Abstract: Stability Analysis for 3 by 3 Matrices Under Perturbation

The study of autonomous systems finds applications in many branches of sciences, and stability analysis stands out as a crucial part to understand this kind of dynamical systems. In 1952, Alan Turing proposed that spatial patterns in embryonic morphogenesis were driven by diffusion-induced instability in a system of nonlinear diffusion equations. This problem involves stability analysis of two by two matrices, while more realistic scenarios may introduce a third variable to the system. Inspired by the Turing Instability and the needs for realistic analysis, I will try to determine the features of stable regions for three by three matrices under a perturbation that increases linearly with time. Furthermore, I will examine the procession of eigenvectors and trajectories of solutions, in order to understand to the development of the dynamical system. With the already established Hurwitz condition of matrix stability, we can transform the problem into polynomial analysis, and solve it both analytically and theoretically.