the attractor
DESCRIPTION
CCCN talk by Wang XiongTRANSCRIPT
Mystery and Beauty of Attractor
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
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Outline
A gameConcepts: state space, evolution rule, attractor
Some applications
In numerical analysis, economics,
PageRank, system identification ...
Different kinds of attractors
Point attractor, Periodic attractor, Torus
Attractor, Strange attractor
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Let’s start by playing a game…
Take any three-digit number, using at least two different digits. (Leading zeros are allowed.)
Arrange the digits in ascending and then in descending order to get two three-digit numbers, adding leading zeros if necessary.
Subtract the smaller number from the bigger number.
Go back to step 2.
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An example
For example, choose 211:
211 – 112 = 099
990 – 099 = 891 (rather than 99 - 99 = 0)
981 – 189 = 792
......
Have a try… I am pretty sure what you will
get
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The answer must be 495
211 – 112 = 099 990 – 099 = 891 (rather than
99 - 99 = 0) 981 – 189 = 792 972 – 279 = 693
963 – 369 = 594 954 -459 = 495
What about your result?
You may wonder why….
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In this game…
The rule of the game is a iteration, or discrit time dynamic system
The dynamic space is all the three-digit numbers
Given any number is called initial condition
The 495, 000 are two fixed point attractors
These two attractors have different attraction basins
And point attractor is the only story in this game…
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Four-digit number game
For example, choose 3524:
5432 – 2345 = 3087 8730 – 0378 = 8352
8532 – 2358 = 6174
2111 – 1112 = 0999 9990 – 0999 = 8991
(rather than 999 – 999 = 0) 9981 – 1899 =
8082 8820 – 0288 = 8532 8532 – 2358 =
6174
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Another more challenging
game…
Can you fill the right numbers in the following
sentence
In this sentence, the number of occurrences
of 0 is _, of 1 is _, of 2 is _, of 3 is _, of 4 is _,
of 5 is _, of 6 is _, of 7 is _, of 8 is _, and of 9
is _.
To make a true statement about the number
of occurrences of each of the digits 0 to 9 that
it contains9
One possible answer is ...
In this sentence, the
number of occurrences of 0
is 1, of 1 is 11, of 2 is 2, of
3 is 1, of 4 is 1, of 5 is 1, of
6 is 1, of 7 is 1, of 8 is 1,
and of 9 is 1.10
Stranger still...
What about the following pair of sentences?
In the next sentence, the number of
occurences of 0 is 1, of 1 is 7, of 2 is 4, of 3
is 1, of 4 is 1, of 5 is 1, of 6 is 1, of 7 is 1, of 8
is 2, and of 9 is 1.
In the previous sentence, the number of
occurences of 0 is 1, of 1 is 8, of 2 is 2, of 3
is 1, of 4 is 2, of 5 is 1, of 6 is 1, of 7 is 2, of 8
is 1, and of 9 is 1.11
Question...
How is it possible to come up with these
strangely introspective sentences?
Can you give the answer directly?
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Construction
Firstly, consider the blank `template
sentence':
In this sentence, the number of occurences of
0 is _, of 1 is _, of 2 is _, of 3 is _, of 4 is _, of
5 is _, of 6 is _, of 7 is _, of 8 is _, and of 9 is
_.
What is needed is some way of filling in the
gaps.
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Choosing an initial value
Suppose that we begin by putting in
any ten numbers, even though the
resulting sentence is almost bound to
be false.
For example, choosing all of the
numbers to be zero gives the
vector:(0,0,0,0,0,0,0,0,0,0)
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Choosing an initial value
The corresponding sentence would
be:
In this sentence, the number of
occurences of 0 is 0, of 1 is 0, of 2 is
0, of 3 is 0, of 4 is 0, of 5 is 0, of 6 is
0, of 7 is 0, of 8 is 0, and of 9 is 0.
In this case, the sentence is certainly
false 15
Applying a iteration process
Count up the proper numbers of digits that really occur in the corresponding sentence. This gives the new vector:(11,1,1,1,1,1,1,1,1,1)
Unfortunately, applying this process has again resulted in a false sentence:
In this sentence, the number of occurences of 0 is 11, of 1 is 1, of 2 is 1, of 3 is 1, of 4 is 1, of 5 is 1, of 6 is 1, of 7 is 1, of 8 is 1, and of 9 is 1.
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This is known as iterating the process. Here
are the vectors that result:
(11,1,1,1,1,1,1,1,1,1)
(1,12,1,1,1,1,1,1,1,1)
(1,11,2,1,1,1,1,1,1,1)
(1,11,2,1,1,1,1,1,1,1)
...
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After a few steps, the numbers are no longer
changing. The resulting vector
(1,11,2,1,1,1,1,1,1,1)is called a fixed-point
of the process, since its value does not
change when the process is applied. (We
might also call it a 1-cycle, i.e. a cycle which
repeats every 1 step of the process.)
What's more, the corresponding sentence is
actually true !18
Another point attractor
In fact, the above fixed point is not the only one, there is another (corresponding to the second self-documenting sentence shown above) as can be seen from the following sequence of vectors:
(243000,645,9,2225,234,0,23445987,23434,2,34)
(5,1,9,7,9,4,2,2,2,3)
(1,2,4,2,2,2,1,2,1,3)
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(1,4,6,2,2,1,1,1,1,1)
(1,7,3,1,2,1,2,1,1,1)
(1,7,3,2,1,1,1,2,1,1)
(1,7,3,2,1,1,1,2,1,1)
...
And this one is also true!
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Cycles (periodic orbits)
For some initial vectors, the process does not
lead to a fixed point, but instead gives an
alternating pair of values. For example:
(243,645,9765,2225,2340,300,234,23434,
2,34)
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After a few iteration
(4,1,9,8,8,4,3,2,1,2)
(1,3,3,2,3,1,1,1,3,2)
(1,5,3,5,1,1,1,1,1,1)
(1,8,1,2,1,3,1,1,1,1)
(1,8,2,2,1,1,1,1,2,1)
(1,7,4,1,1,1,1,1,2,1)
(1,8,2,1,2,1,1,2,1,1)
(1,7,4,1,1,1,1,1,2,1)
(1,8,2,1,2,1,1,2,1,1)
...
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This gives you the right
answer
(1,7,4,1,1,1,1,1,2,1)
(1,8,2,1,2,1,1,2,1,1)
In the next sentence, the number of occurences
of 0 is 1, of 1 is 7, of 2 is 4, of 3 is 1, of 4 is 1, of
5 is 1, of 6 is 1, of 7 is 1, of 8 is 2, and of 9 is 1.
In the previous sentence, the number of
occurences of 0 is 1, of 1 is 8, of 2 is 2, of 3 is 1,
of 4 is 2, of 5 is 1, of 6 is 1, of 7 is 2, of 8 is 1,
and of 9 is 1.
This is a periodic-2 attractor23
The philosophy of this game…
As with all matters of the heart, you’ll know
when you find it.
So keep looking. Don’t settle.
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The philosophy of this game…
Notice how an apparently difficult problem
(namely, coming up with correct values to
place into the template) has been solved by
iterating a simple process.
Decompose a big difficult problem into few
simple repeated easy tasks… exactly suitable
for computers.
In this case it worked because the fixed point
was `attracting' other values when the
process was iterated. 25
Applications
In numerical analysis
In economics
In PageRank
In system identification
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In numerical analysis
Use this philosophy to solve equation…
A first simple and useful example is the
Babylonian method for computing the square
root of a
from whatever starting point.
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An example
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In economics
A Nash
equilibrium of a
game is a fixed
point of the
game's best
response
correspondenc
e.
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PageRank
The vector
of PageRank values
of all web pages is
the fixed point of
alinear
transformation deriv
ed from the World
Wide Web's link
structure.
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In system identification
In this system, a is unknown parameter we
must identify from what we can observe
By some parameter identification algorithm,
we start with arbitrary a, and iterate it …then
we can get the true a31
Different initial estimate a0 all approach the
same real a Ref: Charles K. Chui, Guanrong Chen: Kalman Filtering: with Real-Time Applications
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Different kinds of attractors
Point attractor,
Periodic attractor,
Torus Attractor,
Strange attractor
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Review some concepts
Dynamic space
Time
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2D ODE System
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Chaotic transient: Point
attractor1
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Chaotic transient: Point
attractor2
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Chaotic transient: Point
attractor1
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Chaotic transient: Point
attractor1
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E. N. Lorenz, “Deterministic non-periodic flow,” J. Atmos. Sci., 20,
130-141, 1963.
Lorenz System
,
)(
bzxyz
yxzcxy
xyax
28,3/8,10 cba
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Chen System
28;3;35 cba
G. Chen and T. Ueta, “Yet another chaotic attractor,” Int J. of Bifurcation and Chaos, 9(7),
1465-1466, 1999.
T. Ueta and G. Chen, “Bifurcation analysis of Chen’s equation,” Int J. of Bifurcation and
Chaos, 10(8), 1917-1931, 2000.
T. S. Zhou, G. Chen and Y. Tang, “Chen's attractor exists,” Int. J. of Bifurcation and Chaos,
14, 3167-3178, 2004.
,
)(
)(
bzxyz
cyxzxacy
xyax
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Xiong Wang: Summary of Recent Work 4242
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Concluding Remarks
Done:
Lorenz system has been extended to a
one-parameter family
Rossler system is being extended
To be Done:
Are all 3-D autonomous systems with 1
or 2 quadratic terms intrinsically related?
Can they be extended?
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Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email: [email protected]
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