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< r3[2| cos3 â | + 2cos2 â | sin â | + | cos â | sin2 â + 2| sin3 â |
since | sin â |, | cos â | < 1. To satisfy Eq. (23) it is now certainly sufficient to satisfy the more stringent requirement
7r3 < 0.25(u2 + v2) = 0.25r2,
9.6 Liapunov’s Second Method
which yields r < 1/28. Thus, at least in this disk, the hypotheses of Theorem 9.6.1 are satisfied, so the origin is an asymptotically stable critical point of the system (20). The same is then true of the critical point (0.5, 0.5) of the original system (18).
If we refer to Theorem 9.6.3, the preceding argument also shows that the disk with center (0.5, 0.5) and radius 1/28 is a region of asymptotic stability for the system (18). This is a severe underestimate of the full basin of attraction, as the discussion in Section
9.4 shows. To obtain a better estimate of the actual basin of attraction from Theorem 9.6.3, one would have to estimate the terms in Eq. (23) more accurately, use a better (and presumably more complicated) Liapunov function, or both.
dx/dt = y — xf(x, y), dy/dt = — x — yf(x, y),
where f is continuous and has continuous first partial derivatives. Show that if f (x, y) > 0 in some neighborhood of the origin, then the origin is an asymptotically stable critical point, and if f (x, y) < 0 in some neighborhood of the origin, then the origin is an unstable critical point.
Hint: Construct a Liapunov function of the form c(x2 + y2).
6. A generalization of the undamped pendulum equation is
d2u/dt2 + g(u) = 0, (i)
where g(0) = 0, g(u) > 0 for 0 < u < k, and g(u) < 0 for — k < u < 0; that is, ug(u) > 0 for u = 0, — k < u < k. Notice that g(u) = sin u has this property on (—ï/2, ï/2).
(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 — sinx > 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 =— 1 x3 + 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 + 2 y3; unstable
5. Consider the system of equations
Chapter 9. Nonlinear Differential Equations and Stability
(c) Using the energy function V(x, y) = 2y2 + (1 — cos x) 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 Taylor’s formula with a remainder it follows that sin x = x — ax3 /3!, where à depends on x but 0 < à < 1 for —n/2 < x < n/2. Then letting x = r cos 9, y = r sin9, show that V(rcos9,rsin9) = — r2[1 + h(r,9)], where lh(r,9)l < 1 if r is sufficiently small.
The Lienard equation (Problem 28 of Section 9.3) is
1? +c(u) d+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.
(a) A special case of the Lienard equation of Problem 8 is
-2 g(u) = 0,
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.