A Few Attractors...

Following with the Mathematical trend to my threads... here is another extremely interesting branch of Mathematics.

Relating to my other Chaos Theory thread, here are a few attractors that you can create in ACAD.

The attractors are created by solving various Non-Linear Differential Equations numerically, (using Euler's method in this case).

I have modelled three attractors for you to experiment with:

The Lorenz Attractor:

This stems from a simplification of the equations used to model Weather systems, and is the general attractor that most people think of when they think about Chaos Theory... as Edward Lorenz was the pioneer of Chaos theory.

The Rössler Attractor:

The Duffing's Attractor:

This attractor is obtained when numerically solving the non-linear ordinary differential equation that is Duffing's Equation.

The equation models a damped oscillator, such as a weighted non-uniform spring. To solve the equation numerically, I have reduced the second order differential equation to two first order equations, and then used Euler's method to generate the attractor.

Various Attractors can be generated by varying the input parameters (a,b and c in the code).

(defun c:lorenz (/ iLim i h a b c x0 y0 z0 x y z)

 (setq iLim 10000 i -1 h 0.01 a 10. b 28. c (/ 8. 3.) x0 0.1 y0 0. z0 0.)

 (entmake '((0 . "POLYLINE") (70 . ))
 (while (< (setq i (1+ i)) iLim)

   (setq x (+ x0 (* h a (- y0 x0)))
         y (+ y0 (* h (- (* x0 (- b z0)) y0)))
         z (+ z0 (* h (- (* x0 y0) (* c z0)))) x0 x y0 y z0 z)
   (entmake (list '(0 . "VERTEX") '(70 . 32) (cons 10 (list x y z)))))
 
 (entmake '((0 . "SEQEND")))
 (princ))



(defun c:rossler (/ iLim i h a b c x0 y0 z0 x y z)

 (setq iLim 10000 i -1 h 0.01 a 0.2 b 0.2 c 5.7 x0 0.1 y0 0. z0 0.)

 (entmake '((0 . "POLYLINE") (70 . ))
 (while (< (setq i (1+ i)) iLim)

   (setq x (+ x0 (* h (- (- y0) z0)))
         y (+ y0 (* h (+ x0 (* a y0))))
         z (+ z0 (* h (+ b (* z0 (- x0 c))))) x0 x y0 y z0 z)
   (entmake (list '(0 . "VERTEX") '(70 . 32) (cons 10 (list x y z)))))

 (entmake '((0 . "SEQEND")))
 (princ))


(defun c:duffings (/ iLim i h a b x0 y0 z0 x y z)

 (setq iLim 10000 i -1 h 0.04 a 0.2 b 0.3 x0 0. y0 0. z0 0.)

 (entmake '((0 . "POLYLINE") (70 . ))
 (while (< (setq i (1+ i)) iLim)

   (setq x (+ x0 (* h y0))
         y (+ y0 (* h (+ (- x0 (* x0 x0 x0) (* a y0)) (* b (cos z0)))))
         z (+ z0 h) x0 x y0 y z0 z)    
   (entmake (list '(0 . "VERTEX") '(70 . 32) (cons 10 (list x y z)))))
 
 (entmake '((0 . "SEQEND")))
 (princ))

I hope you enjoy this little exploration into this fascinating area of mathematics, and, of course, if you have any questions - please ask.

Lee

Amazing....

Cool ! oh maybe it was the 1/2 empty bottle of scotch

Incredible...How...?...What..?..Why..?

Good intellectual exercise Lee

Thanks all

This thread is now quite old - a better write-up may be found on my site here.

This is just one of a few Mathematical Endeavours:

Attractors

Fractals

Iterated Function Systems

Koch Snowflake

Logistic Map

Sierpinski Triangle

Enjoy!

Lee

Lee, you should also add the Fibonacci Sequence in there.

Lee, you should also add the Fibonacci Sequence in there.

Thanks for the suggestion Tannar - I'll see what I can do!

Going with the theme of this thread, I was recently inspired to write the following new addition to my site:

Sierpinski Triangle

The 3D one is spectacular!! Man I love seeing what math can do.

The 3D one is spectacular!! Man I love seeing what math can do.

Thanks Tannar! As you can tell, I'm also fascinated by fractal geometry

These things are great!

These things are great!

Thanks, I'm glad you like them!

Id love to eventually get to the point where i would be able to create things like these.

Practice, practice, practice...