The qualitative analysis of a dynamical system of differential equations arising from the study of multilayer scales on pure metals II

Abstract:

We provide a qualitative analysis of the $n$-dimensional dynamical system: \[ \dot q_i=-\sum _{j=1}^n \frac {a_{ij}}{q_j^k},\quad q_i(t)>0,\qquad i=1,\dots , n, \] where $k$ is an arbitrary positive integer. Under mild algebraic conditions on the constant matrix $A=(a_{ij})$, we show that every solution $\mathbf q(t)$, $t\in [0,a)$, extends to a solution on $[0,+\infty )$, such that $\lim _{t\to +\infty } q_i(t)=+\infty$, for $i=1,\dots , n$. Moreover, the difference between any two solutions approaches $0$ as $t\to +\infty$. We then use this result to give a complete qualitative analysis of a 3-dimensional dynamical system introduced by F. Gesmundo and F. Viani in modeling parabolic growth of three-oxide scales on pure metals.