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

Published electronically:
July 19, 1999

MathSciNet review:
1475685

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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.**Nora C. Hopkins,*Quadratic differential equations in graded algebras*, Non-associative algebra and its applications (Oviedo, 1993) Math. Appl., vol. 303, Kluwer Acad. Publ., Dordrecht, 1994, pp. 179–182. MR**1338177****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.**Michael K. Kinyon,*Quadratic differential equations on graded structures*, Non-associative algebra and its applications (Oviedo, 1993) Math. Appl., vol. 303, Kluwer Acad. Publ., Dordrecht, 1994, pp. 215–218. MR**1338183****5.**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**6.**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**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.**Lawrence Markus,*Quadratic differential equations and non-associative algebras*, Contributions to the theory of nonlinear oscillations, Vol. V, Princeton Univ. Press, Princeton, N.J., 1960, pp. 185–213. MR**0132743****9.**Arthur A. Sagle and Ralph E. Walde,*Introduction to Lie groups and Lie algebras*, Academic Press, New York-London, 1973. Pure and Applied Mathematics, Vol. 51. MR**0360927****10.**Richard D. Schafer,*An introduction to nonassociative algebras*, Pure and Applied Mathematics, Vol. 22, Academic Press, New York-London, 1966. MR**0210757****11.**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

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