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April  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

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 & Continuous Dynamical Systems - S, 2013, 6 (4) : 999-1016. doi: 10.3934/dcdss.2013.6.999
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show all references

References:
[1]

Academic Press, New York, 1975.  Google Scholar

[2]

Springer Verlag, London, 1995.  Google Scholar

[3]

Prentice Hall, Upper Saddle River, New Jersey, 2002. Google Scholar

[4]

Automation and Remote Control, 60 (1999), 1117-1125.  Google Scholar

[5]

Differential Equations, 33 (1997), 59-66.  Google Scholar

[6]

Mathematical and Computer Modelling, 32 (2000), 1445-1455 doi: 10.1016/S0895-7177(00)00216-8.  Google Scholar

[7]

Doklady Mathematics, 78 (2008), 660-664. doi: 10.1134/S1064562408050049.  Google Scholar

[8]

Springer-Verlag, Berlin, Heidelberg, 1984. doi: 10.1007/978-3-642-69409-7.  Google Scholar

[9]

New York, Academic Press, 1974.  Google Scholar

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