Gradus
VOL 3, NO 2 (2016): AUTUMN (NOVEMBER)
A HINTÁZÁS LINEÁRIS, LÉPCSŐSFÜGGVÉNY-EGYÜTTHATÓS MODELLJÉNEK PERIODIKUS MEGOLDÁSAIRÓL
ON THE LINEAR, STEP-FUNCTION COEFFICIENT MODEL OF SWINGIG
Abstract
Az alábbi egyenletet vizsgáljuk:......,ahol g és l jelöli rendre a gravitációs gyorsulás értékét, illetve az inga hosszát. Az ?>0 paraméter méri a hintázás intenzitását. Be- vezetjük a közeledo? és távolodó megoldás fogalmát. Szükséges és elegendo? föltételt adunk a T és az ? paraméterek segítségével a 2T -periodikus és a 4T -periodikus megoldások létezésére.
The equation......is considered, where g and l denote the constant of gravity and the length of the pendulum, respectively; ? > 0 is a parameter measuring the intensity of swinging. Concepts of solutions going away from the origin and approaching to the origin are introduced. Necessary and sufficient conditions are given in terms of T and ? for the existence of solutions of these types, which yield condi- tions for the existence of 2T -periodic and 4T -periodic solutions as special cases. The domain of instability, i.e. the Arnold tongues of parametric resonance are deduced from these results.
Keywords
Kulcsszavak: másodrenduű lineáris differenciálegyenletek, lépcsősfüggvény-, együtthatók, periodikus együtthatók, impulzív hatások, periodikus, megoldások, parametrikus rezonancia, hintázás,
Keywords: second order linear differential, equations, step-function coefficients, periodic coefficients, parametric resonance, swinging,
References
| [1] | Arnold, V. I. Ordinary Differential Equations. Springer-Verlag, Berlin, 2006. |
| [2] | Butikov, E. I. Parametric resonance in a linear oscillator at square-wave modulation. Eur. J. Phys.26 (2005), 157–174. |
| [3] | Csizmadia, L. ; Hatvani, L. On the linear model of seinging with square wave coefficient by anelementary geometric method. Acta Sci. Math. 81 (2015), 483–502. |
| [4] | Curry, S. M. How chidren swing. Amer. J. Phys. 44 (1976), 924–926. |
| [5] | Csörgo?, S.; Hatvani, L. Stability properties of solutions of linear second order differential equa-tions with random coefficients. J. Differential Equations 248(2010), no. 1, 21–49. |
| [6] | Hatvani, L. On the existence of a small solution to linear second order differential equations withstep function coefficients. Dynam. Contin. Discrete Impuls. Systems 4 (1998), 321–330. |
| [7] | Hatvani, L. An elementary method for the study of Meissner’s equation and its application toproving the Oscillation Theorem. Acta Sci. Math. 79(2013), no. 1–2, 87–105. |
| [8] | Hatvani, L. On the parametrically excited pendulum equation with a square wave coefficient (inpreparation). |
| [9] | Haupt, O. Über eine Methode zum Bewise von Oszillationstheoremen. Math. Ann. 76(1914),67–104. |
| [10] | Haupt, O. Über lineare homogene Differentialgleichungen 2. Ordnung mit periodischen Koeffizi-enten. Math. Ann. 79(1917), 278–285. |
| [11] | Hill, G. W. On the part of the motion of the lunar perigee which is a function of the mean motionsof the Sun and Moon. Acta Mathematica 8(1886), 1–36. |
| [12] | Hochstadt, H. A special Hill’s equation with discontinuous coefficients. Amer. Math. Monthly70(1963), 18–26. |
| [13] | Landau, L. D.; Lifshitz, E. M. Mechanics. Course of Theoretical Physics. Vol. 1, Elsevier, Am-sterdam, 1976. |
| [14] | Lyapunov, A. M. The general problem of the stability of motion. Lyapunov centenary issue. In-ternat. J. Control 55(1992), no. 3, 521–790. |
| [15] | Magnus, W.; Winkler, S. Hill’s equation. Dover Publications, Inc., New York, 1979. |
| [16] | Meissner, E. Über Schüttel-schwingungen in Systemen mit Periodisch Veränderlicher Elastizität.Schweizer Bauzeitung 72(1918), no. 10, 95–98. |
| [17] | D. R. Merkin, Introduction to the Theory of Stability. Springer-Verlag, 1997. |
| [18] | Seyranian, A. P.; Seyranian, A. A. Breakup of resonance zones for Meissner’s equation withsmall damping. Moscow University Mechanics Bulletin (2003), No. 5, 53–59. |