Previous |  Up |  Next


Chaotic and periodic solutions; differential inclusions; relay hysteresis
Chaos generated by the existence of Smale horseshoe is the well-known phenomenon in the theory of dynamical systems. The Poincaré-Andronov-Melnikov periodic and subharmonic bifurcations are also classical results in this theory. The purpose of this note is to extend those results to ordinary differential equations with multivalued perturbations. We present several examples based on our recent achievements in this direction. Singularly perturbed problems are studied as well. Applications are given to ordinary differential equations with both dry friction and relay hysteresis terms.
[1] Andronow A. A., Witt A. A., Chaikin S. E.: Theorie der Schwingungen I. Akademie Verlag, Berlin, 1965
[2] Bliman P. A., Krasnosel’skii A. M.: Periodic solutions of linear systems coupled with relay. Proc. 2nd. World Congr. Nonl. Anal., Athens – 96, Nonl. Anal., Th., Meth., Appl., 30 (1997), 687–696 MR 1487651 | Zbl 0888.34036
[3] Butenin N. V., Nejmark Y. I., Fufaev N. A.: An Introduction to the Theory of Nonlinear Oscillations. Nauka, Moscow, 1987, (in Russian) MR 0929029
[4] Chicone C.: Lyapunov-Schmidt reduction and Melnikov integrals for bifurcation of periodic solutions in coupled oscillators. J. Differential Equations, 112 (1994), 407–447 MR 1293477
[5] Deimling K.: Multivalued Differential Equations. W. De Gruyter, Berlin, 1992 MR 1189795 | Zbl 0820.34009
[6] Deimling K.: Multivalued differential equations and dry friction problems. in Proc. Conf. Differential and Delay Equations, Ames, Iowa 1991 (A. M. Fink, R. K. Miller, W. Kliemann, eds.), World Scientific, Singapore 1992, 99–106 MR 1170147
[7] Deimling K., Szilágyi P.: Periodic solutions of dry friction problems. Z. angew. Math. Phys. (ZAMP), 45 (1994), 53–60 MR 1259526
[8] Deimling K., Hetzer G., Shen W.: Almost periodicity enforced by Coulomb friction. Adv. Differential Equations, 1 (1996), 265–281 MR 1364004 | Zbl 0838.34016
[9] den Hartog J. P.: Mechanische Schwingungen. 2nd ed., Springer-Verlag, Berlin, 1952 Zbl 0046.17201
[10] Fečkan M.: Bifurcation of periodic solutions in differential inclusions. Appl. Math., 42 (1997), 369–393 MR 1467555 | Zbl 0903.34036
[11] Fečkan M.: Bifurcation from homoclinic to periodic solutions in singularly perturbed differential inclusions. Proc. Royal Soc. Edinburgh, 127A (1997), 727–753 MR 1465417 | Zbl 0990.34038
[12] Fečkan M.: Chaotic solutions in differential inclusions: Chaos in dry friction problems. Trans. Amer. Math. Soc. (to appear) MR 1473440 | Zbl 0921.34016
[13] Fečkan M.: Bifurcation from homoclinic to periodic solutions in ordinary differential equations with multivalued perturbations. J. Differential Equations, 130 (1996), 415-450 MR 1410897
[14] Fečkan M.: Chaos in ordinary differential equations with multivalued perturbations: applications to dry friction problems. Proc. 2nd. World Congr. Nonl. Anal., Athens – 96, Nonl. Anal., Th., Meth., Appl., 30 (1997), 1355–1364 MR 1490058 | Zbl 0894.34010
[15] Fečkan M.: Periodic solutions in systems at resonances with small relay hysteresis. Math. Slovaca (to appear) MR 1804472 | Zbl 1047.34011
[16] Fečkan M., Gruendler J.: Bifurcation from homoclinic to periodic solutions in ordinary differential equations with singular perturbations. preprint
[17] Gruendler J.: Homoclinic solutions for autonomous ordinary differential equations with nonautonomous perturbations. J. Differential Equations, 122 (1996), 1–26 MR 1356127
[18] Guckenheimer J., Holmes P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, New York, 1983 MR 0709768 | Zbl 0515.34001
[19] Kauderer H.: Nichtlineare Mechanik. Springer-Verlag, Berlin, 1958 MR 0145709 | Zbl 0080.17409
[20] Kunze M.: On Lyapunov exponents for non-smooth dynamical systems with an application to a pendulum with dry friction. preprint, 1997 MR 1758290
[21] Kunze M., Michaeli B.: On the rigorous applicability of Oseledet’s ergodic theorem to obtain Lyapunov exponents for non-smooth dynamical systems. submitted to the Proc. 2nd. Marrakesh Inter. Conf. Differential Eq., Ed. A. Vanderbauwhede, 1995
[22] Kunze M., Küpper T.: Qualitative bifurcation analysis of a non-smooth friction-oscillator model. Z. angew. Math. Phys. (ZAMP), 48 (1997), 87–101 MR 1439737 | Zbl 0898.70013
[23] Kunze M., Küpper T., You J.: On the application of KAM theory to non-smooth dynamical systems. Differential Equations, 139 (1997), 1–21 MR 1467350
[24] Macki J. W., Nistri P., Zecca P.: Mathematical models for hysteresis. SIAM Review, 35 (1993), 94–123 MR 1207799 | Zbl 0771.34018
[25] Macki J. W., Nistri P., Zecca P.: Periodic oscillations in systems with hysteresis. Rocky Mountain J. Math. 22 (1992), 669–681 MR 1180729 | Zbl 0759.34013
[26] Popp K.: 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.) DE–49 (1992), 1–12
[27] Popp K., Hinrichs N., Oestreich M.: Dynamical behaviour of a friction oscillator with simultaneous self and external excitation. Sādhanā 20, 2–4 (1995), 627–654 MR 1375904 | Zbl 1048.70503
[28] Popp K., Stelter P.: Stick-slip vibrations and chaos. Philos. Trans. R. Soc. London A 332 (1990), 89–105 Zbl 0709.70019
[29] Reissig R.: Erzwungene Schwingungen mit zäher Dämpfung und starker Gleitreibung. II. Math. Nachr. 12 (1954), 119–128 MR 0069996
[30] Reissig R.: Über die Stabilität gedämpfter erzwungener Bewegungen mit linearer Rückstellkraft. Math. Nachr. 13 (1955), 231–245 MR 0078535 | Zbl 0066.33503
[31] Rumpel R. J.: Singularly perturbed relay control systems. preprint, 1996
[32] Rumpel R. J.: On the qualitative behaviour of nonlinear oscillators with dry friction. ZAMM 76 (1996), 665–666 Zbl 0900.34041
Partner of
EuDML logo