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 | References | Similar Articles | Additional Information

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: \[ \frac {{d{q_1}}}{{dt}} = - \frac {{m - 1}}{m}{\mkern 1mu} \frac {{{K_2}}}{2}{q_1} + m\frac {{{K_2}}}{2}{q_2},\qquad \frac {{d{q_2}}}{{dt}} = \frac {{{K_2}}}{2}{q_1} + m\frac {{{K_1}}}{2}{q_2}.\]

- H. C. Akuezue, R. L. Baker, and M. W. Hirsch,
*The qualitative analysis of a dynamical system modeling the formation of multilayer scales on pure metals*, SIAM J. Math. Anal.**25**(1994), no. 4, 1167–1175. MR**1278897**, DOI https://doi.org/10.1137/S003614109119853X
F. Gesmundo and F. Viani,

*The formation of multilayer scales in the parabolic oxidation of pure metals*, J. Corrosion Sci.

**18**(1978), 217-230.

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Keywords:
Differential equations,
dynamical system,
nonlinear dynamical system

Article copyright:
© Copyright 1995
American Mathematical Society