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The coincidence of a pitchfork and Hopf bifurcation at a Takens-Bogdanov (TB)bifurcation occurs in many physical systems such as double-diffusiveconvection, binary convection and magnetoconvection. Analysis of the associatednormal form, in one dimension with periodic boundary condition, shows theexistence of steady patterns, standing waves, modulated waves and travellingwaves, where the values of coefficients of the terms in the normal formclassify all possible different bifurcation scenarios in the neighbourhood ofthe TB bifurcation (Dangelmayr & Knobloch, 1987). In this work we develop a newand 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 thedynamics in a wide range of bifurcation scenarios, including in domains muchwider than the lengthscale of the pattern. We identify two bifurcationscenarios in which coexistence between different types of solutions isindicated from the analysis of the normal form equation. In these scenarios, welook for spatially localised solutions by examining pattern formation in widedomains. We recover two types of localised states, that of a localised steadystate in the background of the trivial state and that of a spatially localisedtravelling wave in the background of the trivial state which have previouslybeen observed in other systems. Additionally, we identify two new types ofspatially localised states: that of a localised steady state in a modulatedwave background and that of a localised travelling wave in a steady statebackground. The PDE model is easy to solve numerically in large domains and sowill allow further investigation of pattern formation with a TB bifurcation inone or more dimensions and the exploration of a range of background andforeground pattern combinations beyond steady states.

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