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The behavior of the pendulum near the critical points (±nn, 0), with n even, is the same as its behavior near the origin. We expect this on physical grounds since all these critical points correspond to the downward equilibrium position of the pendulum. The conclusion can be confirmed by repeating the analysis carried out above for the origin. Figure 9.3.3 shows the clockwise spirals at a few of these critical points.
Now let us consider the critical point (n, 0). Here the nonlinear equations (8) are approximated by the linear system (17), whose eigenvalues are
— Y ± VY2 + 4«2 ,im
r1> r2 = 2 • (19)
One eigenvalue (zj) is positive and the other (z^) is negative. Therefore, regardless of the amount of damping, the critical point x = n, y = 0 is an unstable saddle point of both the linear system (17) and of the almost linear system (8).
To examine the behavior of trajectories near the saddle point (n, 0) in more detail we write down the general solution of Eqs. (17), namely,
^ ^ + CJD &, (20)
V 1W 2 V2/
where Cj and C2 are arbitrary constants. Since rx > 0 and r2 < 0, it follows that the solution that approaches zero as t corresponds to Cj = 0. For this solution
v/u = r2, so the slope of the entering trajectories is negative; one lies in the second quadrant (C2 < 0) and the other lies in the fourth quadrant (C2 > 0). For C2 = 0 we obtain the pair of trajectories “exiting” from the saddle point. These trajectories have slope rj > 0; one lies in the first quadrant (Cj > 0) and the other lies in the third quadrant (Cj < 0).
The situation is the same at other critical points (nn, 0) with n odd. These all correspond to the upward equilibrium position of the pendulum, so we expect them to be unstable. The analysis at (n, 0) can be repeated to show that they are saddle points oriented in the same way as the one at (n, 0). Diagrams of the trajectories in the neighborhood of two saddle points are shown in Figure 9.3.4.
FIGURE 9.3.3 Asymptotically stable spiral points for the damped pendulum.
Chapter 9. Nonlinear Differential Equations and Stability
The equations of motion of a certain pendulum are
dx/dt = y, dy/dt = —9sinx — 1 y, (21)
where x = Q and y = dd/dt. Draw a phase portrait for this system and explain how it shows the possible motions of the pendulum.
By plotting the trajectories starting at various initial points in the phase plane we obtain the phase portrait shown in Figure 9.3.5. As we have seen, the critical points (equilibrium solutions) are the points (nn, 0), where n = 0, ±1, ±2,.... Even values of n, including zero, correspond to the downward position of the pendulum, while odd values of n correspond to the upward position. Near each of the asymptotically stable critical points the trajectories are clockwise spirals that represent a decaying oscillation about the downward equilibrium position. The wavy horizontal portions of the trajectories that occur for larger values of | y| represent whirling motions of the pendulum. Note that a whirling motion cannot continue indefinitely, no matter how large |y| is; eventually the angular velocity is so much reduced by the damping term that the pendulum can no longer go over the top, and instead begins to oscillate about its downward position.
FIGURE 9.3.5 Phase portrait for the damped pendulum of Example 4.
9.3 Almost Linear Systems
The trajectories that enter the saddle points separate the phase plane into regions. Such a trajectory is called a separatrix. Each region contains exactly one of the asymptotically stable spiral points. The initial conditions on d and dd / dt determine the position of an initial point ( x, y) in the phase plane. The subsequent motion of the pendulum is represented by the trajectory passing through the initial point as it spirals toward the asymptotically stable critical point in that region. The set of all initial points from which trajectories approach a given asymptotically stable critical point is called the basin of attraction or the region of asymptotic stability for that critical point. Each asymptotically stable critical point has its own basin of attraction, which is bounded by the separatrices through the neighboring unstable saddle points. The basin of attraction for the origin is shown in blue in Figure 9.3.5. Note that it is mathematically possible (but physically unrealizable) to choose initial conditions on a separatrix so that the resulting motion leads to a balanced pendulum in a vertically upward position of unstable equilibrium.
An important difference between nonlinear autonomous systems and the linear systems discussed in Section 9.1 is illustrated by the pendulum equations. Recall that the linear system (1) has only the single critical point x = 0 if det A = 0. Thus, if the origin is asymptotically stable, then not only do trajectories that start close to the origin approach it, but, in fact, every trajectory approaches the origin. In this case the critical point x = 0 is said to be globally asymptotically stable. This property of linear systems is not, in general, true for nonlinear systems. For nonlinear systems an important question is to determine (or to estimate) the basin of attraction for each asymptotically stable critical point.