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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
DOI: https://doi.org/10.1090/S0002-9939-1995-1264803-4
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: \[ \frac {{d{q_1}}}{{dt}} = m\frac {{{K_1}}}{{2{q_1}}} - \frac {{m - 1}}{m} \frac {{{K_2}}}{{2{q_2}}},\qquad \frac {{d{q_2}}}{{dt}} = - m\frac {{{K_1}}}{{2{q_1}}} + \frac {{{K_2}}}{{2{q_2}}}.\] We provide a complete qualitative analysis of (1.1) by making use of known results about the general n-dimensional dynamical system: \[ \frac {{d{p_i}}}{{dt}} = - \sum \limits _{j = 1}^n {\frac {{{a_{ij}}}}{{{p_j}}},\quad {p_i}(t) > 0,\qquad i = 1, \ldots ,n.} \] 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: \[ \frac {{d{q_1}}}{{dt}} = - \frac {{m - 1}}{m}{\mkern 1mu} \frac {{{K_2}}}{2}{q_1} + m\frac {{{K_2}}}{2}{q_2},\qquad \frac {{d{q_2}}}{{dt}} = \frac {{{K_2}}}{2}{q_1} + m\frac {{{K_1}}}{2}{q_2}.\]


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