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Periodic Solutions of Nonlinear Systems

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Principles of Discontinuous Dynamical Systems

Abstract

In this part of the book, we investigate, by applying methods developed in the previous chapters, existence and stability of periodic solutions of quasilinear systems with variable moments of impulses.

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References

  1. M.U. Akhmetov, Periodic solutions of non-autonomous systems of differential equations with impulse action in the critical case (russian), Izv. Akad. Nauk Kazakh. SSR, Seria Fiz.-Mat., (1991a) no. 3, 62–65.

    Google Scholar 

  2. M.U. Akhmetov, Periodic solutions of systems of differential equations with a non classical right-hand side containing a small parameter (russian), TIC: Collection: asymptotic solutions of non linear equations with small parameter. 1991d, 11–15. UBL: Akad. Nauk Ukr. SSR, Inst. Mat., Kiev.

    Google Scholar 

  3. M.U. Akhmetov, R.F. Nagaev, Periodic solutions of a nonlinear impulse system in a neighborhood of a generating family of quasiperiodic solutions, Differ. Equ., 36 (2000) 799–806.

    Article  MATH  MathSciNet  Google Scholar 

  4. M.U. Akhmetov, N.A. Perestyuk, Asymptotic representation of solutions of regularly perturbed systems of differential equations with a non-classical right-hand side, Ukrainian Math. J., 43 (1991) 1209–1214.

    Article  MATH  MathSciNet  Google Scholar 

  5. M.U. Akhmetov, N.A. Perestyuk, Periodic and almost periodic solutions of strongly nonlinear impulse systems, J. Appl. Math. Mech., 56 (1992b) 829–837.

    Article  MathSciNet  Google Scholar 

  6. J. Awrejcewicz, C.H. Lamarque, Bifurcation and chaos in nonsmooth mechanical systems, World Scientific, Singapore, 2003.

    MATH  Google Scholar 

  7. V.I. Babitsky, Theory of vibro-impact systems and applications, Springer, Berlin, 1998.

    MATH  Google Scholar 

  8. A. Balanov, N. Janson, D. Postnov, O. Sosnovtseva, Synchronization: from simple to complex, Springer, Berlin, 2009.

    MATH  Google Scholar 

  9. D.D. Bainov, V. Govachev, Impulsive differential equations with a small parameter, World Scientific, Singapore, New Jersey, London, Hong Kong, 1994.

    Book  MATH  Google Scholar 

  10. M. di Bernardo, C.J. Budd, A.R. Champneys, P. Kowalczyk, Piecewise-smooth dynamical systems, Springer, London, 2008a.

    MATH  Google Scholar 

  11. I.I. Blekhman, Synchronization of dynamical systems (russian), Nauka, Moscow, 1971.

    Google Scholar 

  12. B. Brogliato, Nonsmooth impact mechanics, Springer, London, 1996.

    MATH  Google Scholar 

  13. B. Brogliato, Impacts in mechanical systems – Analysis and modeling, Springer, New York, 2000.

    Book  Google Scholar 

  14. P.J. Holmes, The dynamics of repeated impacts with a sinusoidal vibrating table, J. Sound Vib., 84 (1982) 173–189.

    MATH  Google Scholar 

  15. F.C. Hoppensteadt, C.S. Peskin, Mathematics in Medicine and in the Life Sciences, Springer, New York, 1992.

    Book  MATH  Google Scholar 

  16. A.E. Kobrinskii, A.A. Kobrinskii, Vibro-shock systems (russian), Nauka, Moscow, 1971.

    Google Scholar 

  17. Y. Kuramoto, Chemical oscillations, Springer, Berlin, 1984.

    Book  MATH  Google Scholar 

  18. A.C.J. Luo, Global transversality, resonance and chaotic dynamics, World Scientific, Hackensack, NJ, 2008.

    Book  MATH  Google Scholar 

  19. A.M. Lyapunov, Probléme général de la stabilité du mouvement, Princeton University Press, Princeton, N.J., 1949.

    Google Scholar 

  20. I.G. Malkin, Theory of stability of motion, U.S. Atomic Energy Commission, Office of Technical Information, 1958.

    Google Scholar 

  21. R.E. Mirollo, S.H. Strogatz, Synchronization of pulse-coupled biological oscillators, SIAM J. Appl. Math., 50 (1990) 1645–1662.

    Article  MATH  MathSciNet  Google Scholar 

  22. E. Mosekilde, Zh. Zhusubalyev, Bifurcations and chaos in piecewise-smooth dynamical systems, World Scientific, River Edge, NJ, 2003.

    MATH  Google Scholar 

  23. R.F. Nagaev, Periodic solutions of piecewise continuous systems with a small parameter (russian), Prikl. Mat. Mech., 36 (1972) 1059–1069.

    MathSciNet  Google Scholar 

  24. R.F. Nagaev, Mechanical processes with repeated and decaying impacts (russian), Nauka, Moscow, 1985.

    Google Scholar 

  25. R.F. Nagaev, Dynamics of synchronising systems. Springer, Berlin, 2003.

    Book  MATH  Google Scholar 

  26. R.F. Nagaev, D.G. Rubisov, Impulse motions in a one-dimensional system in a gravitational force field, Soviet Appl. Mech., 26 (1990) 885–890.

    Article  MATH  MathSciNet  Google Scholar 

  27. Yu.I. Neimark, The method of point transformations in the theory of nonlinear oscillations (russian), Nauka, Moscow, 1972.

    Google Scholar 

  28. M. Oestreich, N. Hinrichs, K. Popp, C.J. Budd, Analytical and experimental investigation of an impact oscillator. Proceedings of the ASME 16th Biennal Conf. on Mech. Vibr. and Noise, DETC97VIB-3907: 1–11, 1997.

    Google Scholar 

  29. F. Peterka, Part I: Theoretical analysis of n-multiple (1 ∕ n)-impact solutions, CSAV Acta Technica, 26 (1974) 462–473.

    Google Scholar 

  30. F. Pfeiffer, Multibody systems with unilateral constraints (russian), J. Appl. Math. Mech., 65 (2001) 665–670.

    Article  MathSciNet  Google Scholar 

  31. F. Pfeiffer, Chr. Glocker, Multibody dynamics with unilateral contacts. Wiley, New York, 1996.

    Book  MATH  Google Scholar 

  32. A. Pikovsky, Synchronization : a universal concept in nonlinear sciences, Cambridge University Press, Cambridge, 2001.

    Book  MATH  Google Scholar 

  33. V.N. Pilipchuk, R.A. Ibrahim, Dynamics of a two-pendulum model with impact interaction and an elastic support, Nonlinear Dynam., 21 (2000) 221–247.

    Article  MATH  MathSciNet  Google Scholar 

  34. H. Poincaré, Les méthodes nouvelles de la mécanique céleste, 2,3, Gauthier-Villars, Paris, 1892.

    Google Scholar 

  35. V.F. Zhuravlev, A method for analyzing vibration-impact systems by means of special functions, Mech. Solids, 11 (1976) 23–27.

    MathSciNet  Google Scholar 

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Correspondence to Marat Akhmet .

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Akhmet, M. (2010). Periodic Solutions of Nonlinear Systems. In: Principles of Discontinuous Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-6581-3_7

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