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Quadratic differential equations in the complex domain I

Author: Nora C. Hopkins
Journal: Trans. Amer. Math. Soc. 367 (2015), 6771-6782
MSC (2010): Primary 34M99, 34C14, 17A36
Published electronically: June 16, 2015
MathSciNet review: 3378813
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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}$.

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  • [1] Einar Hille, Ordinary differential equations in the complex domain, Dover Publications, Inc., Mineola, NY, 1997. Reprint of the 1976 original. MR 1452105
  • [2] N. C. Hopkins, Quadratic differential equations in the complex domain II, forthcoming.
  • [3] 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
  • [4] E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944. MR 0010757
  • [5] N. Jacobson, Forms of algebras, Some Recent Advances in the Basic Sciences, Vol. 1 (Proc. Annual Sci. Conf., Belfer Grad. School Sai., Yeshiva Univ., New York, 1962-1964) Belfer Graduate School of Science, Yeshiva Univ., New York, 1966, pp. 41–71. MR 0214628
  • [6] Michael K. Kinyon and Arthur A. Sagle, Quadratic dynamical systems and algebras, J. Differential Equations 117 (1995), no. 1, 67–126. MR 1320184, 10.1006/jdeq.1995.1049
  • [7] 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, 10.1216/rmjm/1181072457
  • [8] 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
  • [9] 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, 10.1006/jmaa.1997.5668
  • [10] 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.
  • [11] Richard D. Schafer, An introduction to nonassociative algebras, Pure and Applied Mathematics, Vol. 22, Academic Press, New York-London, 1966. MR 0210757
  • [12] Sebastian Walcher, Algebras and differential equations, Hadronic Press Monographs in Mathematics, Hadronic Press, Inc., Palm Harbor, FL, 1991. MR 1143536

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Additional Information

Nora C. Hopkins
Affiliation: Department of Mathematics and Computer Science, Indiana State University, Terre Haute, Indiana 47809

Received by editor(s): March 11, 2009
Published electronically: June 16, 2015
Article copyright: © Copyright 2015 American Mathematical Society