Example 5.  Solve the Rössler attractor I. V. P.   
        [Graphics:Images/LorenzAttractorMod_gr_243.gif]          with  [Graphics:Images/LorenzAttractorMod_gr_244.gif],    
          [Graphics:Images/LorenzAttractorMod_gr_245.gif]          with  [Graphics:Images/LorenzAttractorMod_gr_246.gif],       
    
    [Graphics:Images/LorenzAttractorMod_gr_247.gif]      with  [Graphics:Images/LorenzAttractorMod_gr_248.gif].   

Solution 5.

Enter the vector function  [Graphics:../Images/LorenzAttractorMod_gr_249.gif].  

[Graphics:../Images/LorenzAttractorMod_gr_250.gif]
[Graphics:../Images/LorenzAttractorMod_gr_251.gif]



[Graphics:../Images/LorenzAttractorMod_gr_252.gif]

[Graphics:../Images/LorenzAttractorMod_gr_253.gif]
[Graphics:../Images/LorenzAttractorMod_gr_254.gif]
[Graphics:../Images/LorenzAttractorMod_gr_255.gif]

[Graphics:../Images/LorenzAttractorMod_gr_256.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_257.gif]

[Graphics:../Images/LorenzAttractorMod_gr_258.gif]

[Graphics:../Images/LorenzAttractorMod_gr_259.gif]
[Graphics:../Images/LorenzAttractorMod_gr_260.gif]
[Graphics:../Images/LorenzAttractorMod_gr_261.gif]
[Graphics:../Images/LorenzAttractorMod_gr_262.gif]

[Graphics:../Images/LorenzAttractorMod_gr_263.gif]
[Graphics:../Images/LorenzAttractorMod_gr_264.gif]

We can plot the projections on the coordinate planes.

[Graphics:../Images/LorenzAttractorMod_gr_265.gif]

[Graphics:../Images/LorenzAttractorMod_gr_266.gif]

[Graphics:../Images/LorenzAttractorMod_gr_267.gif]
[Graphics:../Images/LorenzAttractorMod_gr_268.gif]


[Graphics:../Images/LorenzAttractorMod_gr_269.gif]

[Graphics:../Images/LorenzAttractorMod_gr_270.gif]

[Graphics:../Images/LorenzAttractorMod_gr_271.gif]
[Graphics:../Images/LorenzAttractorMod_gr_272.gif]


[Graphics:../Images/LorenzAttractorMod_gr_273.gif]

[Graphics:../Images/LorenzAttractorMod_gr_274.gif]

[Graphics:../Images/LorenzAttractorMod_gr_275.gif]
[Graphics:../Images/LorenzAttractorMod_gr_276.gif]


[Graphics:../Images/LorenzAttractorMod_gr_277.gif]

[Graphics:../Images/LorenzAttractorMod_gr_278.gif]

[Graphics:../Images/LorenzAttractorMod_gr_279.gif]


[Graphics:../Images/LorenzAttractorMod_gr_280.gif]

[Graphics:../Images/LorenzAttractorMod_gr_281.gif]

[Graphics:../Images/LorenzAttractorMod_gr_282.gif]


[Graphics:../Images/LorenzAttractorMod_gr_283.gif]

[Graphics:../Images/LorenzAttractorMod_gr_284.gif]

[Graphics:../Images/LorenzAttractorMod_gr_285.gif]

We can plot the coordinate functions.

[Graphics:../Images/LorenzAttractorMod_gr_286.gif]

[Graphics:../Images/LorenzAttractorMod_gr_287.gif]

[Graphics:../Images/LorenzAttractorMod_gr_288.gif]


[Graphics:../Images/LorenzAttractorMod_gr_289.gif]

[Graphics:../Images/LorenzAttractorMod_gr_290.gif]

[Graphics:../Images/LorenzAttractorMod_gr_291.gif]


[Graphics:../Images/LorenzAttractorMod_gr_292.gif]

[Graphics:../Images/LorenzAttractorMod_gr_293.gif]

[Graphics:../Images/LorenzAttractorMod_gr_294.gif]


[Graphics:../Images/LorenzAttractorMod_gr_295.gif]

[Graphics:../Images/LorenzAttractorMod_gr_296.gif]

[Graphics:../Images/LorenzAttractorMod_gr_297.gif]
[Graphics:../Images/LorenzAttractorMod_gr_298.gif]
[Graphics:../Images/LorenzAttractorMod_gr_299.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004