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Abstract

We review mathematical results on non-smooth dynamical systems, i.e., systems that incorporate effects of friction and/or impacts.

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References

  1. Andronov A.A., Vitt A.A., Khaikin S.E. (1966) Theory of Oscillators. Dover Publications, New York

    MATH  Google Scholar 

  2. Arnold L. (1998) Random Dynamical Systems. Springer, Berlin Heidelberg New York

    Book  MATH  Google Scholar 

  3. Arrowsmith D.K., Place C.M. (1990) An Introduction to Dynamical Systems. Cambridge University Press, Cambridge New York

    MATH  Google Scholar 

  4. Aubin J.P., Cellina A. (1984) Differential Inclusions. Springer, Berlin Heidelberg New York

    Book  MATH  Google Scholar 

  5. Awrejcewicz J., Holicke M.M. (1999) Melnikov’s method and stick-slip chaotic oscillations in very weakly forced mechanical systems. Int J Bifurcation and Chaos 9: 505–518

    Article  MATH  Google Scholar 

  6. Deimling K. (1992) Multivalued Differential Equations. de Gruyter, Berlin New York

    Google Scholar 

  7. Dellnitz M., Hohmann A. (1997) A subdivision algorithm for the computation of unstable manifolds and global attractors. Numer Mathematik 75: 293–317

    Article  MathSciNet  MATH  Google Scholar 

  8. Dellnitz M., Froyland G., Junge O. (2000) The algorithms behind GAIO-set oriented numerical methods for dynamical systems. In this DANSE book volume

    Google Scholar 

  9. Feckan M. (1996) Bifurcations from homoclinic to periodic solutions in ordinary differential equations with multivalued perturbations. J Differential Equations 130: 415–450

    Article  MathSciNet  MATH  Google Scholar 

  10. Feckan M. (1999) Chaotic solutions in differential inclusions: chaos in dry friction problems. Trans Amer Math Soc 351: 2861–2873

    Article  MathSciNet  MATH  Google Scholar 

  11. Filippov A.F. (1988) Differential Equations with Discontinuous Right-Hand Sides. Kluwer, Dordrecht Boston London

    MATH  Google Scholar 

  12. Giannakopoulos F., Kaul A., Pliete K. (1999) Qualitative analysis of a planar system of piecewise linear differential equations with a line of discontinuity. Preprint

    Google Scholar 

  13. Hubbuch F. (1995) Die Dynamik des periodisch erregten Reibschwingers. Z Ang Math Mech 75, Supplement 1: S51–S52

    Article  Google Scholar 

  14. Hubbuch F., Müller A. (1995) Lyapunov Exponenten in dynamischen Systemen mit Unstetigkeiten. Z Ang Math Mech 75, Supplement 1: S91–S92

    MATH  Google Scholar 

  15. Kunze M. (1998) Unbounded solutions in non-smooth dynamical systems at resonance. Z Angew Math Mech 78, Supplement 3: S985–S986

    Article  MathSciNet  MATH  Google Scholar 

  16. Kunze M. (1999) Periodic solutions of conservative non-smooth dynamical systems. Z Angew Math Mech 79, Supplement 1: S97–S100

    Article  MATH  Google Scholar 

  17. Kunze M. (1999) Non-Smooth Dynamical Systems. Habilitation Thesis, Universität Köln

    Google Scholar 

  18. Kunze M. (2000) On Lyapunov exponents for non-smooth dynamical systems with an application to a pendulum with dry friction. J Dynamics Differential Equations 12: 31–116

    Article  MathSciNet  MATH  Google Scholar 

  19. Kunze M. (2000) Remarks on boundedness of semilinear oscillators. In: Sanchez L. (Ed.) Proc. Autumn School Nonlinear Analysis and Differential Equations, Lisbon 1998. Birkhäuser, Basel Boston

    Google Scholar 

  20. Kunze M., Küpper T. (1997) Qualitative analysis of a non-smooth frictionoscillator model. Z Angew Math Phys 48: 1–15

    Article  MathSciNet  Google Scholar 

  21. Kunze M., Küpper T., Li J. (2000) On the application of Conley index theory to non-smooth dynamical systems. Differential Integral Equations 13: 479–502

    MathSciNet  MATH  Google Scholar 

  22. Kunze M., Küpper T., You J. (1997) On the application of KAM theory to discontinuous dynamical systems. J Differential Equations 139: 1–21

    Article  MathSciNet  MATH  Google Scholar 

  23. Kunze M., Michaeli B. (1995) On the rigorous applicability of Oseledets’ ergodic theorem to obtain Lyapunov exponents for non-smooth dynamical systems. To appear in: Arino O. (Ed.) Proc 2nd Marrakesh International Conference on Differential Equations

    Google Scholar 

  24. Kunze M., Neumann J. (1997) Linear complementary problems and the simulation of the motion of rigid body systems subject to Coulomb friction. Z Angew Math Mech 77: 833–838

    Article  MathSciNet  MATH  Google Scholar 

  25. Lefschetz S. (1965) Stability of Nonlinear Control Systems. Academic Press, New York London

    MATH  Google Scholar 

  26. Littlewood J. (1968) Some Problems in Real and Complex Analysis. Heath. Lexington, Massachusetts

    MATH  Google Scholar 

  27. Liu B. (1999) Boundedness in nonlinear oscillations at resonance. J Differential Equations 153: 142–174

    Article  MathSciNet  MATH  Google Scholar 

  28. Michaeli B. (1998) Lyapunov-Exponenten bei nichtglatten dynamischen Systemen. Ph D Thesis, Universität Köln

    Google Scholar 

  29. Mischaikow K. (1995) Conley Index Theory. In: Johnson R. (Ed.) Dynamical Systems, Montecatini Terme 1994. Lecture Notes in Mathematics Vol 1609. Springer, Berlin Heidelberg New York, 119-20.

    Google Scholar 

  30. Moritz S. (2000) Hopf-Verzweigung bei unstetigen planaren Systemen. MA Thesis, Universität Köln

    Google Scholar 

  31. Müller A. (1994) Lyapunov Exponenten in nicht-glatten dynamischen Systemen. MA Thesis, Universität Köln

    Google Scholar 

  32. Oseledets V.I. (1968) A multiplicative ergodic theorem. Ljapunov characteristic numbers for dynamical systems. Trans Moscow Math Soc 19: 197–231

    MATH  Google Scholar 

  33. Pliete K. (1998) Über die Anzahl der geschlossenen Orbits bei unstetigen stückweise linearen dynamischen Systemen in der Ebene. MA Thesis, Universität Köln

    Google Scholar 

  34. Pliete K. Ph D Thesis, in preparation

    Google Scholar 

  35. Popp K., Stelter P. (1990) Nonlinear oscillations of structures induced by dry friction. In: Schiehlen W. (Ed.) Nonlinear Dynamics in Engineering Systems-IUTAM Symposium Stuttgart 1989. Springer, Berlin Heidelberg New York, 233–240

    Google Scholar 

  36. Popp K., Stelter P. (1990) Stick-slip vibrations and chaos. Phil Trans Roy Soc London A 332: 89–105

    Article  MATH  Google Scholar 

  37. Rybakowski K.P. (1987) The Homotopy Index and Partial Differential Equations. Springer, Berlin Heidelberg New York

    Book  MATH  Google Scholar 

  38. Scharstein H. (1998) Kräfte-und Leistungsbilanz bei der künstlichen Schlagbewegung einzelner Insektenflügel. In: Nachtigall W., Wisser A. (Eds.) Technische Biologie und Bionik 4, München 1998. Gustaf Fischer Verlag, Stuttgart

    Google Scholar 

  39. Voßhage Ch. (2000) Visualisierung von Attraktoren und invarianten Maßen in nichtglatten dynamischen Systemen. MA Thesis, Universität Köln

    Google Scholar 

  40. Ward J.R., Jr. (1998) Global bifurcation of periodic solutions to ordinary differential equations. J Differential Equations 142: 1–16

    Article  MathSciNet  MATH  Google Scholar 

  41. Wiederhöft A. (1994) Der periodisch erregte Einmassenreibschwinger. MA Thesis, Universität Köln

    Google Scholar 

  42. Wiggins S. (1990) Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, Berlin Heidelberg New York

    Book  MATH  Google Scholar 

  43. Zou Y.-K., Küpper T. (2000) Melnikov method and detection of chaos for nonsmooth systems. Submitted to SIAM J Math Anal

    Google Scholar 

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Kunze, M., Küpper, T. (2001). Non-Smooth Dynamical Systems: An Overview. In: Fiedler, B. (eds) Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56589-2_19

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  • DOI: https://doi.org/10.1007/978-3-642-56589-2_19

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-62524-4

  • Online ISBN: 978-3-642-56589-2

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