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
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 110 (1990), 125-133
- MSC: Primary 34C25; Secondary 34D20, 70K40
- DOI: https://doi.org/10.1090/S0002-9939-1990-1013974-9
- MathSciNet review: 1013974