Systems of linear ordinary differential equations with bounded coefficients may have very oscillating solutions

Author:
D. Novikov

Journal:
Proc. Amer. Math. Soc. **129** (2001), 3753-3755

MSC (1991):
Primary 34C10, 34M10; Secondary 34C07

Published electronically:
June 27, 2001

MathSciNet review:
1860513

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Abstract | References | Similar Articles | Additional Information

An elementary example shows that the number of zeroes of a component of a solution of a system of linear ordinary differential equations cannot be estimated through the norm of coefficients of the system.

**1.**Yuliĭ Il′yashenko and Sergeĭ Yakovenko,*Counting real zeros of analytic functions satisfying linear ordinary differential equations*, J. Differential Equations**126**(1996), no. 1, 87–105. MR**1382058**, 10.1006/jdeq.1996.0045**2.**D. Novikov and S. Yakovenko,*Meandering of trajectories of polynomial vector fields in the affine 𝑛-space*, Proceedings of the Symposium on Planar Vector Fields (Lleida, 1996), 1997, pp. 223–242. MR**1461653**, 10.5565/PUBLMAT_41197_14**3.**-, Trajectories of polynomial vector fields and ascending chains of polynomial ideals, Ann. Inst. Fourier**49**(1999), no. 2, 563-609. CMP**99:14****4.**S. Yakovenko,*On functions and curves defined by ordinary differential equations*, Proceedings of the Arnoldfest (Ed. by E. Bierstone, B. Khesin, A. Khovanskii, J. Marsden), Fields Institute Communications, 1999, pp. 203-219. CMP**2000:08**

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Additional Information

**D. Novikov**

Affiliation:
Department of Mathematics, Toronto University, Toronto, Ontario, Canada M5S 3G3

Email:
dmitry@math.toronto.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-01-06120-2

Keywords:
Bounded oscillation,
linear differential equations

Received by editor(s):
July 31, 2000

Received by editor(s) in revised form:
September 11, 2000

Published electronically:
June 27, 2001

Additional Notes:
The author is grateful to S. Yakovenko for drawing his attention to this problem and for many stimulating discussions, and to C. Chicone for amelioration of the final text. This research was supported by the Killam grant of Prof. Milman.

Communicated by:
Carmen C. Chicone

Article copyright:
© Copyright 2001
American Mathematical Society