*(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

Using a probability approach, we assume that ${\left({a}_{k}\right)}_{k=1}^{\infty}$ is given, and ${\left\{{t}_{k}\right\}}_{k=1}^{\infty}$ 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 \phantom{\rule{4.pt}{0ex}}\text{as}\phantom{\rule{4.pt}{0ex}}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

*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.

##### MSC:

34C15 | Nonlinear oscillations, coupled oscillators (ODE) |

34C28 | Complex behavior, chaotic systems (ODE) |

65G30 | Interval and finite arithmetic |

34F05 | ODE with randomness |

34A36 | Discontinuous equations |

##### Keywords:

random coefficients; parametric resonance; forced damped pendulum; computer-aided proof; interval arithmetic##### References:

[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. |

[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. |

[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 02306793 · 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. |

[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. |

[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. |

[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. |

[13] | P. Hartman, Ordinary Differential Equations, Birkhäuser, Boston, 1982. |

[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. |

[16] | L. Hatvani and V. Totik, Asymptotic stability of the equilibrium of the damped oscillator, Differential Integral Equations, 6 (1993), 835–848. |

[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. |

[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. |

[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. |

[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. |

[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 |