Analytical Characterization - Amplitude Equations

Abstract

Our main focus in this CHAPTER is analyzing the simple patterns. Simple patterns are when the number of unknowns is less than or equal to one. All of these simple patterns are in 3D however, the variation of these patterns can happen in only one direction, such as stripes, or along two directions, such as hexagons. Others can be along three directions, such as a simple cube, body-cent red cubic, and face-cent red cubic. Each pattern demonstrates the stability of the system at certain values of the dependent parameter. In this chapter, our initial focus is on analyzing the amplitude equations and determining their equilibrium solutions. Once we have obtained the solutions for each pattern, we proceed to assess their stability using the Jacobian matrix. This step is necessary because the stable equilibria of a pattern can become unstable along modes that are not accounted for in the ODEs. Consequently, it is essential to examine the solutions of patterns, such as z-stripes, along other possible directions, such as the z-direction and w-direction. Even if the pattern amplitude is zero along certain directions, we still need to evaluate stability along those directions because perturbations along new directions may amplify. For instance, a mode such as w, which is not considered in the simplified system for the z-stripes pattern, can destabilize a striped pattern associated with wavevectors k1 and k1. Thus, conducting stability analysis using the Jacobian matrix is of utmost importance.