An analysis of a nonlinear pendulum-type equation arising in smectic C liquid crystals

We consider a planar sample of non-chiral smectic C liquid crystal to which we impose a tilted static electric field which is augmented by a weak low-frequency alternating field. Under certain conditions it is known from the work by Stewart er nl (Stewart IW, Carlsson T and Leslie FM 1994 Phys. Rev. E 49 2130) that the resulting motion of the c-director may be chaotic. This problem has been studied in detail in Stewart et al (Stewart IW, Carlsson T and Leslie FM 1994 Phys. Rev. E 49 2130, Stewart IW, Carlsson T and Ardill R W B 1996 Phys. Rev. E 54 6413) using a Melnikov analysis approach for a particular form of perturbation when the dielectric anisotropy is assumed to be positive. The addition of the oscillatory term to the field is therefore the cause of more complicated behaviour. In this paper we shall discuss the case when the dielectric anisotropy is assumed to be negative. We shall show that, by considering a linear approximation to the equation of motion, the stability of the c-director cannot be guaranteed. Furthermore, we shall employ the harmonic balance technique to the nonlinear equation in order to determine approximations for the anticipated location of an 'escape' region in parameter space. The corresponding Melnikov criteria for negative dielectric anisotropy will also be found and compared with the approximate 'escape' region.