lorenz3D

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Uri_dolphin3 Uri Wilensky (Author)

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Model was written in NetLogo 3D Preview 4 • Viewed 162 times • Downloaded 21 times • Run 0 times
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WHAT IS IT?

The Lorenz system is a well known example of a simple system showing chaos. Its dynamics depend on many parameters. Here named a b c mu gamma and deltat and the initial conditions x0 y0 z0.

HOW IT WORKS

We calculate the values of k points x, y, and z (k = 2000) iterating the 3 main equations:

x(n)= x(n - 1) + gamma * y(n - 1) - x(n - 1) * deltat

y(n)= y(n - 1) + ( - (x(n - 1) * z(n - 1)) + mu * x(n - 1) - (y(n - 1)) * deltat)

z(n)= z(n - 1) + ((x(n - 1) * y(n - 1) - b * z(n - 1) * deltat

Initial points are x(0)=x0, y(0)=y0 and z(0)=z0 This give the trajectory.

HOW TO USE IT

deltat slider: sets deltat between 0.0010 and 0.015

SETUP: Clears all the parameter and start again the the 3D graphic window.

GO: Build the trajectory for the choosen values.

BallColor slider: sets foundamental color of the points.

BallSize slider: sets the size of points between 0.03 to 2

SameColor switch: "On" gradually changes the color of points otherways the color is fixed.

Screen slider: virtually sets the edge of 3D.

THINGS TO TRY

Identify the attractor for different values of "deltat".

Try to lok at different representation in the XY, XZ, YZ and Zt (where t is time) spaces. Especially the last one, if compared with the 3D graph, can be interesting for the novice.

CREDITS AND REFERENCES

Program by Massimo Salzano, www.ecople.org, http://www.dise.unisa.it/docenti/salzano.htm

14 May 2005.

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Click to Run Model

globals [a b c mu gamma x0 y0 z0 k n x y z SC]
breed [ tails ]
breed [ balls ]
breed [ valu ]


balls-own [ z-vel x-vel y-vel ]
turtles-own [ col ]

to setup
  ca

  create-balls 1
  [
    set shape "circle"
    set color 45 set size BallSize
  ]
  setxyz -10 -17 20
  face patch 0 0 0

   set n 1 set a .28 set b 2.11 set c 4.07 set k 2000
   set mu 27.56 set gamma 9
end 

to go
  let speed 0
  ask balls [set x-vel [] set y-vel [] set z-vel[]
  set x0 1 set y0 1 set z0 1
  set x-vel lput x0 x-vel set y-vel lput y0 y-vel set z-vel lput z0 z-vel
  set shape "circle"
  ]

  while [n < k][ask balls[
    set x-vel lput ((item (n - 1) x-vel) + gamma * ((item (n - 1) y-vel) - (item (n - 1) x-vel)) * deltat ) x-vel
    set y-vel lput (((item (n - 1) y-vel)) + ((( - (item (n - 1) x-vel)) * (item (n - 1) z-vel)) + mu * (item (n - 1) x-vel) - (item (n - 1) y-vel)) * deltat) y-vel
    set z-vel lput ((item (n - 1) z-vel) + ((item (n - 1) x-vel) * (item (n - 1) y-vel) - b * (item (n - 1) z-vel)) * deltat) z-vel
    set xcor ((item (n - 1) x-vel) / (screen / 2)) set ycor ((item (n - 1) y-vel) / (screen / 2)) set zcor ((item (n - 1) z-vel) / (screen / 2)) - (screen * 1.5)
    set x item (n - 1) x-vel set y item (n - 1) y-vel set z item (n - 1) z-vel
    if (Same_Color = false) [set SC (n / 150)]
    set color (BallColor + SC)
    plot1 plot2 plot3 plotz
    set speed sqrt( item (n - 1) x-vel * item (n - 1) x-vel + item (n - 1) y-vel * item (n - 1) y-vel + item (n - 1) z-vel * item (n - 1) z-vel )
    if tails? and speed > 0.03
    [
      hatch-tails 2
      [ set col color ]
    ]
    ]set n n + 1 ]
end 

to plot1
   set-current-plot "PlotXY"
   plotxy x y
end 

to plot2
   set-current-plot "PlotXZ"
   plotxy x z
end 

to plot3
   set-current-plot "PlotYZ"
   plotxy y z
end 

to plotz
   set-current-plot "Plot Z"
   plotxy n - 1 z
end 


; Modification of an idea of 2005 Uri Wilensky.  This code may be freely copied, distributed,
; altered, or otherwise used by anyone for any legal purpose.
;
; THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
; "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
; LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
; A PARTICULAR PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL THE COPYRIGHT
; OWNERS OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
; SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
; LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
; DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
; THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
; (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
; OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

There are 2 versions of this model.

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Uri Wilensky over 12 years ago Updated from NetLogo 4.1 Download this version
Uri Wilensky over 12 years ago lorenz3D Download this version

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