quences of increasing Jupiter's mass to speed

up perturbation effects.

II. METHOD

The dynamical system utilized for this study

is the three-dimensional elliptic restricted three-

body (ERTB) problem, with Jupiter as the sec-

ondary mass m

2

. We integrated the equations

of motion in the rotating-pulsating frame (cf.

Szebehely 1967, Szebehely and Giacaglia 1964)

with a Bulirsch-Stoer extrapolation method

(e.g. Stoer and Bulirsch 1980, Press et al.

1992). See Murison (1989) for details on the

numerical performance of the integrator. The

(constant) eccentricity used for Jupiter in all

cases was e

J

=0.04848. This model is conven-

ient because of its simplicity. The ERTB prob-

lem is the simplest dynamical model which is

still complex enough to exhibit all the important

behavior required for this study. Comparisons

with a model that includes the effects of Saturn

on the motion of Jupiter (cf. Lecar and Frank-

lin, 1973) yielded no significant differences that

would affect our results in the region of the

asteroid belt considered here. In particular, an

integration of (2311) El Leoncito yielded a T

L

of 460 T

J

, fully consistent with our value of 422

T

J

. Relying on a reasonable and simple statisti-

cal interpretation, as well as indications from

recent nonlinear dynamics literature, we conjec-

ture that the property of certain orbits to

exhibit long-term "stability" despite short

Lyapunov times is in fact to be expected, as we

argue in the next section.

Along with each orbit, a second "test" orbit

was integrated. This second orbit started a dis-

tance 10

-6

in phase space from the reference

orbit and was used for calculating the

Lyapunov exponent. Renormalization of the

test orbit with respect to the reference orbit

occurred whenever the phase space distance

exceeded 10

-4

, thus avoiding saturation prob-

lems. See e.g. Benettin et al. (1976) and Wolf

et al. (1985) for details on calculating Lyapu-

nov exponents.

We obtained orbital elements for the 25