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Theorems 9.6.1 and 9.6.2 give sufficient conditions for stability and instability, respectively. However, these conditions are not necessary, nor does our failure to determine a suitable Liapunov function mean that there is not one. Unfortunately, there are no general methods for the construction of Liapunov functions; however, there has been extensive work on the construction of Liapunov functions for special classes of equations. An elementary algebraic result that is often useful in constructing positive definite or negative definite functions is stated without proof in the following theorem.
Theorem 9.6.4 The function
V(x, y) = ax2 + bxy + cy2 (14)
is positive definite if, and only if,
a > 0 and 4ac b2 > 0, (15)
and is negative definite if, and only if,
a < 0 and 4ac b2 > 0. (16)
The use of Theorem 9.6.4 is illustrated in the following example.
Show that the critical point (0, 0) of the autonomous system
dx/dt =x xy2, dy/dt = y x2 y (17)
is asymptotically stable.
We try to construct a Liapunov function of the form (14).Then V (x, y) = 2ax + by,
Vy(x, y) = bx + 2cy, so
V(x, y) = (2ax + by)(x xy2) + (bx + 2cy)(y x2y)
= [2a(x2 + x2 y2) + b(2xy + xy3 + x3 y) + 2c(y2 + x2 y2)].
If we choose b = 0, and a and c to be any positive numbers, then VVis negative definite and V is positive definite by Theorem 9.6.4. Thus by Theorem 9.6.1 the origin is an asymptotically stable critical point.
Chapter 9. Nonlinear Differential Equations and Stability
Consider the system
dx/dt = x (1 x y),
dy/dt = y (0.75 y 0.5x).
In Example 1 of Section 9.4 we found that this system models a certain pair of competing species, and that the critical point (0.5, 0.5) is asymptotically stable. Confirm this conclusion by finding a suitable Liapunov function.
It is helpful to transform the point (0.5, 0.5) to the origin. To this end let
x = 0.5 + u, y = 0.5 + v. (19)
Then, substituting for x and y in Eqs. (18), we obtain the new system
du/ dt = 0.5u 0.5v u2 uv,
d v/dt = 0.25u 0.5v 0.5 u v v2.
To keep the calculations relatively simple, consider the function V(u, v) = u2 + v2 as a possible Liapunov function. This function is clearly positive definite, so we only need to determine whether there is a region containing the origin in the uv-plane where the derivative Vwith respect to the system (20) is negative definite. We compute V(u, v) and find that
V du dv
V(u, v) = V + V
v 2 udt v dt
= 2u(0.5u 0.5v u2 uv) + 2v(0.25u 0.5v 0.5uv v2),
V(u, v) = [(u2 + 1.5uv + v2) + (2u3 + 2u2v + uv2 + 2v3)], (21)
where we have collected together the quadratic and cubic terms. We want to show that the expression in square brackets in Eq. (21) is positive definite, at least for u and v sufficiently small. Observe that the quadratic terms can be written as
u2 + 1.5uv + v2 = 0.25(u2 + v2) + 0.75(u + v)2, (22)
so these terms are positive definite. On the other hand, the cubic terms in Eq. (18) may be of either sign. Thus we must show that, in some neighborhood of u = 0, v = 0, the cubic terms are smaller in magnitude than the quadratic terms; that is,
|2u3 + 2u2v + uv2 + 2v3| < 0.25(u2 + v2) + 0.75(u + v)2. (23)
To estimate the left side of Eq. (23) we introduce polar coordinates u = r cos Q, v = r sin Q. Then
|2u3 + 2u2v + uv2 + 2v3| = r3|2cos3 Q + 2 cos2 Q sinQ + cosQ sin2 Q + 2 sin3 Q\
< r3[2\ cos3 Q \ + 2 cos2 Q \ sin Q \ + \ cos Q \ sin2 Q + 2 \ sin3 Q \
since \ sin Q \, \ cos Q \ < 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 Liapunovs 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 (n/2, n/2).