Previous |  Up |  Next

Article

Full entry | Fulltext not available (moving wall 24 months)      Feedback
Keywords:
second-order nonlinear differential equation; stability; instability; Floquet multiplier; Lyapunov exponent; periodic solution
Summary:
We discuss Lyapunov stability/instability of both lower and upper equilibria of free damped pendulum with periodically oscillating suspension point. We recall the results of Bogolyubov and Kapitza, provide new effective criteria of stability/instability of the equilibria of pendulum equation, and give the exact and complete proofs. The criteria obtained are formulated in terms of positivity/negativity of Green's functions of the periodic boundary value problems for linearized equations. Furthermore, we show that if both lower and upper equilibria are stable, then the pendulum considered may possess a periodic motion that corresponds to the ``quasistatic solution'' of Bogolyubov as well as to the ``quasistatic balance'' of Kapitza.
References:
[1] Barteneva, I. V., Cabada, A., Ignatyev, A. O.: Maximum and anti-maximum principles for the general operator of second order with variable coefficients. Appl. Math. Comput. 134 (2003), 173-184. DOI 10.1016/S0096-3003(01)00280-6 | MR 1928973 | Zbl 1037.34014
[2] Bartuccelli, M. V., Gentile, G., Georgiou, K. V.: On the dynamics of a vertically driven damped planar pendulum. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 457 (2001), 3007-3022. DOI 10.1098/rspa.2001.0841 | MR 1875091 | Zbl 1001.70023
[3] Blackburn, J. A., Smith, H. J. T., Grønbech-Jensen, N.: Stability and Hopf bifurcations in an inverted pendulum. Am. J. Phys. 60 (1992), 903-908. DOI 10.1119/1.17011 | MR 1181951 | Zbl 1219.70056
[4] Bogatov, E. M., Mukhin, R. R.: The averaging method, a pendulum with a vibrating suspension: N. N. Bogolyubov, A. Stephenson, P. L. Kapitza and others. Izv. VUZ, Appl. Nonlinear Dyn. 25 (2017), 69-87 Russian. DOI 10.18500/0869-6632-2017-25-5-69-87
[5] Bogolyubov, N. N.: Theory of perturbations in nonlinear mechanics. Collection of Works 14 Institute of Construction Mechanics, Ukrainian Academy of Sciences, Kiev (1950), 9-34 Russian.
[6] Cabada, A., Cid, J. Á., López-Somoza, L.: Maximum Principles for the Hill's Equation. Academic Press, London (2018). DOI 10.1016/C2015-0-00688-8 | MR 3751358 | Zbl 1393.34003
[7] Dancer, E. N., Ortega, R.: The index of Lyapunov stable fixed points in two dimensions. J. Dyn. Differ. Equations 6 (1994), 631-637. DOI 10.1007/BF02218851 | MR 1303278 | Zbl 0811.34018
[8] Coster, C. De, Habets, P.: Two-Point Boundary Value Problems: Lower and Upper Solutions. Mathematics in Science and Engineering 205. Elsevier, Amsterdam (2006). DOI 10.1016/s0076-5392(06)x8055-4 | MR 2225284 | Zbl 1330.34009
[9] Demidovich, B. P.: Lectures on Mathematical Stability Theory. Nauka, Moscow (1967), Russian. MR 0226126 | Zbl 0155.41601
[10] Hakl, R., Torres, P. J.: Maximum and antimaximum principles for a second order differential operator with variable coefficients of indefinite sign. Appl. Math. Comput. 217 (2011), 7599-7611. DOI 10.1016/j.amc.2011.02.053 | MR 2799774 | Zbl 1235.34064
[11] Hartman, P.: Ordinary Differential Equations. John Wiley & Sons, New York (1964). MR 0171038 | Zbl 0125.32102
[12] Holubová, G.: Optimal conditions for the maximum principle for second-order periodic problems. Electron. J. Differential Equations, Special Issue 2023 (2023), 151-160. DOI 10.58997/ejde.sp.02.h2 | MR 4803584
[13] Kapitza, P. L.: Dynamic stability of the pendulum with vibrating suspension point. Sov. Phys. JETP 21 (1951), 588-597 Russian.
[14] Kapitza, P. L.: Pendulum with an oscillating pivot. Usp. fiz. nauk 44 (1951), 7-20 Russian.
[15] Komlenko, J. V., Tonkov, E. L.: A periodic boundary value problem for an ordinary second-order differential equation. Dokl. Akad. Nauk SSSR 179 (1968), 17-19 Russian. MR 0226119 | Zbl 0172.11604
[16] Leonov, G. A.: On stability in the first approximation. J. Appl. Math. Mech. 62 (1998), 511-517. DOI 10.1016/S0021-8928(98)00067-7 | MR 1680316
[17] Leonov, G. A.: First-approximation instability criteria for non-stationary linearizations. J. Appl. Math. Mech. 68 (2004), 827-838. DOI 10.1016/j.jappmathmech.2004.11.004 | MR 2125024 | Zbl 1095.34031
[18] Leonov, G. A., Kuznetsov, N. V.: Time-varying linearization and the Perron effects. Int. J. Bifurcation Chaos Appl. Sci. Eng. 17 (2007), 1079-1107. DOI 10.1142/S0218127407017732 | MR 2329516 | Zbl 1142.34033
[19] Lomtatidze, A.: Theorems on differential inequalities and periodic boundary value problem for second-order ordinary differential equations. Mem. Differ. Equ. Math. Phys. 67 (2016), 1-129. MR 3472904 | Zbl 1352.34033
[20] Sansone, G.: Ordinary differential equations. Vol. I. Izd. Inostrannoj Literatury, Moscow (1953), Russian. MR 0064221
[21] Seyranian, A. A., Seyranian, A. P.: The stability of an inverted pendulum with a vibrating suspension point. J. Appl. Math. Mech. 70 (2006), 754-761. DOI 10.1016/j.jappmathmech.2006.11.009 | MR 2319534 | Zbl 1126.70361
[22] Tonkov, E. L.: The second order periodic equation. Dokl. Akad. Nauk SSSR 184 (1969), 296-299 Russian. MR 0237880 | Zbl 0184.12102
[23] Torres, P. J.: Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem. J. Differ. Equations 190 (2003), 643-662. DOI 10.1016/S0022-0396(02)00152-3 | MR 1970045 | Zbl 1032.34040
Partner of
EuDML logo