IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v128y2019icp306-312.html
   My bibliography  Save this article

Lie symmetry and invariants for a generalized Birkhoffian system on time scales

Author

Listed:
  • Zhang, Yi

Abstract

The Lie symmetry and invariants for a generalized Birkhoffian system on time scales are studied, which include exact invariants and adiabatic invariants. First, the generalized Pfaff-Birkhoff principle on time scales is established, and by using Dubois-Reymond lemma the generalized Birkhoff’s equations on time scale are derived. Secondly, the determining equations of Lie symmetry for the generalized Birkhoffian system on time scales are established. We prove that if the Lie symmetry satisfies the structural equation, it leads to a conserved quantity, which is an exact invariant of the system. Again, the perturbation of Lie symmetry under the action of small disturbance is considered, the determining equations and the structural equations of disturbed system are established, and the adiabatic invariants led by the Lie symmetry perturbation for the generalized Birkhoffian system on time scales are given. Because of the arbitrariness of selecting time scales and the generality of the generalized Birkhoffian system, the results of this paper are of universal significance. The results of this paper contain the corresponding results for Birkhoffian system on time scales and classical generalized Birkhoffian system as its special cases. At the end of the paper, an example is given to illustrate the validity of the method and the results.

Suggested Citation

  • Zhang, Yi, 2019. "Lie symmetry and invariants for a generalized Birkhoffian system on time scales," Chaos, Solitons & Fractals, Elsevier, vol. 128(C), pages 306-312.
  • Handle: RePEc:eee:chsofr:v:128:y:2019:i:c:p:306-312
    DOI: 10.1016/j.chaos.2019.08.014
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077919303248
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.chaos.2019.08.014?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Sahadevan, R. & Prakash, P., 2017. "On Lie symmetry analysis and invariant subspace methods of coupled time fractional partial differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 107-120.
    2. Akbulut, Arzu & Taşcan, Filiz, 2017. "Lie symmetries, symmetry reductions and conservation laws of time fractional modified Korteweg–de Vries (mkdv) equation," Chaos, Solitons & Fractals, Elsevier, vol. 100(C), pages 1-6.
    3. Inc, Mustafa & Yusuf, Abdullahi & Aliyu, Aliyu Isa & Baleanu, Dumitru, 2018. "Lie symmetry analysis, explicit solutions and conservation laws for the space–time fractional nonlinear evolution equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 496(C), pages 371-383.
    4. Song, Chuan-Jing & Zhang, Yi, 2017. "Conserved quantities for Hamiltonian systems on time scales," Applied Mathematics and Computation, Elsevier, vol. 313(C), pages 24-36.
    5. Guha, Partha & Ghose-Choudhury, A., 2015. "Lie symmetries, Lagrangians and Hamiltonian framework of a class of nonlinear nonautonomous equations," Chaos, Solitons & Fractals, Elsevier, vol. 75(C), pages 204-211.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Jin, Shi-Xin & Chen, Xiang-Wei & Li, Yan-Min, 2024. "Approximate Noether theorem and its inverse for nonlinear dynamical systems with approximate nonstandard Lagrangian," Chaos, Solitons & Fractals, Elsevier, vol. 182(C).
    2. Zhang, Yi & Jia, Yun-Die, 2023. "Generalization of Mei symmetry approach to fractional Birkhoffian mechanics," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    3. Tanwar, Dig Vijay, 2022. "Lie symmetry reductions and generalized exact solutions of Date–Jimbo–Kashiwara–Miwa equation," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Stanislav Yu. Lukashchuk, 2022. "On the Property of Linear Autonomy for Symmetries of Fractional Differential Equations and Systems," Mathematics, MDPI, vol. 10(13), pages 1-17, July.
    2. Wang, Lingyu & Huang, Tingwen & Xiao, Qiang, 2018. "Global exponential synchronization of nonautonomous recurrent neural networks with time delays on time scales," Applied Mathematics and Computation, Elsevier, vol. 328(C), pages 263-275.
    3. Tian, Xue & Zhang, Yi, 2021. "Fractional time-scales Noether theorem with Caputo Δ derivatives for Hamiltonian systems," Applied Mathematics and Computation, Elsevier, vol. 393(C).
    4. Yusuf, Abdullahi & Inc, Mustafa & Isa Aliyu, Aliyu & Baleanu, Dumitru, 2018. "Efficiency of the new fractional derivative with nonsingular Mittag-Leffler kernel to some nonlinear partial differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 220-226.
    5. Aljohani, A.F. & Hussain, Q. & Zaman, F.D. & Kara, A.H., 2021. "On a study of some classes of the fourth-order KdV–Klein/Gordon equation and its time fractional forms," Chaos, Solitons & Fractals, Elsevier, vol. 148(C).
    6. Nass, Aminu M., 2019. "Lie symmetry analysis and exact solutions of fractional ordinary differential equations with neutral delay," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 370-380.
    7. Aljohani, A.F. & Alqurashi, Bader Mutair & Kara, A.H., 2021. "Solitons, travelling waves, invariance, conservation laws and ‘approximate’ conservation of the extended Jimbo-Miwa equation," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).
    8. Song, Chuan-Jing & Cheng, Yao, 2020. "Noether's theorems for nonshifted dynamic systems on time scales," Applied Mathematics and Computation, Elsevier, vol. 374(C).
    9. Hashemi, M.S., 2021. "A novel approach to find exact solutions of fractional evolution equations with non-singular kernel derivative," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    10. Miranda-Colorado, Roger, 2020. "Parameter identification of conservative Hamiltonian systems using first integrals," Applied Mathematics and Computation, Elsevier, vol. 369(C).
    11. Khalique, Chaudry Masood & Motsepa, Tanki, 2018. "Lie symmetries, group-invariant solutions and conservation laws of the Vasicek pricing equation of mathematical finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 505(C), pages 871-879.
    12. Inc, Mustafa & Yusuf, Abdullahi & Aliyu, Aliyu Isa & Baleanu, Dumitru, 2018. "Investigation of the logarithmic-KdV equation involving Mittag-Leffler type kernel with Atangana–Baleanu derivative," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 506(C), pages 520-531.
    13. Tanwar, Dig Vijay, 2022. "Lie symmetry reductions and generalized exact solutions of Date–Jimbo–Kashiwara–Miwa equation," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    14. Ávalos-Ruiz, L.F. & Zúñiga-Aguilar, C.J. & Gómez-Aguilar, J.F. & Escobar-Jiménez, R.F. & Romero-Ugalde, H.M., 2018. "FPGA implementation and control of chaotic systems involving the variable-order fractional operator with Mittag–Leffler law," Chaos, Solitons & Fractals, Elsevier, vol. 115(C), pages 177-189.
    15. Fan Qin & Wei Feng & Songlin Zhao, 2022. "Lie Symmetry Group, Invariant Subspace, and Conservation Law for the Time-Fractional Derivative Nonlinear Schrödinger Equation," Mathematics, MDPI, vol. 10(13), pages 1-15, June.
    16. Bruzón, M.S. & Garrido, T.M. & de la Rosa, R., 2016. "Conservation laws and exact solutions of a Generalized Benjamin–Bona–Mahony–Burgers equation," Chaos, Solitons & Fractals, Elsevier, vol. 89(C), pages 578-583.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:128:y:2019:i:c:p:306-312. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.