Skip to main content
SpringerLink
Log in

Periodic and Unbounded Solutions of Periodic Systems

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

Abstract

This paper deals with various cases of resonance, which is a fundamental concept of science and engineering. Specifically, we study the connections between periodic and unbounded solutions for several classes of equations and systems. In particular, we extend the classical Massera’s theorem, dealing with periodic systems of the type

$$\begin{aligned} x'=A(t)x+f(t) , \end{aligned}$$

and clarify that this theorem deals with a case of resonance. Then we provide instability results for the corresponding semilinear systems, with the linear part at resonance. We also use the solution curves developed previously by the author to establish the instability results for pendulum-like equations, and for first-order periodic equations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Alonso, J.M., Ortega, R.: Unbounded solutions of semilinear equations at resonance. Nonlinearity 9(5), 1099–1111 (1996)

    Article  MathSciNet  Google Scholar 

  2. Bellman, R.: Stability Theory of Differential Equations, 2nd edn. McGraw-Hill Book Company Inc, New York-Toronto-London (1970)

    Google Scholar 

  3. Boscaggin, A., Dambrosio, W., Papini, D.: Unbounded solutions to systems of differential equations at resonance. J. Dynam. Differ. Eq. 34(1), 637–650 (2022)

    Article  MathSciNet  Google Scholar 

  4. Castro, A.: Periodic solutions of the forced pendulum equation, Differential equations (Proc. Eighth Fall Conf., Oklahoma State Univ., Stillwater, Okla., 1979), pp. 149-160, Academic Press, New York-London-Toronto, Ont., 1980

  5. Čepička, J., Drábek, P., Jenšiková, J.: On the stability of periodic solutions of the damped pendulum equation. J. Math. Anal. Appl. 209, 712–723 (1997)

    Article  MathSciNet  Google Scholar 

  6. Dambrosio, W., Ortega, R.: Topological translations and nonlinear resonance. Adv. Nonlinear Stud. 7(1), 97–106 (2007)

    Article  MathSciNet  Google Scholar 

  7. Demidovič, B.P.: Lectures on the Mathematical Theory of Stability. Izdat. Nauka, Moscow (1967). ((in Russian))

    Google Scholar 

  8. Korman, P.: Lectures on differential equations. AMS/MAA Textbooks 54, MAA Press, Providence, RI, 2019

  9. Korman, P.: A global solution curve for a class of periodic problems, including the pendulum equation. Z. Angew. Math. Phys. ZAMP 58(5), 749–766 (2007)

    Article  MathSciNet  Google Scholar 

  10. Korman, P.: Global solution curves in harmonic parameters, and multiplicity of solutions. J. Differ. Eq. 296, 186–212 (2021)

    Article  MathSciNet  Google Scholar 

  11. Korman, P., Schmidt, D.S.: Calculating global solution curves for boundary value problems, Wolfram Notebook Archive, online: notebookarchive.org/calculating-global-solution-curves-for-boundary-value-problems–2022-08-eb98nqk/

  12. Landesman, E.M., Lazer, A.C.: Nonlinear perturbations of linear elliptic boundary value problems at resonance. J. Math. Mech. 19, 609–623 (1970)

    MathSciNet  Google Scholar 

  13. Lazer, A.C., Leach, D.E.: Bounded perturbations of forced harmonic oscillators at resonance. Ann. Mat. Pura Appl. 82(4), 49–68 (1969)

    Article  MathSciNet  Google Scholar 

  14. Massera, J.L.: The existence of periodic solutions of systems of differential equations. Duke Math. J. 17, 457–475 (1950)

    Article  MathSciNet  Google Scholar 

  15. Ortega, R.: Periodic Differential Equations in the Plane. A topological perspective. De Gruyter Series in Nonlinear Analysis and Applications, 29. De Gruyter, Berlin, 2019

  16. Seifert, G.: Resonance in undamped second-order nonlinear equations with periodic forcing. Quart. Appl. Math. 48(3), 527–530 (1990)

    Article  MathSciNet  Google Scholar 

  17. Tarantello, G.: On the number of solutions of the forced pendulum equations. J. Differ. Eq. 80, 79–93 (1989)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH, 45221-0025, USA

    Philip Korman

Authors
  1. Philip Korman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Philip Korman.

Additional information

Communicated by Shangjiang Guo.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Korman, P. Periodic and Unbounded Solutions of Periodic Systems. Bull. Malays. Math. Sci. Soc. 47, 107 (2024). https://doi.org/10.1007/s40840-024-01700-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40840-024-01700-9

Keywords

Mathematics Subject Classification

Search

Navigation