Maple Text Commands for Chapter 4 - LIMIT CYCLES 

>	with(DEtools):with(plots):

 

Fitzhugh-Nagumo Oscillator: A model of the action potential of a neuron  

>	theta:=0.14:w:=0.112:epsilon:=0.01:gama:=2.54: sys1:=diff(u(t),t)=-u(t)*(u(t)-theta)*(u(t)-1)-v(t)+w,diff(v(t),t)=epsilon*(u(t)-gama*v(t)): iniset:=seq(seq([0,i,j],i=-2..2),j=-2..2):

 

Figure 4.2: Periodic behavior representing the integrate and fire neuron. 

>	DEplot([sys1],[u(t),v(t)],t=50..150,[[0,0.5,0.1]],stepsize=0.01,u=-0.4..1.2,v=0..0.4,color=black,linecolor=blue,thickness=2);

 

Example 4: A phase portrait of a Lienard system  

>	a2:=90:a4:=-882:a6:=2598.4:a8:=-3359.997:a10:=2133.34:a12:=-651.638:a14:=76.38:epsilon:=1:a0:=0: sys2:=diff(x(t),t)=y(t),diff(y(t),t)=-x(t)-epsilon*y(t)*(a14*x(t)^14+a12*x(t)^12+a10*x(t)^10+a8*x(t)^8+a6*x(t)^6+a4*x(t)^4+a2*x(t)^2-a0):

 

Figure 4.8: The limit cycle is a nonconvex closed curve. 

>	DEplot([sys2],[x(t),y(t)],t=30..60,[[x(0)=1.5,y(0)=0]],x=-2..2,y=-3..3,stepsize=0.05,linecolor=blue,thickness=2);

 

Figure 4.12: Trajectory for a van der Pol system. (Refer to the method of multiple scales in perturbation theory). 

>

with(DEtools):with(plots):epsilon:=0.05: sys3:=diff(x(t),t)=y(t),diff(y(t),t)=-x(t)-epsilon*y(t)*((x(t))^2-1): p1:=DEplot([sys3],[x(t),y(t)],t=0..500,[[x(0)=0.1,y(0)=0]],x=-3..3,y=-3..3,stepsize=0.05,linecolor=blue,thickness=1): t1:=textplot([0.2,3,"dx/dt"],align={above,right}): t2:=textplot([2.8,0.2,"x"],align={above,right}): display({p1,t1,t2},font=[TIMES,ROMAN,20]);

 

End of Chapter 4 Commands