formation in wide
domains. We recover two types of localised states, that of a localised steady
state in the background of the trivial state and that of a spatially localised
travelling wave in the background of the trivial state which have previously
been observed in other systems. Additionally, we identify two new types of
spatially localised states: that of a localised steady state in a modulated
wave background and that of a localised travelling wave in a steady state
background. The PDE model is easy to solve numerically in large domains and so
will allow further investigation of pattern formation with a TB bifurcation in
one or more dimensions and the exploration of a range of background and
foreground pattern combinations beyond steady states.