Quadratic differential equations in graded algebras
Authors:
Nora C. Hopkins and Michael K. Kinyon
Journal:
Trans. Amer. Math. Soc. 351 (1999), 45454559
MSC (1991):
Primary 34C35, 17A60, 34C20, 17A36
Published electronically:
July 19, 1999
MathSciNet review:
1475685
Fulltext PDF Free Access
Abstract 
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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, Nonassociative algebra and its applications (Oviedo, 1993)
Math. Appl., vol. 303, Kluwer Acad. Publ., Dordrecht, 1994,
pp. 179–182. MR 1338177
(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), 121138. CMP 99:04
 4.
Michael
K. Kinyon, Quadratic differential equations on graded
structures, Nonassociative algebra and its applications (Oviedo,
1993) Math. Appl., vol. 303, Kluwer Acad. Publ., Dordrecht, 1994,
pp. 215–218. MR 1338183
(96f:17005)
 5.
Michael
K. Kinyon and Arthur
A. Sagle, Quadratic dynamical systems and algebras, J.
Differential Equations 117 (1995), no. 1,
67–126. MR
1320184 (96e:34018), http://dx.doi.org/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 (95d:34015), http://dx.doi.org/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), 180196. CMP 98:05
 8.
Lawrence
Markus, Quadratic differential equations and nonassociative
algebras, Contributions to the theory of nonlinear oscillations, Vol.
V, Princeton Univ. Press, Princeton, N.J., 1960, pp. 185–213.
MR
0132743 (24 #A2580)
 9.
Arthur
A. Sagle and Ralph
E. Walde, Introduction to Lie groups and Lie algebras,
Academic Press, New York, 1973. Pure and Applied Mathematics, Vol. 51. MR 0360927
(50 #13374)
 10.
Richard
D. Schafer, An introduction to nonassociative algebras, Pure
and Applied Mathematics, Vol. 22, Academic Press, New York, 1966. MR 0210757
(35 #1643)
 11.
Sebastian
Walcher, Algebras and differential equations, Hadronic Press
Monographs in Mathematics, Hadronic Press Inc., Palm Harbor, FL, 1991. MR 1143536
(93e:34002)
 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. 179182. 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), 121138. 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. 215218. MR 96f:17005
 5.
 M. K. Kinyon and A. A. Sagle, Quadratic dynamical systems and algebras, J. Diff. Eq 117 (1995), 67126. 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), 135154. 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), 180196. CMP 98:05
 8.
 L. Markus, Quadratic differential equations and nonassociative 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. 185213. 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
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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:
http://dx.doi.org/10.1090/S0002994799022126
PII:
S 00029947(99)022126
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
