American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Existence and enclosure of solutions to noncoercive systems of inequalities with multivalued mappings and non-power growths
  • DCDS Home
  • This Issue
  • Next Article
    On the principal eigenvalues of some elliptic problems with large drift
January  2013, 33(1): 239-254. doi: 10.3934/dcds.2013.33.239

Resonant forced oscillations in systems with periodic nonlinearities

Alexander Krasnosel'skii 1,

1. 

Institute for Information Transmission Problems, 19 Bolshoi Karetny, 127994, GSP-4 Moscow, Russian Federation

Received  August 2011 Revised  February 2012 Published  September 2012

We present an approach to study degenerate ODE with periodic nonlinearities; for resonant higher order nonlinear equations $L(p)x=f(x)+b(t),\;p=d/dt$ with $2\pi$-periodic forcing $b$ and periodic $f$ we give multiplicity results, in particular, conditions of existence of infinite and unbounded sets of $2\pi$-periodic solutions.
Keywords: higher order ODE, resonance, unbounded set of solutions, method of stationary phase., Forced periodic oscillations, periodic nonlinearities.
Mathematics Subject Classification: Primary: 34C15, 34C25; Secondary: 93C1.
Citation: Alexander Krasnosel'skii. Resonant forced oscillations in systems with periodic nonlinearities. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 239-254. doi: 10.3934/dcds.2013.33.239
References:
[1]

J. M. Alonso and R. Ortega, Unbounded solutions of semilinear equations at resonance,, Nonlinearity, 9 (1996), 1099.  doi: 10.1088/0951-7715/9/5/003.  Google Scholar

[2]

V. I. Arnold, "Arnold's Problems,'', Problem 1996-5, (2005), 1996.   Google Scholar

[3]

D. Bonheure, C. Fabry and D. Ruiz, Problems at resonance for equations with periodic nonlinearities,, Nonlinear Analysis, 55 (2003), 557.  doi: 10.1016/j.na.2003.07.005.  Google Scholar

[4]

A. Cańada and F. Roca, Existence and multiplicity of solutions of someconservative pendulum-type equations with homogeneous Dirichlet conditions,, Diff. and Int. Eq., 10 (1997), 1113.   Google Scholar

[5]

E. N. Dancer, On the use of asymptotics in nonlinear boundary value problems,, Ann. Mat. Pura Appl., 131 (1982), 167.  doi: 10.1007/BF01765151.  Google Scholar

[6]

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

[7]

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

[8]

A. M. Krasnosel'skii and M. A. Krasnosel'skii, Vector fields in the direct product of spaces, and applications todifferential equations,, Differential Equations, 33 (1997), 59.   Google Scholar

[9]

A. M. Krasnosel'skii and J. Mawhin, The index at infinity of some twice degenerate compact vector fields,, Discrete Contin. Dyn. Syst., 1 (1995), 207.  doi: 10.3934/dcds.1995.1.207.  Google Scholar

[10]

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

[11]

M. A. Krasnosel'skii and P. P. Zabreiko, "Geometrical Methods of Nonlinear Analysis,'', Springer-Verlag, (1984).   Google Scholar

[12]

F. W. S. Olver, "Asymptotics and Special Functions,'', Academic Press, (1974).   Google Scholar

[13]

G. Pólya and G. Szegö, "Aufgaben and Lehrsätze aus der Analysis,'', Springer, B I-II (1925).   Google Scholar

[14]

R. Schaaf and K. Schmitt, A class of nonlinear Sturm-Liouville problems with infinitely many solutions,, Trans. Amer. Math. Soc., 306 (1988), 853.  doi: 10.1090/S0002-9947-1988-0933322-5.  Google Scholar

show all references

References:
[1]

J. M. Alonso and R. Ortega, Unbounded solutions of semilinear equations at resonance,, Nonlinearity, 9 (1996), 1099.  doi: 10.1088/0951-7715/9/5/003.  Google Scholar

[2]

V. I. Arnold, "Arnold's Problems,'', Problem 1996-5, (2005), 1996.   Google Scholar

[3]

D. Bonheure, C. Fabry and D. Ruiz, Problems at resonance for equations with periodic nonlinearities,, Nonlinear Analysis, 55 (2003), 557.  doi: 10.1016/j.na.2003.07.005.  Google Scholar

[4]

A. Cańada and F. Roca, Existence and multiplicity of solutions of someconservative pendulum-type equations with homogeneous Dirichlet conditions,, Diff. and Int. Eq., 10 (1997), 1113.   Google Scholar

[5]

E. N. Dancer, On the use of asymptotics in nonlinear boundary value problems,, Ann. Mat. Pura Appl., 131 (1982), 167.  doi: 10.1007/BF01765151.  Google Scholar

[6]

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

[7]

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

[8]

A. M. Krasnosel'skii and M. A. Krasnosel'skii, Vector fields in the direct product of spaces, and applications todifferential equations,, Differential Equations, 33 (1997), 59.   Google Scholar

[9]

A. M. Krasnosel'skii and J. Mawhin, The index at infinity of some twice degenerate compact vector fields,, Discrete Contin. Dyn. Syst., 1 (1995), 207.  doi: 10.3934/dcds.1995.1.207.  Google Scholar

[10]

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

[11]

M. A. Krasnosel'skii and P. P. Zabreiko, "Geometrical Methods of Nonlinear Analysis,'', Springer-Verlag, (1984).   Google Scholar

[12]

F. W. S. Olver, "Asymptotics and Special Functions,'', Academic Press, (1974).   Google Scholar

[13]

G. Pólya and G. Szegö, "Aufgaben and Lehrsätze aus der Analysis,'', Springer, B I-II (1925).   Google Scholar

[14]

R. Schaaf and K. Schmitt, A class of nonlinear Sturm-Liouville problems with infinitely many solutions,, Trans. Amer. Math. Soc., 306 (1988), 853.  doi: 10.1090/S0002-9947-1988-0933322-5.  Google Scholar

[1]

D. Bonheure, C. Fabry, D. Smets. Periodic solutions of forced isochronous oscillators at resonance. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 907-930. doi: 10.3934/dcds.2002.8.907
[2]

Anna Capietto, Walter Dambrosio, Tiantian Ma, Zaihong Wang. Unbounded solutions and periodic solutions of perturbed isochronous Hamiltonian systems at resonance. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1835-1856. doi: 10.3934/dcds.2013.33.1835
[3]

José Luis Bravo, Manuel Fernández, Antonio Tineo. Periodic solutions of a periodic scalar piecewise ode. Communications on Pure & Applied Analysis, 2007, 6 (1) : 213-228. doi: 10.3934/cpaa.2007.6.213
[4]

Xuelei Wang, Dingbian Qian, Xiying Sun. Periodic solutions of second order equations with asymptotical non-resonance. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4715-4726. doi: 10.3934/dcds.2018207
[5]

Giovanni Cimatti. Forced periodic solutions for piezoelectric crystals. Communications on Pure & Applied Analysis, 2005, 4 (2) : 475-485. doi: 10.3934/cpaa.2005.4.475
[6]

C. Rebelo. Multiple periodic solutions of second order equations with asymmetric nonlinearities. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 25-34. doi: 10.3934/dcds.1997.3.25
[7]

Zaihong Wang. Periodic solutions of the second order differential equations with asymmetric nonlinearities depending on the derivatives. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 751-770. doi: 10.3934/dcds.2003.9.751
[8]

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
[9]

George J. Bautista, Ademir F. Pazoto. Decay of solutions for a dissipative higher-order Boussinesq system on a periodic domain. Communications on Pure & Applied Analysis, 2020, 19 (2) : 747-769. doi: 10.3934/cpaa.2020035
[10]

V. Barbu. Periodic solutions to unbounded Hamiltonian system. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 277-283. doi: 10.3934/dcds.1995.1.277
[11]

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
[12]

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
[13]

Pablo Amster, Mónica Clapp. Periodic solutions of resonant systems with rapidly rotating nonlinearities. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 373-383. doi: 10.3934/dcds.2011.31.373
[14]

Pedro J. Torres. Non-collision periodic solutions of forced dynamical systems with weak singularities. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 693-698. doi: 10.3934/dcds.2004.11.693
[15]

Feng Wang, Jifeng Chu, Zaitao Liang. Prevalence of stable periodic solutions in the forced relativistic pendulum equation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4579-4594. doi: 10.3934/dcdsb.2018177
[16]

Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098
[17]

Shiwang Ma. Nontrivial periodic solutions for asymptotically linear hamiltonian systems at resonance. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2361-2380. doi: 10.3934/cpaa.2013.12.2361
[18]

Laura Olian Fannio. Multiple periodic solutions of Hamiltonian systems with strong resonance at infinity. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 251-264. doi: 10.3934/dcds.1997.3.251
[19]

J. R. Ward. Periodic solutions of first order systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 381-389. doi: 10.3934/dcds.2013.33.381
[20]

Shouchuan Hu, Nikolaos S. Papageorgiou. Double resonance for Dirichlet problems with unbounded indefinite potential and combined nonlinearities. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2005-2021. doi: 10.3934/cpaa.2012.11.2005

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (21)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences