Tricritical point as a crossover between type-I$_s$ and type-II$_s$ bifurcations

A tricritical point as a crossover between (stationary finite-wavelength) type-I$_s$ and (stationary longwave) type-II$_s$ bifurcations is identified in the study of diffusive-thermal (Turing) instability of flames propagating in a Hele-Shaw channel in a direction transverse to a shear flow. Three regimes exhibiting different scaling laws are identified in the neighbourhood of the tricritical point. For these three regimes, sixth-order partial differential equations are obtained governing the weakly nonlinear evolution of unstable solutions near the onset of instability. These sixth-order PDES may be regarded as the substitute for the classical fourth-order Kuramoto--Sivashinsky equation which is not applicable near the tricritical point.