Example 6.  Solve the Rössler attractor I. V. P.   
    [Graphics:Images/LorenzAttractorMod_gr_300.gif]              with  [Graphics:Images/LorenzAttractorMod_gr_301.gif],    
      [Graphics:Images/LorenzAttractorMod_gr_302.gif]          with  [Graphics:Images/LorenzAttractorMod_gr_303.gif],       
    [Graphics:Images/LorenzAttractorMod_gr_304.gif]      with  [Graphics:Images/LorenzAttractorMod_gr_305.gif].          

Solution 6.

Enter the vector function  [Graphics:../Images/LorenzAttractorMod_gr_306.gif].  

[Graphics:../Images/LorenzAttractorMod_gr_307.gif]
[Graphics:../Images/LorenzAttractorMod_gr_308.gif]



[Graphics:../Images/LorenzAttractorMod_gr_309.gif]

[Graphics:../Images/LorenzAttractorMod_gr_310.gif]
[Graphics:../Images/LorenzAttractorMod_gr_311.gif]
[Graphics:../Images/LorenzAttractorMod_gr_312.gif]

[Graphics:../Images/LorenzAttractorMod_gr_313.gif]

Compute the Runge-Kutta solution. Since this curve is known to twist a lot in 3D space, it will require about 1000 subintervals to get good resolution.

[Graphics:../Images/LorenzAttractorMod_gr_314.gif]

[Graphics:../Images/LorenzAttractorMod_gr_315.gif]

[Graphics:../Images/LorenzAttractorMod_gr_316.gif]
[Graphics:../Images/LorenzAttractorMod_gr_317.gif]
[Graphics:../Images/LorenzAttractorMod_gr_318.gif]
[Graphics:../Images/LorenzAttractorMod_gr_319.gif]

[Graphics:../Images/LorenzAttractorMod_gr_320.gif]
[Graphics:../Images/LorenzAttractorMod_gr_321.gif]

We can plot the projections on the coordinate planes.

[Graphics:../Images/LorenzAttractorMod_gr_322.gif]

[Graphics:../Images/LorenzAttractorMod_gr_323.gif]

[Graphics:../Images/LorenzAttractorMod_gr_324.gif]
[Graphics:../Images/LorenzAttractorMod_gr_325.gif]


[Graphics:../Images/LorenzAttractorMod_gr_326.gif]

[Graphics:../Images/LorenzAttractorMod_gr_327.gif]

[Graphics:../Images/LorenzAttractorMod_gr_328.gif]
[Graphics:../Images/LorenzAttractorMod_gr_329.gif]


[Graphics:../Images/LorenzAttractorMod_gr_330.gif]

[Graphics:../Images/LorenzAttractorMod_gr_331.gif]

[Graphics:../Images/LorenzAttractorMod_gr_332.gif]
[Graphics:../Images/LorenzAttractorMod_gr_333.gif]


[Graphics:../Images/LorenzAttractorMod_gr_334.gif]

[Graphics:../Images/LorenzAttractorMod_gr_335.gif]

[Graphics:../Images/LorenzAttractorMod_gr_336.gif]


[Graphics:../Images/LorenzAttractorMod_gr_337.gif]

[Graphics:../Images/LorenzAttractorMod_gr_338.gif]

[Graphics:../Images/LorenzAttractorMod_gr_339.gif]


[Graphics:../Images/LorenzAttractorMod_gr_340.gif]

[Graphics:../Images/LorenzAttractorMod_gr_341.gif]

[Graphics:../Images/LorenzAttractorMod_gr_342.gif]

We can plot the coordinate functions.

[Graphics:../Images/LorenzAttractorMod_gr_343.gif]

[Graphics:../Images/LorenzAttractorMod_gr_344.gif]

[Graphics:../Images/LorenzAttractorMod_gr_345.gif]


[Graphics:../Images/LorenzAttractorMod_gr_346.gif]

[Graphics:../Images/LorenzAttractorMod_gr_347.gif]

[Graphics:../Images/LorenzAttractorMod_gr_348.gif]


[Graphics:../Images/LorenzAttractorMod_gr_349.gif]

[Graphics:../Images/LorenzAttractorMod_gr_350.gif]

[Graphics:../Images/LorenzAttractorMod_gr_351.gif]


[Graphics:../Images/LorenzAttractorMod_gr_352.gif]

[Graphics:../Images/LorenzAttractorMod_gr_353.gif]

[Graphics:../Images/LorenzAttractorMod_gr_354.gif]
[Graphics:../Images/LorenzAttractorMod_gr_355.gif]
[Graphics:../Images/LorenzAttractorMod_gr_356.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004