Previous |  Up |  Next

Article

Full entry | Fulltext not available (moving wall 24 months)      Feedback
Keywords:
periodic solution; local averaging theorem; forced autoparametric vibrating system; tuned mass absorber
Summary:
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.
References:
[1] Bajaj, A. K., Chang, S. I., Johnson, J. M.: Amplitude modulated dynamics of a resonantly excited autoparametric two degree-of-freedom system. Nonlinear Dyn. 5 (1994), 433-457. DOI 10.1007/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.
[3] Brzeski, P., Perlikowski, P., Kapitaniak, T.: Numerical optimization of tuned mass absorbers attached to strongly nonlinear Duffing oscillator. Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 298-310. DOI 10.1016/j.cnsns.2013.06.001 | MR 3142467 | Zbl 1344.70009
[4] Buică, A., Françoise, J.-P., Llibre, J.: Periodic solutions of nonlinear periodic differential systems with a small parameter. Commun. Pure Appl. Anal. 6 (2007), 103-111. DOI 10.3934/cpaa.2007.6.103 | MR 2276332 | Zbl 1139.34036
[5] Buică, A., Giné, J., Llibre, J.: A second order analysis of the periodic solutions for nonlinear periodic differential systems with a small parameter. Physica D 241 (2012), 528-533. DOI 10.1016/j.physd.2011.11.007 | MR 2878933 | Zbl 1247.34060
[6] Bustos, M. T. de, López, M. A., Martínez, R.: On the periodic solutions of a linear chain of three identical atoms. Nonlinear Dyn. 76 (2014), 893-903. DOI 10.1007/s11071-013-1176-1 | MR 3192186 | Zbl 1306.34071
[7] Euzébio, R. D., Llibre, J.: Periodic solutions of {\it El Niño} model through the Vallis differential system. Discrete Contin. Dyn. Syst. 34 (2014), 3455-3469. DOI 10.3934/dcds.2014.34.3455 | MR 3190988 | Zbl 1302.86010
[8] Gus'kov, A. M., Panovko, G. Ya., Bin, C. V.: Analysis of the dynamics of a pendulum vibration absorber. J. Mach. Manuf. Reliab. 37 (2008), 321-329. DOI 10.3103/s105261880804002x
[9] Hatwal, H., Mallik, A. K., Ghosh, A.: Nonlinear vibrations of a harmonically excited autoparametric system. J. Sound Vib. 81 (1982), 153-164. DOI 10.1016/0022-460X(82)90201-2 | MR 0650937 | Zbl 0483.70018
[10] Hatwal, H., Mallik, A. K., Ghosh, A.: Forced nonlinear oscillations of an autoparametric system. I. Periodic responses. J. Appl. Mech. 50 (1983), 657-662. DOI 10.1115/1.3167106 | Zbl 0537.70024
[11] Hatwal, H., Mallik, A. K., Ghosh, A.: Forced nonlinear oscillations of an autoparametric system. II. Chaotic responses. J. Appl. Mech. 50 (1983), 663-668. DOI 10.1115/1.3167107 | Zbl 0537.70025
[12] Huang, R., Chu, D.: Relative perturbation analysis for eigenvalues and singular values of totally nonpositive matrices. SIAM J. Matrix Anal. Appl. 36 (2015), 476-495. DOI 10.1137/140995702 | MR 3340200 | Zbl 1328.65089
[13] Lembarki, F. E., Llibre, J.: Periodic orbits for the generalized Yang-Mills Hamiltonian system in dimension 6. Nonlinear Dyn. 76 (2014), 1807-1819. DOI 10.1007/s11071-014-1249-9 | MR 3192700 | Zbl 1314.70032
[14] Li, Z., Liu, Q., Zhang, K.: Harmonic motions of a weakly forced autoparametric vibrating system. J. Phys., Conf. Ser. 1053 (2018), Article ID 012088. DOI 10.1088/1742-6596/1053/1/012088
[15] Liu, Q., Cai, L.: Averaging methods for nonlinear systems with a small parameter via reduction and topological degree. Nonlinear Anal., Real World Appl. 45 (2019), 461-471. DOI 10.1016/j.nonrwa.2018.07.019 | MR 3854316 | Zbl 07030113
[16] Liu, Q., Qian, D.: Modulated amplitude waves with nonzero phases in Bose-Einstein condensates. J. Math. Phys. 52 (2011), Article ID 082702, 11 pages. DOI 10.1063/1.3623415 | MR 2858048 | Zbl 1272.82025
[17] Liu, Q., Qian, D.: Construction of modulated amplitude waves via averaging in collisionally inhomogeneous Bose-Einstein condensates. J. Nonlinear Math. Phys. 19 (2012), 255-268. DOI 10.1142/S1402925112500179 | MR 2949250 | Zbl 1254.35209
[18] Liu, Q., Xing, M., Li, X., Wang, C.: Unstable and exact periodic solutions of three-particles time-dependent FPU chains. Chin. Phys. B 24 (2015), 246-252. DOI 10.1088/1674-1056/24/12/120401
[19] Llibre, J., Moeckel, R., Simó, C.: Central Configurations, Periodic Orbits, and Hamiltonian Systems. Advanced Courses in Mathematics, CRM Barcelona. Birkhäuser/Springer, Basel (2015). DOI 10.1007/978-3-0348-0933-7 | MR 3381578 | Zbl 1336.37002
[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. MR 2768130 | Zbl 1223.34039
[21] Llibre, J., Yu, J., Zhang, X.: Limit cycles for a class of third-order differential equations. Rocky Mt. J. Math. 40 (2010), 581-594. DOI 10.1216/RMJ-2010-40-2-581 | MR 2646459 | Zbl 1196.37087
[22] Llibre, J., Zhang, X.: On the Hopf-zero bifurcation of the Michelson system. Nonlinear Anal., Real World Appl. 12 (2011), 1650-1653. DOI 10.1016/j.nonrwa.2010.10.019 | MR 2781884 | Zbl 1220.34070
[23] Rabanal, R.: On the limit cycles of a class of Kukles type differential systems. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 95 (2014), 676-690. DOI 10.1016/j.na.2013.10.013 | MR 3130553 | Zbl 1297.34046
[24] Verhulst, F.: Nonlinear Differential Equations and Dynamical Systems. Universitext. Springer, Berlin (1996). DOI 10.1007/978-3-642-61453-8 | MR 1422255 | Zbl 0854.34002
[25] Vyas, A., Bajaj, A. K.: Dynamics of autoparametric vibration absorbers using multiple pendulums. J. Sound Vib. 246 (2001), 115-135. DOI 10.1006/jsvi.2001.3616 | MR 1895251 | Zbl 1237.70118
[26] Vyas, A., Bajaj, A. K., Raman, A.: Dynamics of structures with wideband autoparametric vibration absorbers: theory. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 460 (2004), 1547-1581. DOI 10.1098/rspa.2003.1204 | MR 2067554 | Zbl 1108.70307
[27] Wang, G., Zhou, Z., Zhu, S., Wang, S.: Ordinary Differential Equations. Higher Education Press, Beijing (2006), Chinese.
Partner of
EuDML logo