# American Institute of Mathematical Sciences

August  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 and Continuous Dynamical Systems - S, 2013, 6 (4) : 999-1016. doi: 10.3934/dcdss.2013.6.999
##### 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.

show all references

##### 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.
 [1] Xuelei Wang, Dingbian Qian, Xiying Sun. Periodic solutions of second order equations with asymptotical non-resonance. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4715-4726. doi: 10.3934/dcds.2018207 [2] Feliz Minhós, Hugo Carrasco. Solvability of higher-order BVPs in the half-line with unbounded nonlinearities. Conference Publications, 2015, 2015 (special) : 841-850. doi: 10.3934/proc.2015.0841 [3] Anna Capietto, Walter Dambrosio, Tiantian Ma, Zaihong Wang. Unbounded solutions and periodic solutions of perturbed isochronous Hamiltonian systems at resonance. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1835-1856. doi: 10.3934/dcds.2013.33.1835 [4] C. Rebelo. Multiple periodic solutions of second order equations with asymmetric nonlinearities. Discrete and Continuous Dynamical Systems, 1997, 3 (1) : 25-34. doi: 10.3934/dcds.1997.3.25 [5] Zaihong Wang. Periodic solutions of the second order differential equations with asymmetric nonlinearities depending on the derivatives. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 751-770. doi: 10.3934/dcds.2003.9.751 [6] Feliz Minhós, João Fialho. Existence and multiplicity of solutions in fourth order BVPs with unbounded nonlinearities. Conference Publications, 2013, 2013 (special) : 555-564. doi: 10.3934/proc.2013.2013.555 [7] Peiguang Wang, Xiran Wu, Huina Liu. Higher order convergence for a class of set differential equations with initial conditions. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3233-3248. doi: 10.3934/dcdss.2020342 [8] Shouchuan Hu, Nikolaos S. Papageorgiou. Double resonance for Dirichlet problems with unbounded indefinite potential and combined nonlinearities. Communications on Pure and Applied Analysis, 2012, 11 (5) : 2005-2021. doi: 10.3934/cpaa.2012.11.2005 [9] Sandra Lucente. Large data solutions for semilinear higher order equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3525-3533. doi: 10.3934/dcdss.2020247 [10] José Luis Bravo, Manuel Fernández, Antonio Tineo. Periodic solutions of a periodic scalar piecewise ode. Communications on Pure and Applied Analysis, 2007, 6 (1) : 213-228. doi: 10.3934/cpaa.2007.6.213 [11] George J. Bautista, Ademir F. Pazoto. Decay of solutions for a dissipative higher-order Boussinesq system on a periodic domain. Communications on Pure and Applied Analysis, 2020, 19 (2) : 747-769. doi: 10.3934/cpaa.2020035 [12] V. Mastropietro, Michela Procesi. Lindstedt series for periodic solutions of beam equations with quadratic and velocity dependent nonlinearities. Communications on Pure and Applied Analysis, 2006, 5 (1) : 1-28. doi: 10.3934/cpaa.2006.5.1 [13] Yuan Guo, Xiaofei Gao, Desheng Li. Structure of the set of bounded solutions for a class of nonautonomous second order differential equations. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1607-1616. doi: 10.3934/cpaa.2010.9.1607 [14] D. Bonheure, C. Fabry, D. Smets. Periodic solutions of forced isochronous oscillators at resonance. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 907-930. doi: 10.3934/dcds.2002.8.907 [15] Pengyu Chen. Periodic solutions to non-autonomous evolution equations with multi-delays. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 2921-2939. doi: 10.3934/dcdsb.2020211 [16] Aliang Xia, Jianfu Yang. Normalized solutions of higher-order Schrödinger equations. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 447-462. doi: 10.3934/dcds.2019018 [17] Alina Gleska, Małgorzata Migda. Qualitative properties of solutions of higher order difference equations with deviating arguments. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 239-252. doi: 10.3934/dcdsb.2018016 [18] Hiroshi Takeda. Global existence of solutions for higher order nonlinear damped wave equations. Conference Publications, 2011, 2011 (Special) : 1358-1367. doi: 10.3934/proc.2011.2011.1358 [19] Belgacem Rahal, Cherif Zaidi. On finite Morse index solutions of higher order fractional elliptic equations. Evolution Equations and Control Theory, 2021, 10 (3) : 575-597. doi: 10.3934/eect.2020081 [20] Zhiguo Wang, Yiqian Wang, Daxiong Piao. A new method for the boundedness of semilinear Duffing equations at resonance. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3961-3991. doi: 10.3934/dcds.2016.36.3961

2020 Impact Factor: 2.425