The steady-state bifurcations near a double-zero eigenvalue of the reaction-diffusion equations associated with a trimolecular chemical reaction (the Brusselator) are analysed for the case of one spatial variable and non-flux boundary conditions. The analysis is based on a previously obtained classification of steady-state mode interactions of codimension two that can occur in non-flux boundary-value problems. All possible codimension-two bifurcations are examined for the coupling of the first two modes. It is also shown that degeneracies of codimension greater than two cannot appear. The structurally stable bifurcation diagrams near a codimension-two bifurcation exhibit a number of non-local phenomena like hysteresis, mode-jumping and bifurcations to time-periodic states.
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