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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
References:
[1]

C. A. Desoer and M. Vidyasagar, "Feedback Systems: Input-Output Properties,", Academic Press, (1975).   Google Scholar

[2]

A. Isidori, "Nonlinear Control Systems,", Springer Verlag, (1995).   Google Scholar

[3]

H. K. Khalil, "Nonlinear Systems,", Prentice Hall, (2002).   Google Scholar

[4]

A. M. Krasnosel'skii, Unbounded sequences of cycles in autonomous control systems,, Automation and Remote Control, 60 (1999), 1117.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

F. W. S. Olver, "Asymptotics and Special Functions,", New York, (1974).   Google Scholar

show all references

References:
[1]

C. A. Desoer and M. Vidyasagar, "Feedback Systems: Input-Output Properties,", Academic Press, (1975).   Google Scholar

[2]

A. Isidori, "Nonlinear Control Systems,", Springer Verlag, (1995).   Google Scholar

[3]

H. K. Khalil, "Nonlinear Systems,", Prentice Hall, (2002).   Google Scholar

[4]

A. M. Krasnosel'skii, Unbounded sequences of cycles in autonomous control systems,, Automation and Remote Control, 60 (1999), 1117.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

F. W. S. Olver, "Asymptotics and Special Functions,", New York, (1974).   Google Scholar

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