Weight functions for a class of Liapunov functions in the plane

Anderson, Larry R.; Ducich, Sarah E.

doi:10.1090/qam/614556

Authors: Larry R. Anderson and Sarah E. Ducich
Journal: Quart. Appl. Math. 38 (1981), 497-504
MSC: Primary 34D20
DOI: https://doi.org/10.1090/qam/614556
MathSciNet review: 614556
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Abstract: In this paper we give a class of weight functions which generate Liapunov functions for a general second-order differential system. In the special case of a Lienard equation we give conditions under which these weight functions may be chosen so as to improve certain known estimates of regions of asymptotic stability. The procedure is applied to a well-known equation and new estimates are obtained.

[1] Larry R. Anderson, Estimates of planar regions of asymptotic stability, Quart. Appl. Math. 36 (1978/79), no. 4, 434–438. MR 520125, https://doi.org/10.1090/S0033-569X-1979-0520125-8
[2] Larry R. Anderson and Walter Leighton, Liapunov functions for autonomous systems of second order, J. Math. Anal. Appl. 23 (1968), 645–664. MR 0229922, https://doi.org/10.1016/0022-247X(68)90145-5
[3] J. P. LaSalle and S. Lefschetz, Stability by Liapunov's direct method, with applications, Academic Press, New York, 1961
[4] Walter Leighton, Morse theory and Liapunov functions, Rend. Circ. Mat. Palermo (2) 13 (1964), 229–238. MR 0180729, https://doi.org/10.1007/BF02849531
[5] Walter Leighton, On the construction of Liapunov functions for certain autonomous nonlinear differential equations, Contributions to Differential Equations 2 (1963), 367–383 (1963). MR 0153927
[6] Raimond A. Struble, Nonlinear differential equations, International Series in Pure and applied Mathematics, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1961. MR 0130408