Maple Text Commands for Chapter 7 - 

THREE-DIMENSIONAL AUTONOMOUS SYSTEMS AND CHAOS 

Exercise 1: Finding eigenvalues and eigenvectors. 

>	with(LinearAlgebra):

 

>	A:=Matrix([[1,0,-4],[0,5,4],[-4,4,3]]);

 

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>	Eigenvectors(A);

 

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Figure 7.10(a): The Rossler Attractor  

Note that you can rotate the 3-D figure in Maple! 

>

with(DEtools): a:=0.2:b:=0.2:c:=6.3: Rossler:=diff(x(t),t)=-y(t)-z(t),diff(y(t),t)=x(t)+a*y(t),diff(z(t),t)=b+x(t)*z(t)-c*z(t): DEplot3d({Rossler},{x(t),y(t),z(t)},t=50..200,[[x(0)=1,y(0)=1,z(0)=1]], scene=[x(t),y(t),z(t)],stepsize=0.05,thickness=1,linecolor=blue,font=[TIMES, ROMAN,15],orientation=[40,120]);

 

Figure 7.10(b): Time series plot. Shows sensitivity to initial conditions. 

>

with(plots): p1:=DEplot({Rossler},{x(t),y(t),z(t)},t=50..100,[[x(0)=1,y(0)=1,z(0)=1]], scene=[t,x(t)],stepsize=0.05,thickness=1,linestyle=1,linecolor=black): p2:=DEplot({Rossler},{x(t),y(t),z(t)},t=50..100,[[x(0)=1.01,y(0)=1,z(0)=1]], scene=[t,x(t)],stepsize=0.05,thickness=1,linestyle=1,linecolor=red): display({p1,p2},font=[TIMES,ROMAN,15]);

 

Figure 7.14: Chua's double scroll attractor  

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a:=15.6:b:=25.58:c:=-5/7:d:=-8/7: Chua:=diff(x(t),t)=a*(y(t)-x(t)-(c*x(t)+0.5*(d-c)*(abs(x(t)+1)-abs(x(t)-1)))),diff(y(t),t)=x(t)-y(t)+z(t),diff(z(t),t)=-b*y(t): p1:=DEplot3d({Chua},{x(t),y(t),z(t)},t=0..80,[[x(0)=1.6,y(0)=0,z(0)=-1.6], [x(0)=-1.6,y(0)=0,z(0)=1.6]],scene=[x(t),y(t),z(t)],stepsize=0.05,thickness=1,linecolor=blue): display(p1,font=[TIMES,ROMAN,15]);

 

Exercise 6: The Chapman cycle (used to model the production of ozone), a stiff system of ODEs.  

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M:=9*10^(17): k1:=3*10^(-12):k2:=1.22*10^(-33):k3:=5.5*10^(-4):k4:=6.86*10^(-16): deq1:=diff(x(t),t)=2*k1*y(t)+k3*z(t)-k2*x(t)*y(t)*M-k4*x(t)*z(t), diff(y(t),t)=k3*z(t)+2*k4*x(t)*z(t)-k1*y(t)-k2*x(t)*y(t)*M, diff(z(t),t)=k2*x(t)*y(t)*M-k3*z(t)-k4*x(t)*z(t): ics:=x(0)=4*10^(16),y(0)=2*10^(16),z(0)=2*10^(16): dsol1:=dsolve({deq1,ics},numeric,range=0..10^8,stiff=true): evalf(dsol1(10^8),5);

 

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End of Chapter 7 Commands