The mathieu differential equation and generalizations to infinite fractafolds

Abstract

One of the well-studied equations in the theory of ODEs is the Mathieu differential equation. A common approach for obtaining solutions is to seek solutions via Fourier series by converting the equation into an infinite system of linear equations for the Fourier coefficients. We study the asymptotic behavior of these Fourier coefficients and discuss the ways in which to numerically approximate solutions. We present both theoretical and numerical results pertaining to the stability of the Mathieu differential equation and the properties of solutions. Further, based on the idea of using Fourier series, we provide a method in which the Mathieu differential equation can be generalized to be defined on the infinite Sierpinski gasket. We discuss the stability of solutions to this fractal differential equation and describe further results concerning properties and behavior of these solutions. 0 2000 Mathematics Subject Classification. Primary: 34D23; Secondary: 28A80.

Generalized parametric resonance in a spin-1 Bose-Einstein condensate

“…Except for Mathieu's equation (ME) describing parametric resonance, there exist many other generalized equations, such as the ones including damping, delay or quasiperiod, etc [20]. The unstable regions show more complicated and interesting patterns [21]. In 2016, Chapman's group studied the dynamics of populations on |f = 1, m f = 0 in nematic space of a spin-1 Bose-Einstein condensate (BEC) by applying a periodically driving second order Zeeman energy [22].…”