This paper considers random Hill’s equations in the limit where the periodic forcing function becomes a Dirac delta function. For this class of equations, the forcing strength qk, the oscillation frequency λk, and the period (Δτ)k are allowed to vary from cycle to cycle. Such equations arise in astrophysical orbital problems in extended mass distributions, in the reheating problem for inflationary cosmologies, and in periodic Schrödinger equations. The growth rates for solutions to the periodic differential equation can be described by a matrix transformation, where the matrix elements vary from cycle to cycle. Working in the delta function limit, this paper addresses several coupled issues. We find the growth rates for the 2×2 matrices that describe the solutions. This analysis is carried out in the limiting regimes of both large qk1 and small qk1 forcing strength parameters. For the latter case, we present an alternate treatment of the dynamics in terms of the Fokker–Planck equation, which allows for a comparison of the two approaches. Finally, we elucidate the relationship between the fundamental parameters (λk,qk) appearing in the stochastic differential equation and the matrix elements that specify the corresponding discrete map. This work provides analytic—and accurate—expressions for the growth rates of these stochastic differential equations in both the qk1 and the qk1 limits.

1.
Abramowitz
,
M.
and
Stegun
,
I. A.
,
Handbook of Mathematical Functions
(
Dover
,
New York
,
1970
).
2.
Adams
,
F. C.
and
Bloch
,
A. M.
, “
Hill’s equation with random forcing terms
,”
SIAM J. Appl. Math.
68
,
947
(
2008
).
3.
Adams
,
F. C.
and
Bloch
,
A. M.
, “
Hill’s equation with random forcing parameters: General treatment including marginally stable cases
,”
J. Stat. Phys.
(submitted).
4.
Adams
,
F. C.
,
Bloch
,
A. M.
,
Butler
,
S. C.
,
Druce
,
J. M.
, and
Ketchum
,
J. A.
, “
Orbits and instabilities in a triaxial cusp potential
,”
Astrophys. J.
670
,
1027
(
2007
).
5.
Anderson
,
P. W.
, “
Absence of diffusion in certain random lattices
,”
Phys. Rev.
109
,
1492
(
1958
).
6.
Binney
,
J.
and
Tremaine
,
S.
,
Galactic Dynamics
(
Princeton University Press
,
Princeton
,
1987
).
7.
Cambronero
,
S.
,
Rider
,
B.
, and
Rameríz
,
J.
, “
On the shape of the ground state eigenvalue density of a random Hill’s equation
,”
Commun. Pure Appl. Math.
59
,
935
(
2006
).
8.
Cohen
,
J. E.
and
Newman
,
C. M.
, “
The stability of large random matrices and their products
,”
Ann. Probab.
12
,
283
(
1984
).
9.
Doering
,
C. R.
and
Gradoua
,
J. C.
, “
Resonant activation over a fluctuating barrier
,”
Phys. Rev. Lett.
69
,
2318
(
1992
).
10.
Furstenberg
,
H.
, “
Noncommuting random products
,”
Trans. Am. Math. Soc.
108
,
377
(
1963
).
11.
Furstenberg
,
H.
and
Kesten
,
H.
, “
Products of random matrices
,”
Ann. Math. Stat.
31
,
457
(
1960
).
12.
Guth
,
A. H.
, “
Inflationary universe: A possible solution to the horizon and flatness problems
,”
Phys. Rev. D
23
,
347
(
1981
).
13.
Hill
,
G. W.
, “
On the part of the motion of the lunar perigee which is a function of the mean motions of the Sun and Moon
,”
Acta Math.
8
,
1
(
1886
).
14.
Ishii
,
K.
, “
Localization of eigenstates and transport phenomena in one-dimensional disordered systems
,”
Prog. Theor. Phys.
45
,
77
(
1973
).
15.
Kofman
,
L.
,
Linde
,
A.
, and
Starobinsky
,
A. A.
, “
Reheating after inflation
,”
Phys. Rev. Lett.
73
,
3195
(
1994
).
16.
Kofman
,
L.
,
Linde
,
A.
, and
Starobinsky
,
A. A.
, “
Towards the theory of reheating after inflation
,”
Phys. Rev. D
56
,
3258
(
1997
).
17.
Kolb
,
E. W.
and
Turner
,
M. S.
,
The Early Universe
(
Addison-Wesley
,
Reading, MA
,
1990
).
18.
Lima
,
R.
and
Rahibe
,
M.
, “
Exact Lyapunov exponent for infinite products of random matrices
,”
J. Phys. A
27
,
3427
(
1994
).
19.
Magnus
,
W.
and
Winkler
,
S.
,
Hill’s Equation
(
Wiley
,
New York
,
1966
).
20.
Pastur
,
L.
and
Figotin
,
A.
,
Spectra of Random and Almost-Periodic Operators
,
A Series of Comprehensive Studies in Mathematics
(
Springer-Verlag
,
Berlin
,
1991
).
21.
Zanchin
,
V.
,
Maia
,
A.
,
Craig
,
W.
, and
Brandenberger
,
R.
, “
Reheating in the presence of noise
,”
Phys. Rev. D
57
,
4651
(
1998
).
You do not currently have access to this content.