6.By introducing suitable dimensionless variables, we can write the system of nonlinear equations for the damped pendulum (Equations (8) of Section 9.3) as dxdt=y,dydt=−y−sin x.dxdt=y,dydt=−y−sin x. a.Show that the origin is a critical point. b.Show that although V(x, y) = x2 + y2 is positive definite, ⋅V(x,y)V⋅x,y takes on both positive and negative values in any domain containing the origin, so that V is not a Liapunov function. Hint: x − sin x > 0 for x > 0, and x − sin x < 0 for x < 0. Consider these cases with y positive but y so small that y2 can be ignored compared to y. c.Using the energy function V(x,y)=12y2+(1−cos x)Vx,y=12y2+1−cos x mentioned in Problem 5b, show that the origin is a stable critical point. Since there is damping in the system, we can expect that the origin is asymptotically stable. However, it is not possible to draw this conclusion using this Liapunov function. d.To show asymptotic stability, it is necessary to construct a better Liapunov function than the one used in part c. Show that V(x,y)=12(x+y)2+x2+12y2Vx,y=12x+y2+x2+12y2 is such a Liapunov function, and conclude that the origin is an asymptotically stable critical point. Hint: From Taylor's formula with a remainder, it follows that sin x = x − αx3/3!, where α depends on x but 0 < α < 1 for −π/2 < x < π/2. Then, letting x = r cos θ, y = r sin θ, show that ⋅V(r cos θ,r sin θ)=−r2(1+h(r,θ))V⋅r cos θ,r sin θ=−r21+hr,θ, where |h(r, θ)| < 1 if r is sufficiently small.
6.By introducing suitable dimensionless variables, we can write the system of nonlinear equations for the damped pendulum (Equations (8) of Section 9.3) as
a.Show that the origin is a critical point.
b.Show that although V(x, y) = x2 + y2 is positive definite, ⋅V(x,y)V⋅x,y takes on both positive and negative values in any domain containing the origin, so that V is not a Liapunov function. Hint: x − sin x > 0 for x > 0, and x − sin x < 0 for x < 0. Consider these cases with y positive but y so small that y2 can be ignored compared to y.
c.Using the energy function V(x,y)=12y2+(1−cos x)Vx,y=12y2+1−cos x mentioned in Problem 5b, show that the origin is a stable critical point. Since there is damping in the system, we can expect that the origin is asymptotically stable. However, it is not possible to draw this conclusion using this Liapunov function.
d.To show asymptotic stability, it is necessary to construct a better Liapunov function than the one used in part c. Show that V(x,y)=12(x+y)2+x2+12y2Vx,y=12x+y2+x2+12y2 is such a Liapunov function, and conclude that the origin is an asymptotically stable critical point. Hint: From Taylor's formula with a remainder, it follows that sin x = x − αx3/3!, where α depends on x but 0 < α < 1 for −π/2 < x < π/2. Then, letting x = r cos θ, y = r sin θ, show that ⋅V(r cos θ,r sin θ)=−r2(1+h(r,θ))V⋅r cos θ,r sin θ=−r21+hr,θ, where |h(r, θ)| < 1 if r is sufficiently small.
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