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Stability problems for the mathematical pendulum. (English) Zbl 1164.34017
Summary: The first part of this review paper is devoted to the simple (undamped, unforced) pendulum with a varying coefficient. If the coefficient is a step function, then small oscillations are described by the equation $$ \ddot x +a^2(t)x=0,\ a(t):=a_k\quad \text{if}\quad t_{k-1}\le t<t_k,\quad k=1,2,\dots\, . $$ Using a probability approach, we assume that $(a_k )^\infty_{k=1}$ is given, and $\{t_k\}_{k=1}^\infty$ is chosen at random so that $t_k - t_{k-1}$ are independent random variables. The first problem is to guarantee that all solutions tend to zero as $t \to \infty$, provided that $a_k\nearrow \infty \text{ as } k \to \infty$. In the problem that the coefficient $a^2$ takes only two different (alternating) values, and $t_k - t_{k-1}$ are identically distributed, one has to find the distributions and their critical expected values such that the amplitudes of the oscillations tend to $\infty$ in some (probabilistic) sense. In the second part we deal with the damped forced pendulum equation $$ \ddot x+10^{-1}\dot x +\sin x=\cos t. $$ {\it J. H. Hubbard} [Am. Math. Mon. 106, No. 8, 741-758 (1999; Zbl 0989.70014)] discovered that some motions of this simple physical model are chaotic. Recently, also using the computer (the method of interval arithmetic), we gave a proof for Hubbard’s assertion. Here we show some tools of the proof.
34C15Nonlinear oscillations, coupled oscillators (ODE)
34C28Complex behavior, chaotic systems (ODE)
65G30Interval and finite arithmetic
34F05ODE with randomness
34A36Discontinuous equations
Full Text: DOI
[1] V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1978; Translated from the Russian by K. Vogtmann and A. Weinstein, Graduate Texts in Mathematics, 60. · Zbl 0386.70001
[2] T. Csendes, B. Bánhelyi and L. Hatvani, Towards a computer-assisted proof for chaos in a forced damped pendulum equation, J. Computational and Applied Mathematics, 199 (2007), 378--383. · Zbl 1108.65121 · doi:10.1016/j.cam.2005.08.046
[3] B. Bánhelyi, T. Csendes, B. M. Garay and L. Hatvani, Computer-assisted proof of chaotic behaviour of the forced damped pendulum, Folia FSN Universitatis Masarykianae Brunensis, Matematica, to appear. · Zbl 1263.34061
[4] B. Bánhelyi, T. Csendes, B. M. Garay and L. Hatvani, A computer-assisted proof for {$\Sigma$}3-chaos in the forced damped pendulum equation, submitted.
[5] R. Borelli and C. Coleman, Computers, lies and the fishing season, College Math. J., 25 (1994), 401--412. · Zbl 1291.97023 · doi:10.2307/2687505
[6] S. Csörgo and L. Hatvani, Stability properties of solutions of linear second order differential equations with random coefficients, in preparation.
[7] Á. Elbert, Stability of some differential equations, Advances in Difference Equations (Veszprém, 1995), Gordon and Breach, Amsterdam, 1997, 165--187. · Zbl 0890.39010
[8] Á. Elbert, On asymptotic stability of some Sturm-Liouville differential equations, General Seminar of Mathematics, University of Patras (1996/97), 22--23.
[9] Á. Elbert, On damping of linear oscillators, Studia Sci. Math. Hungar., 38 (2001), 191--208. · Zbl 0997.34040
[10] A. S. Galbraith, E. J. McShane and G. B. Parrish, On the solutions of linear second-order differential equations, Proc. Nat. Acad. Sci. U.S.A., 53 (1965), 247--249. · Zbl 0133.34103 · doi:10.1073/pnas.53.2.247
[11] J. R. Graef and J. Karsai, On irregular growth and impulses in oscillator equations, Advances in Difference Equations (Veszprém, 1995), Gordon and Breach, Amsterdam, 1997, 253--262. · Zbl 0896.34025
[12] N. Guglielmi and L. Hatvani, On small oscillations of mechanical systems with time-dependent kinetic and potential energy, Discrete Contin. Dyn. Syst., to appear. · Zbl 1146.70011
[13] P. Hartman, Ordinary Differential Equations, Birkhäuser, Boston, 1982. · Zbl 0476.34002
[14] L. Hatvani, On the existence of a small solution to linear second order differential equations with step function coefficients, Dynam. Contin. Discrete Impuls. Systems, 4 (1998), 321--330. · Zbl 0957.35106 · doi:10.3934/dcds.1998.4.321
[15] L. Hatvani, Growth condition guaranteeing small solutions for linear oscillator with increasing elasticity coefficient, Georgian Math. J., to appear. · Zbl 1131.34029
[16] L. Hatvani and V. Totik, Asymptotic stability of the equilibrium of the damped oscillator, Differential Integral Equations, 6 (1993), 835--848. · Zbl 0777.34036
[17] L. Hatvani and L. Stachó, On small solutions of second order differential equations with random coefficients, Arch. Math. (Brno), 34(1) (1998), 119--126. · Zbl 0915.34051
[18] L. Hatvani, On stability properties of solutions of second order differential equations, Proceedings of the 6th Colloquium on the Qualitative Theory of Differential Equations (Szeged, 1999), No. 11, 6 pp. (electronic), Proc. Colloq. Qual. Theory Differ. Equ., Electron. J. Qual. Theory Differ. Equ., Szeged, 2000. · Zbl 0971.34037
[19] L. Hatvani, On small solutions of second order linear differential equations with non-monotonous random coefficients, Acta Sci. Math. (Szeged), 68 (2002), 705--725. · Zbl 1027.34058
[20] L. Hatvani and L. Székely, On the existence of small solutions of linear systems of difference equations with varying coefficients, J. Difference Equ. Appl., 12 (2006), 837--845. · Zbl 1103.39008 · doi:10.1080/10236190600772390
[21] H. Hochstadt, A special Hill’s equation with discontinuous coefficients, Amer. Math. Monthly, 70 (1963), 18--26. · Zbl 0117.05103 · doi:10.2307/2312778
[22] J. H. Hubbard, The forced damped pendulum: chaos, complications and control, Amer. Math. Monthly, 106 (1999), 741--758. · Zbl 0989.70014 · doi:10.2307/2589021
[23] H. Milloux, Sur l’equation differentielle x” + A(t)x = 0, Prace Mat.-Fiz., 41 (1934), 39--54. · Zbl 0009.16402
[24] N. S. Nedialkov, K. R. Jackson and G. F. Corliss, Valiated solutions of initial value problems for ordinary differential equations, Appl. Math. Comput., 105 (1999), 21--68. · Zbl 0934.65073 · doi:10.1016/S0096-3003(98)10083-8
[25] P. Pucci and J. Serrin, Asymptotic stability for intermittently controlled nonlinear oscillators, SIAM J. Math. Anal., 25 (1994), 815--835. · Zbl 0809.34067 · doi:10.1137/S0036141092240679