Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

The qualitative analysis of a dynamical system modeling the formation of two-layer scales on pure metals
HTML articles powered by AMS MathViewer

by R. L. Baker PDF
Proc. Amer. Math. Soc. 123 (1995), 1373-1378 Request permission

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: \[ \frac {{d{q_1}}}{{dt}} = - \frac {{m - 1}}{m}{\mkern 1mu} \frac {{{K_2}}}{2}{q_1} + m\frac {{{K_2}}}{2}{q_2},\qquad \frac {{d{q_2}}}{{dt}} = \frac {{{K_2}}}{2}{q_1} + m\frac {{{K_1}}}{2}{q_2}.\]
References
  • 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 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.
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC: 34C99, 34C35, 34D99
  • Retrieve articles in all journals with MSC: 34C99, 34C35, 34D99
Additional Information
  • © Copyright 1995 American Mathematical Society
  • 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