Maple Text Commands for Chapter 10 -  

THE SECOND PART OF HILBERT'S 16'TH PROBLEM 

Essentially - determine the maximum number of limit cycles of a polynomial system of degree n: 

The problem is still unresolved after over 100 years! 

Lienard system with large parameter. 

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restart:with(plots):with(DEtools): deq1:=diff(x(t),t)=mu*y(t)-mu*(-x(t)+(x(t))^3): deq2:=diff(y(t),t)=-x(t)/mu: bifdeq1:=(parameter)->subs(mu=parameter,deq1): bifdeq2:=(parameter)->subs(mu=parameter,deq2): LimitCycle:=seq(DEplot({bifdeq1('i/10'),bifdeq2('i/10')},[x(t),y(t)],0..80,[[x(0)=0.4,y(0)=0]],y=-2..2,x=-2..2,arrows=NONE,stepsize=0.1,linecolour=blue),i=1..50): LimitCycle:=subs(THICKNESS(3)=THICKNESS(0),[LimitCycle]): display(LimitCycle,insequence=true);

 

Table 10.1: The maximum number of small-amplitude limit cycles that can be bifurcated from the origin for the system  

See: Christopher C.J and Lynch S., Small-amplitude limit cycles of Lienard equations with either quadratic or cubic damping or restoring coefficients, Nonlinearity, 12(4), 1099-1112 (1999). 

End of Chapter 10 Commands