Chap6.mw

Maple Text Commands for Chapter 6 - Bifurcation Theory 

Example 2: Taylor series expansion. 

> series(1/(1+2*u),u=0,4);
 

series(1-2*u+4*u^2-8*u^3+O(u^4),u,4) 

> with(plots):
 

Warning, the name changecoords has been redefined 

Exercise 8(b): Animation of a simple curve. 

Click on the graph and then the play button in the toolbar. 

> animate([r,r*((r-1)^2-mu*r),r=-0.5..2],mu=-0.5..0.5,numpoints=100,frames=100,color=red);
 

Plot 

Exercise 9: The system dx/dt = y+`μx`-xy^2, dy/dt = `μy`-x-y^3undergoes a Hopf bifurcation at mu[0] = 0. 

Think of the origin blowing a smoke ring! 

> with(DEtools):with(plots):
deq1:=diff(x(t),t)=y(t)+mu*x(t)-x(t)*(y(t))^2:
deq2:=diff(y(t),t)=mu*y(t)-x(t)-(y(t))^3:
bifdeq1:=(parameter)->subs(mu=parameter,deq1):
bifdeq2:=(parameter)->subs(mu=parameter,deq2):
Hopf:=seq(DEplot({bifdeq1('i/40-1'),bifdeq2('i/40-1')},[x(t),y(t)],0..80,[[x(0)=0.5,y(0)=0.5]],y=-1..1,x=-1..1,arrows=NONE,stepsize=0.1,linecolour=blue),i=0..48):
Hopf:=subs(THICKNESS(3)=THICKNESS(0),[Hopf]):
display(Hopf,insequence=true);
 

Plot 

The parameter mu varies from -1to 0.2 in the above animation. 

End of Chapter 6 Commands