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2500 double pendulums. Varying initial angles: domains of chaos and order.

Vitaliy Kaurov

Vitaliy Kaurov, WOLFRAM Research

Posted 3 months ago

2500 double pendulums. Varying initial angles: domains of chaos and order.

Double Pendulum is the simplest mechanical system without friction and external forcing that demonstrates chaos . Integrable systems (none-chaotic) require the number of independent conserved quantities to match the number of degrees of freedom. The simple pendulum meets this criterion, whereas the double pendulum does not, resulting in its non-integrable and chaotic nature. Double pendulum has 2 degrees of freedom (2 angles) and just 1 conserved quantity (total mechanical energy).

To observe chaos double pendulum we can build different visualizations. But there is one I highly enjoy -- it is many double pendulums for various initial conditions by Jonathan Nafziger . I want to recreate it with Wolfram Language.

In Wolfram ecosystem we have quite a few examples of double pendulum:

◼

Wolfram Function Repository

◼

Documentation

◼

Wolfram Demonstrations

Start by defining a function to depict a single double pendulum with simple two lines:

I n [  ] : =

C l e a r A l l [ d o u B p e n D ] d o u B p e n D [ a n g l e 1 _ , a n g l e 2 _ , r a d i u s 1 _ , r a d i u s 2 _ , { x 0 _ , y 0 _ } ] : = M o d u l e [ { x 1 , y 1 , x 2 , y 2 } , x 1 = x 0 + r a d i u s 1 S i n [ a n g l e 1 ] ; y 1 = y 0 - r a d i u s 1 C o s [ a n g l e 1 ] ; x 2 = x 1 + r a d i u s 2 S i n [ a n g l e 2 ] ; y 2 = y 1 - r a d i u s 2 C o s [ a n g l e 2 ] ; G r a p h i c s [ { O p a c i t y [ . 9 ] , L i n e [ { { x 0 , y 0 } , { x 1 , y 1 } } ] , L i n e [ { { x 1 , y 1 } , { x 2 , y 2 } } ] } ] ] d o u B p e n D [ P i / 4 , - P i / 4 , 1 , 1 , { 0 , 0 } ]

O u t [  ] =

The next function places 2500 double pendulums on 50x50 grid where each spot corresponds to a different initial angles values:

I n [  ] : =

C l e a r A l l [ v i z ] v i z [ t b l _ ] : = C o l u m n [ { S t y l e [ " 2 , 5 0 0 D o u b l e P e n d u l u m s " , 2 5 , O p a c i t y [ . 8 ] , D a r k e r @ R e d , F o n t F a m i l y - > " C o p p e r p l a t e " ] , S t y l e [ " V a r y i n g I n i t i a l A n g l e s : D o m a i n s o f C h a o s a n d O r d e r " , 1 7 , O p a c i t y [ . 7 ] , D a r k e r @ R e d , F o n t F a m i l y - > " G e o r g i a " ] , S h o w [ t b l , P l o t R a n g e - > P i , P l o t R a n g e P a d d i n g - > . 3 , F r a m e - > T r u e , B a s e S t y l e - > 1 5 , I m a g e S i z e - > 5 4 0 , F r a m e T i c k s - > { R a n g e [ - P i , P i , P i ] , R a n g e [ - P i , P i , P i ] } , E p i l o g - > { I n s e t [ S t y l e [ " l o w e r a n g l e " , I t a l i c ] , S c a l e d [ { . 7 , - . 0 2 } ] ] , I n s e t [ R o t a t e [ S t y l e [ " u p p e r a n g l e " , I t a l i c ] , P i / 2 ] , S c a l e d [ { - . 0 2 , . 7 } ] ] } ] } , A l i g n m e n t - > C e n t e r ] v i z @ F l a t t e n [ T a b l e [ d o u B p e n D [ a 1 , a 2 , . 1 , . 1 , { a 2 , a 1 } ] , { a 1 , - P i , P i , P i / 2 5 } , { a 2 , - P i , P i , P i / 2 5 } ] ]

O u t [  ] =

2 , 5 0 0 D o u b l e P e n d u l u m s

V a r y i n g I n i t i a l A n g l e s : D o m a i n s o f C h a o s a n d O r d e r

A single function from Wolfram Function Repository solves numerically the double pendulum:

I n [  ] : =

R e s o u r c e F u n c t i o n [ " D o u b l e P e n d u l u m F o r m u l a " ]

O u t [  ] =

[ ◼ ]

D o u b l e P e n d u l u m F o r m u l a

Using it we compute evolution of all 2500 double pendulums for 10 seconds:

I n [  ] : =

n = 5 0 ; e v o = P a r a l l e l T a b l e 

[ ◼ ]

D o u b l e P e n d u l u m F o r m u l a

[ A s s o c i a t i o n [ " l 1 " - > 1 , " l 2 " - > 1 , " m 1 " - > 1 , " m 2 " - > 1 ] , A s s o c i a t i o n [ " t h e t a 1 i " - > a 1 , " t h e t a 2 i " - > a 2 , " t " - > 1 0 ] ] , { a 1 , - P i , P i , 2 P i / ( n - 1 ) } , { a 2 , - P i , P i , 2 P i / ( n - 1 ) }  ; / / A b s o l u t e T i m i n g

O u t [  ] =

{ 6 . 2 0 1 3 4 , N u l l }

I n [  ] : =

e v o / / D i m e n s i o n s

O u t [  ] =

{ 5 0 , 5 0 }

Here is angles evolution of a sample single double pendulum from our grid:

I n [  ] : =

P l o t [ { e v o [ [ 2 3 , 1 3 ] ] [ " A n g l e 1 S o l u t i o n " ] [ t ] , e v o [ [ 2 3 , 1 3 ] ] [ " A n g l e 2 S o l u t i o n " ] [ t ] } , { t , 0 , 1 0 } ]

O u t [  ] =

And this is how this double pendulum looks at t=4:

I n [  ] : =

d o u B p e n D [ e v o [ [ 2 3 , 1 3 ] ] [ " A n g l e 1 S o l u t i o n " ] [ 4 ] , e v o [ [ 2 3 , 1 3 ] ] [ " A n g l e 2 S o l u t i o n " ] [ 4 ] , 1 , 1 , { 0 , 0 } ]

O u t [  ] =

Compute the range of 50 angles along single angle axis to use as initial conditions:

I n [  ] : =

r g = R a n g e [ - P i , P i , 2 P i / ( n - 1 ) ]

O u t [  ] =

 - π , -

4 7 π

4 9

, -

4 5 π

4 9

, -

4 3 π

4 9

, -

4 1 π

4 9

, -

3 9 π

4 9

, -

3 7 π

4 9

, -

5 π

, -

3 3 π

4 9

, -

3 1 π

4 9

, -

2 9 π

4 9

, -

2 7 π

4 9

, -

2 5 π

4 9

, -

2 3 π

4 9

, -

3 π

, -

1 9 π

4 9

, -

1 7 π

4 9

, -

1 5 π

4 9

, -

1 3 π

4 9

, -

1 1 π

4 9

, -

9 π

4 9

, -

5 π

4 9

, -

3 π

4 9

, -

4 9

3 π

4 9

5 π

4 9

9 π

4 9

1 1 π

4 9

1 3 π

4 9

1 5 π

4 9

1 7 π

4 9

1 9 π

4 9

3 π

2 3 π

4 9

2 5 π

4 9

2 7 π

4 9

2 9 π

4 9

3 1 π

4 9

3 3 π

4 9

5 π

3 7 π

4 9

3 9 π

4 9

4 1 π

4 9

4 3 π

4 9

4 5 π

4 9

4 7 π

4 9

, π 

Manipulate will show full 10 seconds evolution of all 2500 double pendulums:

I n [  ] : =

M a n i p u l a t e [ v i z [ F l a t t e n [ T a b l e [ d o u B p e n D [ e v o [ [ x , y ] ] [ " A n g l e 2 S o l u t i o n " ] [ t ] , e v o [ [ x , y ] ] [ " A n g l e 1 S o l u t i o n " ] [ t ] , . 1 , . 1 , { r g [ [ x ] ] , r g [ [ y ] ] } ] , { x , 1 , n } , { y , 1 , n } ] ] ] , { t , 0 , 1 0 } ]

O u t [  ] =

Compute 300 rasterized frames:

I n [  ] : =

f r a m e s = T a b l e [ R a s t e r i z e [ v i z [ F l a t t e n [ T a b l e [ d o u B p e n D [ e v o [ [ x , y ] ] [ " A n g l e 2 S o l u t i o n " ] [ t ] , e v o [ [ x , y ] ] [ " A n g l e 1 S o l u t i o n " ] [ t ] , . 1 , . 1 , { r g [ [ x ] ] , r g [ [ y ] ] } ] , { x , 1 , n } , { y , 1 , n } ] ] ] , R a s t e r S i z e - > { A u t o m a t i c , 1 0 8 0 } ] , { t , 0 , 1 0 , 1 / 3 0 } ] ; / / A b s o l u t e T i m i n g

O u t [  ] =

{ 1 6 5 . 4 7 5 , N u l l }

And make a video:

I n [  ] : =

F r a m e L i s t V i d e o [ f r a m e s , R a s t e r S i z e - > { A u t o m a t i c , 1 0 8 0 } , F r a m e R a t e - > 1 5 ]

CITE THIS NOTEBOOK

2500 double pendulums. Varying initial angles: domains of chaos and order.
by Vitaliy Kaurov
Wolfram Community, STAFF PICKS, November 26, 2024
https://community.wolfram.com/groups/-/m/t/3328657

POSTED BY: Vitaliy Kaurov

3 Replies

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Johannes Martin

Johannes Martin, Kopernikus Gymnasium, Germany (high school)

Posted 3 months ago

There are a few surprises even in the chaotic areas. There is a nice video on this: https://youtu.be/OIHyN7TzY6A?si=TbftDs6P_HdYfVea It can be nicely re-created with your code. Thanks a lot. t=17 t=100 So even after 100 s, there is a small patch in the highly non-linear regime, where the pendulum can be predicted very accurately.

POSTED BY: Johannes Martin

Vitaliy Kaurov

Vitaliy Kaurov, WOLFRAM Research

Posted 3 months ago

This is spectacular, @Johannes ! Thank you for sharing.

POSTED BY: Vitaliy Kaurov

EDITORIAL BOARD

EDITORIAL BOARD, WOLFRAM

Posted 3 months ago

-- you have earned *Featured Contributor Badge* Your exceptional post has been selected for our editorial column *Staff Picks* http://wolfr.am/StaffPicks and Your Profile is now distinguished by a *Featured Contributor Badge* and is displayed on the Featured Contributor Board. Thank you!

POSTED BY: EDITORIAL BOARD

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