Quadratic differential equations

in -graded algebras

Authors:
Nora C. Hopkins and Michael K. Kinyon

Journal:
Trans. Amer. Math. Soc. **351** (1999), 4545-4559

MSC (1991):
Primary 34C35, 17A60, 34C20, 17A36

DOI:
https://doi.org/10.1090/S0002-9947-99-02212-6

Published electronically:
July 19, 1999

MathSciNet review:
1475685

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Quadratic differential equations whose associated algebra has an automorphism of order two are studied. Under hypotheses that naturally generalize the cases where the even or odd part of the algebra is one dimensional, the following are examined: structure theory of the associated algebra (ideal structure, simplicity, solvability, and nilpotence), derivations and first integrals, trajectories given by derivations, and Floquet decompositions.

**1.**A. Brauer and C. Noel,*Qualitative Theory of Ordinary Differential Equations*, Dover Press, 1970.**2.**N. C. Hopkins,*Quadratic differential equations in graded algebras*, Nonassociative Algebra and Its Application (S. Gonzalez, ed.), Mathematics and its Applications #303, Kluwer Academic Publishers, 1994, pp. 179-182. MR**96f:17004****3.**N. C. Hopkins and M. K. Kinyon,*Automorphism eigenspaces of quadratic differential equations and qualitative theory*, Diff. Eqs. and Dynamical Systems**5**(1997), 121-138. CMP**99:04****4.**M. K. Kinyon,*Quadratic differential equations on graded structures*, Nonassociative Algebra and Its Applications (S. Gonzalez, ed.), Mathematics and its Applications #303, Kluwer Academic Publisers, 1994, pp. 215-218. MR**96f:17005****5.**M. K. Kinyon and A. A. Sagle,*Quadratic dynamical systems and algebras*, J. Diff. Eq**117**(1995), 67-126. MR**96e:34018****6.**M. K. Kinyon and A. A. Sagle,*Automorphisms and derivations of ordinary differential equations and algebras*, Rocky Mountain Math. J.**24**(1994), 135-154. MR**95d:34015****7.**M. K. Kinyon and S. Walcher,*Ordinary differential equations admitting a finite linear group of symmetries*, J. Math. Anal. Appl.**216**(1997), 180-196. CMP**98:05****8.**L. Markus,*Quadratic differential equations and non-associative algebras*, Contributions to the Theory of Nonlinear Oscillations, Vol. V (L. Cesari, J. P. LaSalle, and S. Lefschetz, eds.), Princeton Univ. Press, Princeton, 1960, pp. 185-213. MR**24:A2580****9.**A. A. Sagle and R. Walde,*Introduction to Lie Groups and Lie Algebras*, Academic Press, New York, 1973. MR**50:13374****10.**R. D. Schafer,*Introduction to Nonassociative Algebras*, Academic Press, New York, 1966. MR**35:1643****11.**S. Walcher,*Algebras and Differential Equations*, Hadronic Press, Palm Harbor, 1991. MR**93e:34002**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
34C35,
17A60,
34C20,
17A36

Retrieve articles in all journals with MSC (1991): 34C35, 17A60, 34C20, 17A36

Additional Information

**Nora C. Hopkins**

Affiliation:
Department of Mathematics and Computer Science, Indiana State University, Terre Haute, Indiana 47809

Email:
hopkins@laurel.indstate.edu

**Michael K. Kinyon**

Affiliation:
Department of Mathematics and Computer Science, Indiana University South Bend, South Bend, Indiana 46634

Email:
mkinyon@iusb.edu

DOI:
https://doi.org/10.1090/S0002-9947-99-02212-6

Received by editor(s):
October 1, 1996

Received by editor(s) in revised form:
June 2, 1997

Published electronically:
July 19, 1999

Article copyright:
© Copyright 1999
American Mathematical Society