Periodic solutions of nonlinear differential systems by the method of averaging

Zhanyong Li; Qihuai Liu; Kelei Zhang

Applications of Mathematics (2020)

  • Volume: 65, Issue: 4, page 511-542
  • ISSN: 0862-7940

Abstract

top
In many engineering problems, when studying the existence of periodic solutions to a nonlinear system with a small parameter via the local averaging theorem, it is necessary to verify some properties of the fundamental solution matrix to the corresponding linearized system along the periodic solution of the unperturbed system. But sometimes, it is difficult or it requires a lot of calculations. In this paper, a few simple and effective methods are introduced to investigate the existence of periodic solutions for a kind of small parametric systems. In order to prove the existence of periodic solutions by these ideas, we also introduce a forced autoparametric vibrating system and a generalized model of the tuned mass absorber with pendulum discussed by Brzeski, Perlikowski, and Kapitaniak. Then, we also propose an averaging method to study the existence of periodic solutions.

How to cite

top

Li, Zhanyong, Liu, Qihuai, and Zhang, Kelei. "Periodic solutions of nonlinear differential systems by the method of averaging." Applications of Mathematics 65.4 (2020): 511-542. <http://eudml.org/doc/297048>.

@article{Li2020,
abstract = {In many engineering problems, when studying the existence of periodic solutions to a nonlinear system with a small parameter via the local averaging theorem, it is necessary to verify some properties of the fundamental solution matrix to the corresponding linearized system along the periodic solution of the unperturbed system. But sometimes, it is difficult or it requires a lot of calculations. In this paper, a few simple and effective methods are introduced to investigate the existence of periodic solutions for a kind of small parametric systems. In order to prove the existence of periodic solutions by these ideas, we also introduce a forced autoparametric vibrating system and a generalized model of the tuned mass absorber with pendulum discussed by Brzeski, Perlikowski, and Kapitaniak. Then, we also propose an averaging method to study the existence of periodic solutions.},
author = {Li, Zhanyong, Liu, Qihuai, Zhang, Kelei},
journal = {Applications of Mathematics},
keywords = {periodic solution; local averaging theorem; forced autoparametric vibrating system; tuned mass absorber},
language = {eng},
number = {4},
pages = {511-542},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Periodic solutions of nonlinear differential systems by the method of averaging},
url = {http://eudml.org/doc/297048},
volume = {65},
year = {2020},
}

TY - JOUR
AU - Li, Zhanyong
AU - Liu, Qihuai
AU - Zhang, Kelei
TI - Periodic solutions of nonlinear differential systems by the method of averaging
JO - Applications of Mathematics
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 4
SP - 511
EP - 542
AB - In many engineering problems, when studying the existence of periodic solutions to a nonlinear system with a small parameter via the local averaging theorem, it is necessary to verify some properties of the fundamental solution matrix to the corresponding linearized system along the periodic solution of the unperturbed system. But sometimes, it is difficult or it requires a lot of calculations. In this paper, a few simple and effective methods are introduced to investigate the existence of periodic solutions for a kind of small parametric systems. In order to prove the existence of periodic solutions by these ideas, we also introduce a forced autoparametric vibrating system and a generalized model of the tuned mass absorber with pendulum discussed by Brzeski, Perlikowski, and Kapitaniak. Then, we also propose an averaging method to study the existence of periodic solutions.
LA - eng
KW - periodic solution; local averaging theorem; forced autoparametric vibrating system; tuned mass absorber
UR - http://eudml.org/doc/297048
ER -

References

top
  1. Bajaj, A. K., Chang, S. I., Johnson, J. M., 10.1007/bf00052453, Nonlinear Dyn. 5 (1994), 433-457. (1994) DOI10.1007/bf00052453
  2. Brzeski, P., Karmazyn, A., Perlikowski, P., Synchronization of two forced double-well Duffing oscillators with attached pendulums, J. Theor. Appl. Mech. 51 (2013), 603-613. (2013) 
  3. Brzeski, P., Perlikowski, P., Kapitaniak, T., 10.1016/j.cnsns.2013.06.001, Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 298-310. (2014) Zbl1344.70009MR3142467DOI10.1016/j.cnsns.2013.06.001
  4. Buică, A., Françoise, J.-P., Llibre, J., 10.3934/cpaa.2007.6.103, Commun. Pure Appl. Anal. 6 (2007), 103-111. (2007) Zbl1139.34036MR2276332DOI10.3934/cpaa.2007.6.103
  5. Buică, A., Giné, J., Llibre, J., 10.1016/j.physd.2011.11.007, Physica D 241 (2012), 528-533. (2012) Zbl1247.34060MR2878933DOI10.1016/j.physd.2011.11.007
  6. Bustos, M. T. de, López, M. A., Martínez, R., 10.1007/s11071-013-1176-1, Nonlinear Dyn. 76 (2014), 893-903. (2014) Zbl1306.34071MR3192186DOI10.1007/s11071-013-1176-1
  7. Euzébio, R. D., Llibre, J., 10.3934/dcds.2014.34.3455, Discrete Contin. Dyn. Syst. 34 (2014), 3455-3469. (2014) Zbl1302.86010MR3190988DOI10.3934/dcds.2014.34.3455
  8. Gus'kov, A. M., Panovko, G. Ya., Bin, C. V., 10.3103/s105261880804002x, J. Mach. Manuf. Reliab. 37 (2008), 321-329. (2008) DOI10.3103/s105261880804002x
  9. Hatwal, H., Mallik, A. K., Ghosh, A., 10.1016/0022-460X(82)90201-2, J. Sound Vib. 81 (1982), 153-164. (1982) Zbl0483.70018MR0650937DOI10.1016/0022-460X(82)90201-2
  10. Hatwal, H., Mallik, A. K., Ghosh, A., 10.1115/1.3167106, J. Appl. Mech. 50 (1983), 657-662. (1983) Zbl0537.70024DOI10.1115/1.3167106
  11. Hatwal, H., Mallik, A. K., Ghosh, A., 10.1115/1.3167107, J. Appl. Mech. 50 (1983), 663-668. (1983) Zbl0537.70025DOI10.1115/1.3167107
  12. Huang, R., Chu, D., 10.1137/140995702, SIAM J. Matrix Anal. Appl. 36 (2015), 476-495. (2015) Zbl1328.65089MR3340200DOI10.1137/140995702
  13. Lembarki, F. E., Llibre, J., 10.1007/s11071-014-1249-9, Nonlinear Dyn. 76 (2014), 1807-1819. (2014) Zbl1314.70032MR3192700DOI10.1007/s11071-014-1249-9
  14. Li, Z., Liu, Q., Zhang, K., 10.1088/1742-6596/1053/1/012088, J. Phys., Conf. Ser. 1053 (2018), Article ID 012088. (2018) DOI10.1088/1742-6596/1053/1/012088
  15. Liu, Q., Cai, L., 10.1016/j.nonrwa.2018.07.019, Nonlinear Anal., Real World Appl. 45 (2019), 461-471. (2019) Zbl07030113MR3854316DOI10.1016/j.nonrwa.2018.07.019
  16. Liu, Q., Qian, D., 10.1063/1.3623415, J. Math. Phys. 52 (2011), Article ID 082702, 11 pages. (2011) Zbl1272.82025MR2858048DOI10.1063/1.3623415
  17. Liu, Q., Qian, D., 10.1142/S1402925112500179, J. Nonlinear Math. Phys. 19 (2012), 255-268. (2012) Zbl1254.35209MR2949250DOI10.1142/S1402925112500179
  18. Liu, Q., Xing, M., Li, X., Wang, C., 10.1088/1674-1056/24/12/120401, Chin. Phys. B 24 (2015), 246-252. (2015) DOI10.1088/1674-1056/24/12/120401
  19. Llibre, J., Moeckel, R., Simó, C., 10.1007/978-3-0348-0933-7, Advanced Courses in Mathematics, CRM Barcelona. Birkhäuser/Springer, Basel (2015). (2015) Zbl1336.37002MR3381578DOI10.1007/978-3-0348-0933-7
  20. Llibre, J., Świrszcz, G., On the limit cycles of polynomial vector fields, Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 18 (2011), 203-214. (2011) Zbl1223.34039MR2768130
  21. Llibre, J., Yu, J., Zhang, X., 10.1216/RMJ-2010-40-2-581, Rocky Mt. J. Math. 40 (2010), 581-594. (2010) Zbl1196.37087MR2646459DOI10.1216/RMJ-2010-40-2-581
  22. Llibre, J., Zhang, X., 10.1016/j.nonrwa.2010.10.019, Nonlinear Anal., Real World Appl. 12 (2011), 1650-1653. (2011) Zbl1220.34070MR2781884DOI10.1016/j.nonrwa.2010.10.019
  23. Rabanal, R., 10.1016/j.na.2013.10.013, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 95 (2014), 676-690. (2014) Zbl1297.34046MR3130553DOI10.1016/j.na.2013.10.013
  24. Verhulst, F., 10.1007/978-3-642-61453-8, Universitext. Springer, Berlin (1996). (1996) Zbl0854.34002MR1422255DOI10.1007/978-3-642-61453-8
  25. Vyas, A., Bajaj, A. K., 10.1006/jsvi.2001.3616, J. Sound Vib. 246 (2001), 115-135. (2001) Zbl1237.70118MR1895251DOI10.1006/jsvi.2001.3616
  26. Vyas, A., Bajaj, A. K., Raman, A., 10.1098/rspa.2003.1204, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 460 (2004), 1547-1581. (2004) Zbl1108.70307MR2067554DOI10.1098/rspa.2003.1204
  27. Wang, G., Zhou, Z., Zhu, S., Wang, S., Ordinary Differential Equations, Higher Education Press, Beijing (2006), Chinese. (2006) 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.