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


Csizmadia László, Hatvani László

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,


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