Hill ’ s equations arise in a wide variety of physical problems , and are specified by a natural frequency , a periodic forcing function , and a forcing strength parameter .
This classic problem is generalized here in two ways : [ A ] to Random Hill ’ s equations which allow the forcing strength q _ { k } , the oscillation frequency { \lambda } _ { k } , and the period ( { \Delta \tau } ) _ { k } of the forcing function to vary from cycle to cycle , and [ B ] to Stochastic Hill ’ s equations which contain ( at least ) one additional term that is a stochastic process \xi .
This paper considers both random and stochastic Hill ’ s equations with small parameter variations , so that p _ { k } = q _ { k } - \langle { q _ { k } } \rangle , \ell _ { k } = { \lambda } _ { k } - \langle { { \lambda } _ { k } } \rangle , and \xi are all { \cal O } ( \epsilon ) , where \epsilon \ll 1 .
We show that random Hill ’ s equations and stochastic Hill ’ s equations have the same growth rates when the parameter variations p _ { k } and \ell _ { k } obey certain constraints given in terms of the moments of \xi .
For random Hill ’ s equations , the growth rates for the solutions are given by the growth rates of a matrix transformation , under matrix multiplication , where the matrix elements vary from cycle to cycle .
Unlike classic Hill ’ s equations where the parameter space ( the { \lambda } - q plane ) displays bands of stable solutions interlaced with bands of unstable solutions , random Hill ’ s equations are generically unstable .
We find analytic approximations for the growth rates of the instability ; for the regime where Hill ’ s equation is classically stable , and the parameter variations are small , the growth rate \gamma = { \cal O } ( p _ { k } ^ { 2 } ) = { \cal O } ( \epsilon ^ { 2 } ) .
Using the relationship between the ( \ell _ { k } ,p _ { k } ) and the \xi , this result for \gamma can be used to find growth rates for stochastic Hill ’ s equations .