Unbounded sequences of cycles in general autonomous equations with periodic nonlinearities

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

Unbounded sequences of cycles in general autonomous equations with periodic nonlinearities

Alexander M. Krasnoselskii ^1,2,3,

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

Abstract
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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 $T_{0}$.

Keywords: Autonomous equations, cycles, higher order ODE, resonance, unbounded set of solutions, periodic nonlinearities, method of stationary phase..

Mathematics Subject Classification: Primary: 34C15, 34C25; Secondary: 93C1.

Citation: Alexander M. Krasnoselskii. Unbounded sequences of cycles in general autonomous equations with periodic nonlinearities. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 999-1016. doi: 10.3934/dcdss.2013.6.999

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References:

[1]	Academic Press, New York, 1975.
[2]	Springer Verlag, London, 1995.
[3]	Prentice Hall, Upper Saddle River, New Jersey, 2002.
[4]	Automation and Remote Control, 60 (1999), 1117-1125.
[5]	Differential Equations, 33 (1997), 59-66.
[6]	Mathematical and Computer Modelling, 32 (2000), 1445-1455 doi: 10.1016/S0895-7177(00)00216-8.
[7]	Doklady Mathematics, 78 (2008), 660-664. doi: 10.1134/S1064562408050049.
[8]	Springer-Verlag, Berlin, Heidelberg, 1984. doi: 10.1007/978-3-642-69409-7.
[9]	New York, Academic Press, 1974.

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