Encyclopedia of the Solar System 2nd ed

Solar System Dynamics: Regular and Chaotic Motion 797

50 100 150 200 250 300

0.675

0.68

0.685

0.69

0.695

Semimajor axis

Time (Jupiter periods)

FIGURE 7 The semimajor axis as a function of time for

an object using the same starting conditions as in Fig. 6.

The units of the semimajor axis are such that Jupiter’s

semimajor axis (5.202 AU) is taken to be unity.

The equations of motion can be integrated with the
same starting conditions to generate a surface of section
by plotting the values ofxandx ̇whenevery=0 withy ̇> 0
(Fig. 8). The pattern of three distorted curves or “islands”
that emerges is a characteristic of resonant motion when dis-
played in such plots. If a resonance is of the form (p+q):p,
wherepandqare integers, thenqis said to be the order
of the resonance. The number of islands seen in a surface
of section plot of a given resonant trajectory is equal toq.
In this case,p=4,q=3 and three islands are visible.
The center of each island would correspond to a starting
condition that placed the asteroid at exact resonance where
the variation ineandawould be minimal. Such points are
said to be fixed points of the Poincar ́e map. If the starting
location was moved farther away from the center, the subse-
quent variations ineandawould get larger, until eventually
some starting values would lead to trajectories that were not
in resonant motion.

0.5 0.6 0.7 0.8 0.9

-0.4

-0.2

0

0.2

0.4

x.

x

FIGURE 8 A surface of section plot for the same (regular) orbit
shown in Figs. 6 and 7. The 2000 points were generated by
plotting the values ofxandx ̇whenevery=0 with positive ̇y.
The three “islands” in the plot are due to the third-order 7:4
resonance.

4.2.2 CHAOTIC ORBITS

Figures 9 and 10 show the plots ofeandaas a func-

tion of time for an asteroid orbit with starting valuesx 0 =

0.56,y 0 =0,x ̇ 0 =0, andy ̇determined from Eq. (41) with

C=3.07. The corresponding orbital elements area 0 =

0.6984 ande 0 =0.1967. These values are only slightly dif-

ferent from those used earlier, indeed the initial behavior

of the plots is quite similar to that seen in Figs. 6 and 7.

However, subsequent variations ineandaare strikingly

different. The eccentricity varies from 0.188 to 0.328 in an

irregular manner, and the value ofais not always close to

the value associated with exact resonance. This is an ex-

ample of a chaotic trajectory where the variations in the

orbital elements have no obvious periodic or quasi-periodic

structure. The anticorrelation ofaandecan be explained

in terms of the Jacobi constant.

The identification of this orbit as chaotic becomes appar-

ent from a study of its surface of section (Fig. 11). Clearly,

this orbit covers a much larger region of phase space than

the previous example. Furthermore, the orbit does not lie

on a smooth curve, but is beginning to fill an area of the

phase space. The points also help to define a number of

empty regions, three of which are clearly associated with

the 7:4 resonance seen in the regular trajectory. There is

also a tendency for the points to “stick” near the edges of

the islands; this gives the impression of regular motion for

short periods of time.

Chaotic orbits have the additional characteristic that

they are sensitively dependent on initial conditions. This is

illustrated in Fig. 12, where the variation ineas a function

of time is shown for two trajectories; the first corresponds to

Fig. 9 (wherex 0 =0.56) and the second hasx 0 =0.56001.

The initial value ofy ̇was chosen so that the same value ofC

was obtained. Although both trajectories show comparable

initial variations ine, after 60 Jupiter periods it is clear that

the orbits have drifted apart. Such a divergence would not

occur for nearby orbits in a regular part of the phase space.