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within the distance S)to x0 stay “close” (within the distance e )to x0. Note that in Figure 9.2.1a the trajectory is within the circle ||x — x0|| = S at t = 0 and, while it soon passes outside of this circle, it remains within the circle ||x — x01| = e for all t > 0. However, the trajectory of the solution does not have to approach the critical point x0 as t ^ to, as illustrated in Figure 9.2.1b. A critical point that is not stable is said to be unstable.
9.2 Autonomous Systems and Stability
FIGURE 9.2.1 (a) Asymptotic stability. (b) Stability.
A critical point x0 is said to be asymptotically stable if it is stable and if there exists
a 80, with 0 < 80 <8, such that if a solution x = ô(^ satisfies
||ô(0) - x°|| <80, (8)
lim ô(^ = x0. (9)
Thus trajectories that start “sufficiently close” to x0 must not only stay “close” but must eventually approach x0 as t ^ to. This is the case for the trajectory in Figure 9.2.1a but not for the one in Figure 9.2.1b. Note that asymptotic stability is a stronger property than stability, since a critical point must be stable before we can even talk about whether it might be asymptotically stable. On the other hand, the limit condition (9), which is an essential feature of asymptotic stability, does not by itself imply even ordinary stability. Indeed, examples can be constructed in which all of the trajectories approach x0 as t ^ to, but for which x0 is not a stable critical point. Geometrically, all that is needed is a family of trajectories having members that start arbitrarily close to x0, then depart an arbitrarily large distance before eventually approaching x0 as t ^ TO.
While we specified originally that the system (5) is of second order, the definitions just given are independent of the order of the system. If you interpret the vectors in Eqs.
(5) through (9) as ë-dimensional, then the definitions of stability, asymptotic stability, and instability apply also to nth order systems. These definitions can be made more concrete by interpreting them in terms of a specific physical problem.
The Oscillating Pendulum. The concepts of asymptotic stability, stability, and instability can be easily visualized in terms of an oscillating pendulum. Consider the configuration shown in Figure 9.2.2, in which a mass m is attached to one end of a rigid, but weightless, rod of length L. The other end of the rod is supported at the origin O, and the rod is free to rotate in the plane of the paper. The position of the pendulum is described by the angle â between the rod and the downward vertical direction, with the counterclockwise direction taken as positive. The gravitational force mg acts downward, while the damping force c\de/dt\, where c is positive, is always opposite to the direction of motion. We assume that â and de/ dt are both positive. The equation of motion can be quickly derived from the principle of angular momentum,
Chapter 9. Nonlinear Differential Equations and Stability
FIGURE 9.2.2 An oscillating pendulum.
which states that the time rate of change of angular momentum about any point is equal to the moment of the resultant force about that point. The angular momentum about the origin is mL2(dd/dt), so the governing equation is
r2 d2â TdQ
mL—T = —cL---------------mgL sin â. (10)
The factors L and L sin â on the right side of Eq. (10) are the moment arms of the resistive force and of the gravitational force, respectively, while the minus signs are due to the fact that the two forces tend to make the pendulum rotate in the clockwise (negative) direction. You should verify, as an exercise, that the same equation is obtained for the other three possible sign combinations of â and de/ dt.
By straightforward algebraic operations we can write Eq. (10) in the standard form
d2â c de g
—2 +--------7— + — sin â = 0, (11)
dt2 mL dt L
d2 â dâ 2 .
—T + ó-----------+ a sin â = 0, (12)
where y = c/ mL and a2 = g/ L .To convert Eq. (12) to a system of two first order equations we let x = â and y = dâ/ dt; then
dx dy 2 .
— = y, — =—a sin x— y y. (13)
Since y and a2 are constants, the system (13) is an autonomous system of the form (1). The critical points of Eqs. (13) are found by solving the equations