Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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.

 

Quadratic differential equations in the complex domain I
HTML articles powered by AMS MathViewer

by Nora C. Hopkins PDF
Trans. Amer. Math. Soc. 367 (2015), 6771-6782 Request permission

Abstract:

By complexifying all of the variables of an ordinary real quadratic vector differential equation to get a differential equation over $\mathbb {C}$, it is shown that the solution to the complex differential equation can be uniquely defined on an open star-shaped subset of $\mathbb {C}$, dependent on the initial point, containing the maximum interval of existence of the real differential equation. Complex conjugation is shown to commute with solving the differential equation on this complex domain, and well-known algebraic properties of the solutions to the real differential equation are generalized to the equation over $\mathbb {C}$.
References
  • Einar Hille, Ordinary differential equations in the complex domain, Dover Publications, Inc., Mineola, NY, 1997. Reprint of the 1976 original. MR 1452105
  • N. C. Hopkins, Quadratic differential equations in the complex domain II, forthcoming.
  • Nora C. Hopkins and Michael K. Kinyon, Automorphism eigenspaces of quadratic differential equations and qualitative theory, Differential Equations Dynam. Systems 5 (1997), no. 2, 121–138. MR 1657250
  • E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944. MR 0010757
  • N. Jacobson, Forms of algebras, Some Recent Advances in the Basic Sciences, Vol. 1 (Proc. Annual Sci. Conf., Belfer Grad. School Sci., Yeshiva Univ., New York, 1962-1964) Yeshiva Univ., Belfer Graduate School of Science, New York, 1966, pp. 41–71. MR 0214628
  • Michael K. Kinyon and Arthur A. Sagle, Quadratic dynamical systems and algebras, J. Differential Equations 117 (1995), no. 1, 67–126. MR 1320184, DOI 10.1006/jdeq.1995.1049
  • Michael K. Kinyon and Arthur A. Sagle, Automorphisms and derivations of differential equations and algebras, Rocky Mountain J. Math. 24 (1994), no. 1, 135–154. 20th Midwest ODE Meeting (Iowa City, IA, 1991). MR 1270032, DOI 10.1216/rmjm/1181072457
  • Michael K. Kinyon and Arthur A. Sagle, Differential systems and algebras, Differential equations, dynamical systems, and control science, Lecture Notes in Pure and Appl. Math., vol. 152, Dekker, New York, 1994, pp. 115–141. MR 1243197
  • Michael K. Kinyon and Sebastian Walcher, On ordinary differential equations admitting a finite linear group of symmetries, J. Math. Anal. Appl. 216 (1997), no. 1, 180–196. MR 1487259, DOI 10.1006/jmaa.1997.5668
  • L. Marcus, Quadratic differential equations and nonassociative algebras, in Contributions to the Theory of Nonlinear Oscillations, Vol. V, L. Cesair, J.P. LaSalle, and S. Lefschetz (eds.), Princeton Univ. Press, Princeton, 1960, 185-213.
  • Richard D. Schafer, An introduction to nonassociative algebras, Pure and Applied Mathematics, Vol. 22, Academic Press, New York-London, 1966. MR 0210757
  • Sebastian Walcher, Algebras and differential equations, Hadronic Press Monographs in Mathematics, Hadronic Press, Inc., Palm Harbor, FL, 1991. MR 1143536
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 34M99, 34C14, 17A36
  • Retrieve articles in all journals with MSC (2010): 34M99, 34C14, 17A36
Additional Information
  • Nora C. Hopkins
  • Affiliation: Department of Mathematics and Computer Science, Indiana State University, Terre Haute, Indiana 47809
  • MR Author ID: 217047
  • Received by editor(s): March 11, 2009
  • Published electronically: June 16, 2015
  • © Copyright 2015 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 367 (2015), 6771-6782
  • MSC (2010): Primary 34M99, 34C14, 17A36
  • DOI: https://doi.org/10.1090/tran/5318
  • MathSciNet review: 3378813