Stability boundaries of a Mathieu equation having PT symmetry

Abstract

I have applied multiple-scale perturbation theory to a generalized complex P T -symmetric Mathieu equation in order to find the stability boundaries between bounded and unbounded solutions. The analysis suggests that the non-Hermitian parameter present in the equation can be used to control the shape and curvature of these boundaries. Although this was suggested earlier by several authors, analytic formulas for the boundary curves were not given. This paper is a first attempt to fill this gap in the theory.

Limit cycles of a generalised Mathieu differential system

“…where b is real parameter, and the dots denote second derivative with respect to time t. This equation was first discussed in 1868 by Mathieu while studying the problem of vibrations on an elliptical drumhead. Matthieu's equation has many applications in engineering [12,13] and also in theoretical and experimental physics [2,7,14].…”

Energy levels for $\mathcal{PT}$-symmetric deformation of the Mathieu equation

“…We intend to explore the consequences of a non-Hermitian, but PT -symmetric, deformation on the Mathieu equation. In the past few years, different approaches were considered in the literature which proposed some kind of deformation [9,10,[15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]. In the context of PT symmetry, the first works to discuss a periodic potential were Refs.…”