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(a) Letting x = u, y = du/dt, write Eq. (i) as a system of two equations, and show that x = 0, y = 0 is a critical point.
(b) Show that
V(x, y) = 1 y2 + ? g(s) ds, k < x < k (ii)
is positive definite, and use this result to show that the critical point (0, 0) is stable. Note that the Liapunov function V given by Eq. (ii) corresponds to the energy function V(x, y) = 2y2 + (1 cos x) for the case g(u) = sin u.
7. By introducing suitable dimensionless variables, the system of nonlinear equations for the damped pendulum [Eqs. (8) of Section 9.3] can be written as
dx/dt = y, dy/dt = y sin x.
(a) Show that the origin is a critical point.
(b) Show that while V(x, y) = x2 + y2 is positive definite, 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.
In each of Problems 1 through 4 construct a suitable Liapunov function of the form ax2 + cy2, where a and c are to be determined. Then show that the critical point at the origin is of the indicated type.
1. dx/dt =x3 + xy2, dy/dt = 2x2y y3; asymptotically stable
2. dx/dt = Ix3 + 2xy2, dy/dt = y3; asymptotically stable
3. dx/dt = x3 + 2y3, dy/dt = 2xy2; stable (at least)
4. dx/dt = x3 y3, dy/dt = 2xy2 + 4x2 y + 2y3; unstable
5. Consider the system of equations
Chapter 9. Nonlinear Differential Equations and Stability
(c) Using the energy function V(x, y) = 2y2 + (1 cosx) mentioned in Problem 6(b), show that the origin is a stable critical point. Note, however, that even though there is damping and we can expect that the origin is asymptotically stable, 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) = 2(x + y)2 + x2 + 2y2 is such a Liapunov function, and conclude that the origin is an asymptotically stable critical point.
Hint: From Taylors formula with a remainder it follows that sin x = x a x3/3!, where a depends on x but 0 < a < 1 for n/2 < x < n/2. Then letting x = r cos 9, y = rsin9, show that V(rcos9,rsin9) = r2[1 + h(r,9)], where lh(r,9)l < 1 if r is sufficiently small.
8. The Lienard equation (Problem 28 of Section 9.3) is
d2 u , du
y + c(u) + g(u) = 0,
where g satisfies the conditions of Problem 6 and c(u) > 0. Show that the point u = 0,
du/dt = 0 is a stable critical point.
9. (a) A special case of the Lienard equation of Problem 8 is
where g satisfies the conditions of Problem 6. Letting x = u, y = du/dt, show that the origin is a critical point of the resulting system. This equation can be interpreted as describing the motion of a spring-mass system with damping proportional to the velocity and a nonlinear restoring force. Using the Liapunov function of Problem 6, show that the origin is a stable critical point, but note that even with damping we cannot conclude asymptotic stability using this Liapunov function.
(b) Asymptotic stability of the critical point (0, 0) can be shown by constructing a better Liapunov function as was done in part (d) of Problem 7. However, the analysis for a general function g is somewhat sophisticated and we only mention that an appropriate form for V is
V(x, y) = 2 y2 + Ayg(x) + j g(s) ds,
where A is a positive constant to be chosen so that V is positive definite and V is negative definite. For the pendulum problem [g(x) = sin x] use V as given by the preceding equation with A = 2 to show that the origin is asymptotically stable.
Hint: Use sin x = x ax3/3! and cos x = 1 x2/2! where a and depend on x, but 0 < a < 1 and 0 < < 1 for n/2 < x < n/2; let x = r cos 9, y = r sin 9, and show that
V(r cos9, rsin9) = 2r2[1 + 2 sin29 + h(r, 9)], where lh(r, 9)| < 2 if r is sufficiently small. To show that V is positive definite use cos x = 1 x2/2 + yx4/4!, where y depends on x, but 0 < y < 1 for n/2 < x < n/2.
In Problems 10 and 11 we will prove part of Theorem 9.3.2: If the critical point (0, 0) of the almost linear system
dx/dt = a11 x + a12y + F1(x, y), dy/dt = a21 x + a22y + G1(x, y) (i)
is an asymptotically stable critical point of the corresponding linear system
dx/dt = a11 x + a12 y, dy/dt = a21 x + a22 y, (ii)
then it is an asymptotically stable critical point of the almost linear system (i). Problem 12 deals with the corresponding result for instability.
9.7 Periodic Solutions and Limit Cycles
10. Consider the linear system (ii).
(a) Since (0, 0) is an asymptotically stable critical point, show that a11 + a22 < 0 and a11 a22 a12a21 > 0. (See Problem 21 of Section 9.1.)
(b) Construct a Liapunov function V(x, y) = Ax2 + Bxy + Cy2 such that V is positive definite and Vis negative definite. One way to ensure that Vis negative definite is to choose A, B, and C so that V(x, y) = x2 y2. Show that this leads to the result