This page contains a maple code for Chapter 3 (Second-order differential equations: Steady-state solutions and their stability).
#liapunov2.txt #26.05.09 # # investigate stability using theorem 3.3 of the lecture notes # To use theorem 3.3 we need the function V(x,y). # to have the property that V(x,y)>0 for at least one point (x1,y1) in # any neighbourhood of (0,0). # Typical functions to try are V=xy and V=x^2-y^2 # Define the candidate Lyapunov function. # You should make sure it has the required properties #V := x*y; # V := x^2 -4*x*y; V := a*x^4-b*y*x; #Define the differential equations xdot := -4*y +x^2; ydot := 4*x +y^2; # Determine Vdot Vx := diff(V,x): Vy := diff(V,y): Vdot := Vx*xdot +Vy*ydot; # If we can show that Vdot>0 then we are finished! # If we need to use the following piece of code then we are not going # to be able to prove global stability, only asymptotic stability or # unstability. # To use theorem 3.3 we need the point (0,0) to be a local minimum # of the function Vdot(x,y). Find out if this is satisfied, or if it # imposes some constraints on the parameters in the function Vdotx := diff(Vdot,x): Vdoty := diff(Vdot,y): VdotA := subs({x=0,y=0},diff(Vdotx,x)); VdotB := subs({x=0,y=0},diff(Vdotx,y)); VdotC := subs({x=0,y=0},diff(Vdoty,y)); disc := VdotA*VdotC-VdotB^2; print("Investigating Vdot(0,0), We always require"); print("inequality three:",disc>0); # Print the requirements for (0,0) to be a local minimum # This means that Vdot > 0 and (0,0) is unstable. print("For a local minimum we require"); print("inequality four-a",VdotA>0);