Encyclopedia of the Solar System 2nd ed

Solar System Dynamics: Regular and Chaotic Motion 799

0.5 0.6 0.7 0.8 0.9

-0.4

-0.2

0

0.2

0.4

x

x

.

FIGURE 11 A surface of section plot for the same

chaotic orbit as shown in Figs. 9 and 10. The 2000

points were generated by plotting the values ofx

andx ̇whenevery=0 with positivey ̇. The points

are distributed over a much wider region of the

(x,x ̇) plane than the points for the regular orbit

shown in Fig. 8, and they help to define the edges

of the regular regions associated with the 7:4 and

other resonances.

52 54 56 58 60 62 64

0.2

0.22

0.24

0.26

0.28

Eccentricity

Time (Jupiter periods)

x 0 = 0.56001

x 0

FIGURE 12 The variation in the eccentricity for

two chaotic orbits started close to one another. One

plot is part of Fig. 9 using the chaotic orbit started

withx 0 =0.56, and the other is for an orbit with

x 0 =0.56001. Although the divergence of the two

orbits is exponential, the effect becomes noticeable

only after 60 Jupiter periods.

2.2 2.4 2.6 2.8 3.0 3.2

-2.5

-2.0

-1.5

-1.0

-0.5

Log

Log t

chaotic orbit

regular orbit

FIGURE 13 The evolution of the quantityγ[defined in Eq. (43)]

as a function of time (in Jupiter periods) for a regular (x 0 =0.55)

and chaotic (x 0 =0.56) orbit. In this log–log plot, the regular orbit

shows a characteristic slope of−1 with no indication of logγ

tending toward a finite value. However, in the case of the chaotic

orbit, logγtends to a limiting value close to− 0 .77.