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We study BPS domain walls of N = 1 supergravity coupled to a chiral multiplet and their Lorentz invariant vacuums which can be viewed as critical points of BPS equations and the scalar potential. Supersymmetry further implies that gradient flows of BPS equations controlled by a holomorphic superpotential and the Kähler geometry are unstable near local maximum of the scalar potential, whereas they are stable around local minimum and saddles of the scalar potential. However, the analysis using renormalization group flows shows that such gradient flows do not always exist particularly in infrared region.1

We investigate properties of attractors for the scalar field in the Lorentz violating scalar-vector-tensor theory of gravity. In this framework, both the effective coupling and potential functions determine the stabilities of the fixed points. In the model, we consider the constants of the slope of the effective coupling and potential functions which lead to the quadratic effective coupling vector with the (inverse) power-law potential. For the case of a purely scalar field, there are only two stable attractor solutions in the inflationary scenario. In the presence of a barotropic fluid, the fluid dominated solution is absent. We find two scaling solutions: the kinetic scaling solution and the scalar field scaling solutions. We show the stable attractors in regions of ð; Þ parameter space and in a phase plane plot for different qualitative evolutions. From the standard nucleosynthesis, we derive the constraints for the value of the coupling parameter.