Skip to main content
SpringerLink
Log in

Existence of Periodic Solutions to Quaternion-Valued Impulsive Differential Equations

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

In this paper, periodic solutions of quaternion-valued impulsive differential equations (QIDEs) are considered. First, the sufficient and necessary conditions to guarantee periodic solutions are given for linear homogeneous QIDEs. Second, the representations of periodic solutions are derived by constructing Green functions in one case, and the sufficient and necessary conditions to guarantee periodic solutions are presented by means of adjoint systems in the other case for linear nonhomogeneous QIDEs. In addition, the existence and uniqueness of periodic solutions are studied by virtue of fixed point theorems for semilinear QIDEs. All results in the sense of complex-valued and quaternion-valued are equivalent to each other due to the adjoint matrix of quaternion matrix and the isomorphism between quaternion vector space and complex variables space. Finally, examples and simulations are provided to demonstrate the validity of our main results.

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
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Data availability statement

No data.

References

  1. Kou, K.I., Xia, Y.H.: Linear quaternion differential equations: Basic theory and fundamental results. Stud. Appl. Math. 141, 3–45 (2018)

    Article  MATH  Google Scholar 

  2. Kou, K.I., Liu, W.K., Xia, Y.H.: Linear quaternion differential equations: basic theory and fundamental results (II). arXiv: 1602.01660v3 (2016)

  3. Kou, K.I., Liu, W.K., Xia, Y.H.: Solve the linear quaternion-valued differential equations having multiple eigenvalues. J. Math. Phys. 60, 023510 (2019)

    MATH  Google Scholar 

  4. Cai, Z.F., Kou, K.I.: Laplace transform: a new approach in solving linear quaternion differential equations. Math. Methods Appl. Sci. 41, 4033–4048 (2018)

    Article  MATH  Google Scholar 

  5. Cai, Z.F., Kou, K.I.: Solving quaternion ordinary differential equations with two-sided coefficients. Qual. Theory Dyn. Syst. 17, 441–462 (2018)

    Article  MATH  Google Scholar 

  6. Xia, Y.H., Huang, H., Kou, K.I.: An algorithm for solving linear nonhomogeneous quaternion-valued differential equations. Discret. Contin. Dyn. Syst. Ser. S 15, 1685–1697 (2022)

    Article  MATH  Google Scholar 

  7. Xia, Y.H., Kou, K.I., Liu, Y.: Theory and Applications of Quaternion-Valued Differential Equations. Science Press, Beijing (2021)

    Google Scholar 

  8. Gasull, A., Llibre, J., Zhang, X.: One-dimensional quaternion homogeneous polynomial differential equations. J. Math. Phys. 50, 082705 (2009)

    Article  MATH  Google Scholar 

  9. Suo, L.P., Fečkan, M., Wang, J.R.: Quaternion-valued linear impulsive differential equations. Qual. Theory Dyn. Syst. 20, 1–78 (2021)

    Article  MATH  Google Scholar 

  10. Zhu, J.W., Sun, J.T.: Existence and uniqueness results for quaternion-valued nonlinear impulsive differential systems. J. Syst. Sci. Complex. 31, 596–607 (2018)

    Article  MATH  Google Scholar 

  11. Pratap, A., Raja, R., Alzabut, J., Dianavinnarasi, J., Cao, J.D., Rajchakit, G.: Finite-time Mittag-Leffler stability of fractional-order quaternion-valued memristive neural networks with impulses. Neural Process. Lett. 51, 1485–1526 (2020)

    Article  Google Scholar 

  12. Udhayakumar, K., Rakkiyappan, R., Cao, J.D., Tan, X.G.: Mittag-Leffler stability analysis of multiple equilibrium points in impulsive fractional-order quaternion-valued neural networks. Front. Inf. Technol. Electron. Eng. 21, 234–246 (2020)

    Article  Google Scholar 

  13. Tu, Z.W., Yang, X.S., Wang, L.W., Ding, N.: Stability and stabilization of quaternion-valued neural networks with uncertain time-delayed impulses: direct quaternion method. Phys. A Stat. Mech. Appl. 535, 122358 (2019)

    Article  MATH  Google Scholar 

  14. Zhu, J.W., Sun, J.T.: Stability of quaternion-valued impulsive delay difference systems and its application to neural networks. Neurocomputing 284, 63–69 (2018)

    Article  Google Scholar 

  15. Campos, J., Mawhin, J.: Periodic solutions of quaternionic-valued ordinary differential equations. Ann. Mat. 185, 109–127 (2006)

    Article  MATH  Google Scholar 

  16. Zoladek, H.: Classification of diffeomorphisms of S4 induced by quaternionic Riccati equations with periodic coefficients. Topol. Methods Nonlinear Anal. 33, 205–215 (2009)

    Article  MATH  Google Scholar 

  17. Mawhin, J.: Degree, quaternions and periodic solutions. Philos. Trans. R. Soc. A 379, 20190378 (2021)

    Article  Google Scholar 

  18. Cheng, D., Kou, K.I., Xia, Y.H.: Floquet theory for quaternion-valued differential equations. Qual. Theory Dyn. Syst. 19, 1–23 (2020)

    Article  MATH  Google Scholar 

  19. Chen, D., Fečkan, M., Wang, J.: On the stability of linear quaternion-valued differential equations. Qual. Theory Dyn. Syst. 21, 1–7 (2022)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the referees for their careful reading of the manuscript and valuable comments. The authors thank the help from the editor too.

Author information

Authors and Affiliations

  1. Department of Mathematics, Guizhou University, Guiyang, 550025, Guizhou, China

    Leping Suo & JinRong Wang

  2. Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynská dolina, 842 48, Bratislava, Slovakia

    Michal Fečkan

  3. Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73, Bratislava, Slovakia

    Michal Fečkan

Authors
  1. Leping Suo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michal Fečkan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. JinRong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to JinRong Wang.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

This work is partially supported by the National Natural Science Foundation of China (12161015), Guizhou Data Driven Modeling Learning and Optimization Innovation Team ([2020]5016), Major Project of Guizhou Postgraduate Education and Teaching Reform (YJSJGKT[2021]041), Postgraduate Education Innovation Program in Guizhou Province (YJSCXJH[2020]084), Supercomputing Algorithm and Application Laboratory of Guizhou University and Gui’an Scientific Innovation Company (No. K22-0116-003), the Slovak Research and Development Agency under the Contract No. APVV-18-0308, and the Slovak Grant Agency VEGA No. 1/0358/20 and No.2/0127/20.

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

Suo, L., Fečkan, M. & Wang, J. Existence of Periodic Solutions to Quaternion-Valued Impulsive Differential Equations. Qual. Theory Dyn. Syst. 22, 1 (2023). https://doi.org/10.1007/s12346-022-00693-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-022-00693-9

Keywords

Search

Navigation