Hill's Equation with Random Forcing Parameters: Determination of Growth Rates Through Random Matrices
Abstract
This paper derives expressions for the growth rates for the random 2×2 matrices that result from solutions to the random Hill's equation. The parameters that appear in Hill's equation include the forcing strength q _{ k } and oscillation frequency λ _{ k }. The development of the solutions to this periodic differential equation can be described by a discrete map, where the matrix elements are given by the principal solutions for each cycle. Variations in the ( q _{ k }, λ _{ k }) lead to matrix elements that vary from cycle to cycle. This paper presents an analysis of the growth rates including cases where all of the cycles are highly unstable, where some cycles are near the stability border, and where the map would be stable in the absence of fluctuations. For all of these regimes, we provide expressions for the growth rates of the matrices that describe the solutions.
 Publication:

Journal of Statistical Physics
 Pub Date:
 April 2010
 DOI:
 10.1007/s109550109931x
 arXiv:
 arXiv:1002.1014
 Bibcode:
 2010JSP...139..139A
 Keywords:

 Hill’s equation;
 Random matrices;
 Lyapunov exponents;
 Hill's equation;
 Mathematical Physics;
 Astrophysics  Cosmology and Nongalactic Astrophysics;
 General Relativity and Quantum Cosmology;
 82B31
 EPrint:
 22 pages, 3 figures