# American Institute of Mathematical Sciences

April  2013, 6(4): 999-1016. doi: 10.3934/dcdss.2013.6.999

## Unbounded sequences of cycles in general autonomous equations with periodic nonlinearities

 1 Institute for Information Transmission Problems, Russian Academy of Sciences 2 19 Bol.Karetny Lane, Moscow GSP-4, 127994, Russia; National Research University Higher School of Economics 3 20 Myasnitskaya Street, Moscow 101000

Received  April 2011 Revised  February 2012 Published  December 2012

Autonomous higher order differential equations with scalarnonlinearities, periodic with respect to the main phasevariable under appropriate generic conditions, have an infinitesequence of isolated cycles with amplitudes growing to infinityand periods converging to some specific value $T_{0}$.
Citation: Alexander M. Krasnoselskii. Unbounded sequences of cycles in general autonomous equations with periodic nonlinearities. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 999-1016. doi: 10.3934/dcdss.2013.6.999
##### References:
 [1] C. A. Desoer and M. Vidyasagar, "Feedback Systems: Input-Output Properties,", Academic Press, (1975). [2] A. Isidori, "Nonlinear Control Systems,", Springer Verlag, (1995). [3] H. K. Khalil, "Nonlinear Systems,", Prentice Hall, (2002). [4] A. M. Krasnosel'skii, Unbounded sequences of cycles in autonomous control systems,, Automation and Remote Control, 60 (1999), 1117. [5] A. M. Krasnosel'skii and M. A. Krasnosel'skii, Vector fields in the direct product of spaces, and applications to differential equations,, Differential Equations, 33 (1997), 59. [6] A. M. Krasnosel'skii and J. Mawhin, Periodic solutions of equations with oscillating nonlinearities,, Mathematical and Computer Modelling, 32 (2000), 1445. doi: 10.1016/S0895-7177(00)00216-8. [7] A. M. Krasnosel'skii and D. I. Rachinskii, On nonconnected unbounded sets of forced oscillations,, Doklady Mathematics, 78 (2008), 660. doi: 10.1134/S1064562408050049. [8] M. A. Krasnosel'skii and P. P. Zabreiko, "Geometrical Methods of Nonlinear Analysis,", Springer-Verlag, (1984). doi: 10.1007/978-3-642-69409-7. [9] F. W. S. Olver, "Asymptotics and Special Functions,", New York, (1974).

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##### References:
 [1] C. A. Desoer and M. Vidyasagar, "Feedback Systems: Input-Output Properties,", Academic Press, (1975). [2] A. Isidori, "Nonlinear Control Systems,", Springer Verlag, (1995). [3] H. K. Khalil, "Nonlinear Systems,", Prentice Hall, (2002). [4] A. M. Krasnosel'skii, Unbounded sequences of cycles in autonomous control systems,, Automation and Remote Control, 60 (1999), 1117. [5] A. M. Krasnosel'skii and M. A. Krasnosel'skii, Vector fields in the direct product of spaces, and applications to differential equations,, Differential Equations, 33 (1997), 59. [6] A. M. Krasnosel'skii and J. Mawhin, Periodic solutions of equations with oscillating nonlinearities,, Mathematical and Computer Modelling, 32 (2000), 1445. doi: 10.1016/S0895-7177(00)00216-8. [7] A. M. Krasnosel'skii and D. I. Rachinskii, On nonconnected unbounded sets of forced oscillations,, Doklady Mathematics, 78 (2008), 660. doi: 10.1134/S1064562408050049. [8] M. A. Krasnosel'skii and P. P. Zabreiko, "Geometrical Methods of Nonlinear Analysis,", Springer-Verlag, (1984). doi: 10.1007/978-3-642-69409-7. [9] F. W. S. Olver, "Asymptotics and Special Functions,", New York, (1974).
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