Mystery and Beauty of Attractor

Xiong Wang 王雄

Centre for Chaos and Complex Networks

City University of Hong Kong

1

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

2

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.

3

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.

29

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

36

Chaotic transient: Point

attractor2

37

Chaotic transient: Point

attractor1

38

Chaotic transient: Point

attractor1

39

40

E. N. Lorenz, “Deterministic non-periodic flow,” J. Atmos. Sci., 20,

130-141, 1963.

Lorenz System

,

)(

bzxyz

yxzcxy

xyax

28,3/8,10 cba

40

41

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

41

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: wangxiong8686@gmail.com

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