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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The qualitative analysis of a dynamical system modeling the formation of two-layer scales on pure metals

Author: R. L. Baker
Journal: Proc. Amer. Math. Soc. 123 (1995), 1373-1378
MSC: Primary 34C99; Secondary 34C35, 34D99
MathSciNet review: 1264803
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Abstract: F. Gesmundo and F. Viani have modeled the growth rates of two-oxide scales by the system:

$\displaystyle \frac{{d{q_1}}}{{dt}} = m\frac{{{K_1}}}{{2{q_1}}} - \frac{{m - 1}... ...\frac{{d{q_2}}}{{dt}} = - m\frac{{{K_1}}}{{2{q_1}}} + \frac{{{K_2}}}{{2{q_2}}}.$

We provide a complete qualitative analysis of (1.1) by making use of known results about the general n-dimensional dynamical system:

$\displaystyle \frac{{d{p_i}}}{{dt}} = - \sum\limits_{j = 1}^n {\frac{{{a_{ij}}}}{{{p_j}}},\quad {p_i}(t) > 0,\qquad i = 1, \ldots ,n.} $

We show that for $ m > 1$, the Gesmundo-Viani system admits a unique parabolic solution $ {q_i}(t) = {c_i}\sqrt t ,{c_i} > 0$. This parabolic solution attracts all other solutions. Every solution extends uniquely to a solution on $ [0, + \infty )$, such that the extended solution is eventually monotonically increasing. Finally, the trajectory of any solution coincides with a trajectory of the following linear system:

$\displaystyle \frac{{d{q_1}}}{{dt}} = - \frac{{m - 1}}{m}{\mkern 1mu} \frac{{{K... ...qquad \frac{{d{q_2}}}{{dt}} = \frac{{{K_2}}}{2}{q_1} + m\frac{{{K_1}}}{2}{q_2}.$

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Keywords: Differential equations, dynamical system, nonlinear dynamical system
Article copyright: © Copyright 1995 American Mathematical Society