On the existence of stable periodic solutions of differential equations of Duffing type

Lazer, A. C.; McKenna, P. J.

doi:10.1090/S0002-9939-1990-1013974-9

On the existence of stable periodic solutions of differential equations of Duffing type
HTML articles powered by AMS MathViewer

by A. C. Lazer and P. J. McKenna PDF

Proc. Amer. Math. Soc. 110 (1990), 125-133 Request permission

Abstract:

We consider a second-order differential equation periodic in $t$ with period $T > 0$ and with linear damping. Bounds are given for the derivative of the restoring force which will guarantee the existence and uniqueness of a $T$-periodic solution such that the unique $T$-periodic solution is asymptotically stable. These conditions also rule out the existence of additional periodic solutions which are subharmonics of order 2.

References

V. I. Arnol′d, Ordinary differential equations, The M.I.T. Press, Cambridge, Mass.-London, 1973. Translated from the Russian and edited by Richard A. Silverman. MR 0361233
Göran Borg, Über die Stabilität gewisser Klassen von linearen Differentialgleichungen, Ark. Mat. Astr. Fys. 31A (1944), no. 1, 31 (German). MR 0016803
Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. MR 0069338
C. L. Dolph, Nonlinear integral equations of the Hammerstein type, Trans. Amer. Math. Soc. 66 (1949), 289–307. MR 32923, DOI 10.1090/S0002-9947-1949-0032923-4
D. E. Leach, On Poincaré’s perturbation theorem and a theorem of W. S. Loud, J. Differential Equations 7 (1970), 34–53. MR 251308, DOI 10.1016/0022-0396(70)90122-1
W. S. Loud, Periodic solutions of nonlinear differential equations of Duffing type, Proc. U.S.-Japan Seminar on Differential and Functional Equations (Minneapolis, Minn., 1967) Benjamin, New York, 1967, pp. 199–224. MR 0223656
Wilhelm Magnus and Stanley Winkler, Hill’s equation, Interscience Tracts in Pure and Applied Mathematics, No. 20, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1966. MR 0197830
Jean Mawhin, Contractive mappings and periodically perturbed conservative systems, Arch. Math. (Brno) 12 (1976), no. 2, 67–73. MR 437858
Rafael Ortega, Stability and index of periodic solutions of an equation of Duffing type, Boll. Un. Mat. Ital. B (7) 3 (1989), no. 3, 533–546 (English, with Italian summary). MR 1010522, DOI 10.1016/0003-4916(61)90076-8
Sylvan Wallach, The stability of differential equations with periodic coefficients, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 203–204. MR 24536, DOI 10.1073/pnas.34.5.203
S. A. Williams, A nonlinear elliptic boundary value problem, Pacific J. Math. 44 (1973), 767–774. MR 320523
Pavel Drábek and Sergio Invernizzi, On the periodic BVP for the forced Duffing equation with jumping nonlinearity, Nonlinear Anal. 10 (1986), no. 7, 643–650. MR 849954, DOI 10.1016/0362-546X(86)90124-0
Chaitan P. Gupta, Juan J. Nieto, and Luis Sanchez, Periodic solutions of some Liénard and Duffing equations, J. Math. Anal. Appl. 140 (1989), no. 1, 67–82. MR 997843, DOI 10.1016/0022-247X(89)90094-2
Patrick Habets and Gerhard Metzen, Existence of periodic solutions of Duffing equations, J. Differential Equations 78 (1989), no. 1, 1–32. MR 986151, DOI 10.1016/0022-0396(89)90073-9
Raúl F. Manásevich, A nonvariational version of a max-min principle, Nonlinear Anal. 7 (1983), no. 6, 565–570. MR 702799, DOI 10.1016/0362-546X(83)90045-7
Rolf Reissig, Contractive mappings and periodically perturbed non-conservative systems, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 58 (1975), no. 5, 696–702 (English, with Italian summary). MR 430423