Stability and Chaos
Jeff Xia
SUSTech
SUSTech,
Indeed, had the title not been already preempted, one
might suggest that the study of the motions of the celestial
bodies is “the world's oldest profession”. If it isn't the
oldest, then, most surely, it is “the second oldest”.
- Donald Saari
• Newton’s discovery of universal
gravitation was based on Kepler’s (15711630) laws of planetary motion, which is
based on Danish astronomer Brahe’s
(1546-1601) observation of planetary
motion.
• Fortunately, mars was one of the major
planets included in Brahe’s observation.
Major Planets
Planet
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto
Average distance
from the Sun
(astronomical units AU1)
0.387
0.723
1.00
1.52
5.20
9.58
19.2
30.0
39.2
Equatorial radius
(Earth=1)
0.38
0.95
1.00
0.53
11.21
9.45
4.01
3.88
0.19
Mass (Earth=1)
0.0553
0.815
1.00
0.107
318
95.2
14.5
17.2
0.0025
Length of sidereal year2
(Earth units)
0.241
year
0.615
year
1.00
year
1.88
years
11.9
years
29.4
years
83.7
years
164
years
248
years
Period of sidereal
rotation3 (Earth units)
58.65
days
243.0
days
23.93
hours
24.62
hours
9.925
hours
10.66
hours
17.24
hours
16.11
hours
6.387
days
Number of known moons
0
0
1
2
16
18
18
8
1
Average density
(g/cubic cm)
5.43
5.24
5.52
3.94
1.33
0.70
1.30
1.76
1.1
Eccentricity of orbit4
0.21
0.01
0.02
0.09
0.05
0.06
0.05
0.01
0.25
Inclination of orbit5
(degrees)
7.0
3.4
0.0
1.9
1.3
2.5
0.77
1.8
17
• One Planet
• Two Planets:
• Nine or More Planets and Satellites
Is Solar System Stable?
Newton (1642-1727):
• NO, the planetary system is unstable
• God solves the problem by controlling the instabilities
It must be a huge
Newton’s head!
apple
that dropped on
Newton’s idea was heavily criticized by Leibniz and
all enlightened 18th century
Lagrange(1736-1813) and Laplace(1749-1847):
• YES, the planetary system is stable
• Based on analysis of secular terms in orbital
computations
Boltzmann(1844-1906) and Maxwell(1831-1879):
• NO, the planetary system is unstable
• They believed in instability in general
I don’t know, but why would I care?
Numerical Results by Jacques
Laskar:
– Stable for millions and billions of years!
– He used averaging method in
computation
– Unstable without the moon
– Stability goes back to 5 billion years ago
Stability and KAM Theory
• One of the most fundamental mathematical
•
•
•
theories of the last century
Kolmogorov stated the main theorem
Arnold provided a complete proof
Moser, using an idea of Russell Crowe (oops,
actually John Nash), proved the theorem for
much more general cases
Main Results of KAM Theory
• If a system is close to something stable,
then most trajectories are regular and
simple
• For such a system, instability is
improbable if it is at all possible
• King Oscar II of Sweden and Norway (1829-1907)
•Lover of Arts and Sciences;
•Majored in Mathematics at Uppsala University;
•Founded Acta Mathematica;
•King Oscar II Prize in1887
Paul Painleve, 1863-1933
• King Oscar II invited Painleve to give a series lectures in
Stockholms in 1895;
• King Oscar II attended the first lecture;
• Painleve proposed the Painleve Conjecture;
• The conjecture was eventually solved in 1988 by Xia
(using chaos!)
84th and 92nd prime minister of France
Henri Poincare
• Awarded King Oscar II Prize;
• A series and fatal mistake was found in
the original paper;
• New version of the paper and the
discovery of CHAOS
"This work cannot indeed be considered as
furnishing the complete solution of the
question proposed, but that it is
nevertheless of such importance that its
publication will inaugurate a new era in
the history of celestial mechanics."
-Karl Weierstrass
Chaos and Instability
--The world is not as stable as we thought
--More often than not, it is totally chaotic
1 + 2 + 22 + 23 + … + 263 = 264 - 1
= 18,446,744,073,709,551,615
@ 4,000 billion bushels
@ 2000 years of world’s wheat
production
Geometric series grows fast!
Box of Gas Molecules
• Small deviation from initial conditions
leads to an error that grows exponentially
as time goes
• However, the error is confined by the size
of the box
What is Chaos?
• Exponential growth at the microscopic
level
• In mathematical terms, existence of
positive Lyaponov exponents
A Mathematical Model
1.
Take a number x1 between 0 and 1, say,
x1 = 0.31415926535…….
2.
Multiply x1 by 10 and drop integer part of the resulting number:
x2 = 0.14159265358…….
3.
Repeat the process:
x3 = 0.41592653589……
4.
This defines a dynamical process: x1, x2, x3, x4, …
x200 = 0.96442881098……
Future is unpredictable!
The Weather System is Chaotic
The Butterfly Effect:
• It supposed to have thunderstorm right now in
Shenzhen
• A butterfly flopped its wings a few weeks ago in
Chicago
• Now, we have a sunny day here!
What We Know:
• Stable systems are simple
• “Most” chaotic systems are simple too
• “Mixed” systems are most complicated
• N-Body problem, as with most conservative systems, is
mixed, hence complicated, stability in certain regions
and chaos in others
Examples of Chaotic Motions
• Poincare showed the restricted three body
problem can be chaotic.
• Many other chaotic phenomena, including
Arnold Diffusion.
• Painleve Conjecture: Unbounded solutions
in finite time.
Conjecture:
Typical conservative systems are (strongly)
unstable; one can reach almost any state by doing
a little perturbation in the initial state
One can switch the position of the Earth and Pluto
by pushing current positions of planets by a tiny
amount and wait long enough
Applications of Chaos
• In April, 1991, Japan lunched a lunar
probe named Hiten (Muse-2)
• It was realized that Hiten did not have
enough fuel to go to the Moon
• Japan asked US-NASA to help
• Edward Belbruno at JPL took the Job
Spacecraft Hiten
• Belbruno redesigned the trajectory
• Hiten reached the Moon in October, 1991
• Idea: use the remaining fuel sent Hiten to a
•
•
chaotic region, then using Chaos to sent the
spacecraft to the Moon.
My work in three-body problem served as a
theoretical foundation of Belbruno’s design
Belbruno saved another satellite, Hughe’s HGS-1
in 1998