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Article

Keywords:
multivalued mappings; differential inclusions; periodic solutions; dry friction terms
Summary:
Ordinary differential inclusions depending on small parameters are considered such that the unperturbed inclusions are ordinary differential equations possessing manifolds of periodic solutions. Sufficient conditions are determined for the persistence of some of these periodic solutions after multivalued perturbations. Applications are given to dry friction problems.
References:
[1] A.A. Andronow, A.A. Witt, S.E. Chaikin: Theorie der Schwingungen I. Akademie Verlag, Berlin, 1965. MR 0216794
[2] N.V. Butenin, Y.I. Nejmark, N.A. Fufaev: An Introduction to the Theory of Nonlinear Oscillations. Nauka, Moscow, 1987. (Russian) MR 0929029
[3] C. Chicone: Lyapunov-Schmidt reduction and Melnikov integrals for bifurcation of periodic solutions in coupled oscillators. J. Differential Equations 112 (1994), 407–447. DOI 10.1006/jdeq.1994.1110 | MR 1293477
[4] K. Deimling: Multivalued Differential Equations. W. De Gruyter, Berlin, 1992. MR 1189795 | Zbl 0820.34009
[5] K. Deimling: Multivalued differential equations and dry friction problems. Proc. Conf. Differential & Delay Equations, Ames, Iowa 1991, A. M. Fink, R. K. Miller, W. Kliemann (eds.), World Scientific, Singapore, 1992, pp. 99–106. MR 1170147 | Zbl 0820.34009
[6] K. Deimling, P. Szilágyi: Periodic solutions of dry friction problems. Z. angew. Math. Phys. (ZAMP) 45 (1994), 53–60. DOI 10.1007/BF00942846 | MR 1259526
[7] K. Deimling, G. Hetzer, W. Shen: Almost periodicity enforced by Coulomb friction. Adv. Differential Equations 1 (1996), 265–281. MR 1364004
[8] J.P. Den Hartog: Mechanische Schwingungen. 2nd ed., Springer-Verlag, Berlin, 1952. Zbl 0046.17201
[9] M. Fečkan: Bifurcation from homoclinic to periodic solutions in ordinary differential equations with multivalued perturbations. J. Differential Equations 130 (1996), 415–450. DOI 10.1006/jdeq.1996.0152 | MR 1410897
[10] M. Fečkan: Bifurcation from homoclinic to periodic solutions in singularly perturbed differential inclusions. Proceedings Royal Soc. Edinburgh A (to appear). MR 1465417
[11] J. Guckenheimer, P. Holmes: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, New York, 1983. MR 0709768
[12] P. Hartman: Ordinary Differential Equations. Wiley, New York, 1964. MR 0171038 | Zbl 0125.32102
[13] H. Kauderer: Nichtlineare Mechanik. Springer-Verlag, Berlin, 1958. MR 0145709 | Zbl 0080.17409
[14] K. Popp, P. Stelter: Stick-slip vibrations and chaos. Philos. Trans. R. Soc. London A 332 (1990), 89–105. DOI 10.1098/rsta.1990.0102
[15] K. Popp: Some model problems showing stick-slip motion and chaos. ASME WAM, Proc. Symp. Friction-Induced Vibration, Chatter, Squeal and Chaos, R.A. Ibrahim and A. Soom (eds.) vol. DE-49, 1992, pp. 1–12.
[16] K. Popp, N. Hinrichs, M. Oestreich: Dynamical behaviour of a friction oscillator with simultaneous self and external excitation. Sādhanā 20, 2–4 (1995), 627–654. DOI 10.1007/BF02823210 | MR 1375904
[17] T. Pruszko: Some applications of the topological degree theory to multi-valued boundary value problems. Dissertationes Math. 229 (1984), 1–48. MR 0741752 | Zbl 0543.34008
[18] T. Pruszko: Topological degree methods in multi-valued boundary value problems. Nonlinear Anal., Th., Meth. Appl. 5 (1981), 959–973. DOI 10.1016/0362-546X(81)90056-0 | MR 0633011 | Zbl 0478.34017
[19] R. Reissig: Erzwungene Schwingungen mit zäher Dämpfung und starker Gleitreibung. II. Math. Nachr. 12 (1954), 119–128. DOI 10.1002/mana.19540120109 | MR 0069996
[20] R. Reissig: Über die Stabilität gedämpfter erzwungener Bewegungen mit linearer Rückstellkraft. Math. Nachr. 13 (1955), 231–245. DOI 10.1002/mana.19550130310 | MR 0078535 | Zbl 0066.33503
[21] R. Rumpel: Singularly perturbed relay control systems. preprint (1996).
[22] K. Yosida: Functional Analysis. Springer-Verlag, Berlin, 1965. Zbl 0126.11504
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