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Fx 0, Fy 1, Gx -co2 cos x, Gy y. (15)
Thus, at the origin the corresponding linear system is d (x\ ( 0 A (x
? ii 2 ii
dt\y) \o> Yj \y,
which agrees with Eq. (9).
Similarly, evaluating the partial derivatives in Eq. (15) at (n, 0), we obtain
d M(0 A (u
dt \v) \m2 Yj\v
where u x n, v y. This is the linear system corresponding to Eqs. (8) near the point (n, 0).
We now return to the almost linear system (4). Since the nonlinear term g(x) is small compared to the linear term Ax when x is small, it is reasonable to hope that the trajectories of the linear system (1) are good approximations to those of the nonlinear system (4), at least near the origin. This turns out to be true in many (but not all) cases, as the following theorem states.
Let /1 and /2 be the eigenvalues of the linear system (1) corresponding to the almost linear system (4). Then the type and stability of the critical point (0, 0) of the linear system (1) and the almost linear system (4) are as shown in Table 9.3.1.
At this stage, the proof of Theorem 9.3.2 is too difficult to give, so we will accept the results without proof. The statements for asymptotic stability and instability follow as a consequence of a result discussed in Section 9.6, and a proof is sketched in Problems 10 to 12 of that section. Essentially, Theorem 9.3.2 says that for small x (or x x0) the nonlinear terms are also small and do not affect the stability and type of critical point as determined by the linear terms except in two sensitive cases: /1 and /2 pure imaginary, and /1 and /2 real and equal. Recall that earlier in this section we stated that small perturbations in the coefficients of the linear system (1), and hence in the eigenvalues /1 and /2, can alter the type and stability of the critical point only in these two sensitive cases. It is reasonable to expect that the small nonlinear term in Eq. (4) might have a similar substantial effect, at least in these two sensitive cases. This is so, but the main significance of Theorem 9.3.2 is that in all othe/cases the small nonlinear term does not alter the type or stability of the critical point. Thus, except in the two sensitive cases, the type and stability of the critical point of the nonlinear system (4) can be determined from a study of the much simpler linear system (1).
Chapter 9. Nonlinear Differential Equations and Stability
TABLE 9.3.1 Stability and Instability Properties of Linear and Almost Linear Systems
r1, r2 Linear System Almost Linear System
Type Stability Type Stability
r1 > r2 > 0 N Unstable N Unstable
ri < r2 < 0 N Asymptotically N Asymptotically
r2 < 0 < r1 SP Unstable SP Unstable
r, = r2 > 0 PN or IN Unstable N or SpP Unstable
o V ' C II ' C PN or IN Asymptotically N or SpP Asymptotically
r,, r2 = k ± i p
k > 0 SpP Unstable SpP Unstable
X < 0 SpP Asymptotically SpP Asymptotically
r, = ip, r2 = ip C Stable C or SpP Indeterminate
Note: N, node; IN, improper node; PN, proper node; SP, saddle point; SpP, spiral point; C, center.
Even if the critical point is of the same type as that of the linear system, the trajectories of the almost linear system may be considerably different in appearance from those of the corresponding linear system, except very near the critical point. However, it can be shown that the slopes at which trajectories enter or leave the critical point are given correctly by the linear equation.
Damped Pendulum. We continue the discussion of the damped pendulum begun in Examples 2 and 3. Near the origin the nonlinear equations (8) are approximated by the linear system (16), whose eigenvalues are
-y ± s/y2 4a>2 2
The nature of the solutions of Eqs. (8) and (16) depends on the sign of y 2 4m2 as
1. If y2 4m2 > 0, then the eigenvalues are real, unequal, and negative. The critical point (0, 0) is an asymptotically stable node of the linear system (16) and of the almost linear system (8).
2. If y2 4m2 = 0, then the eigenvalues are real, equal, and negative. The critical point (0, 0) is an asymptotically stable (proper or improper) node of the linear system (16). It may be either an asymptotically stable node or spiral point of the almost linear system (8).
3. If y2 4m2 < 0, then the eigenvalues are complex with negative real part. The critical point (0, 0) is an asymptotically stable spiral point of the linear system (16) and of the almost linear system (8).
r1, r2 =
Thus the critical point (0, 0) is a spiral point of the system (8) if the damping is small and a node if the damping is large enough. In either case, the origin is asymptotically stable.
9.3 Almost Linear Systems 485
Let us now consider the case y2 4m2 < 0, corresponding to small damping, in more detail. The direction of motion on the spirals near (0, 0) can be obtained directly from Eqs. (8). Consider the point at which a spiral intersects the positive y-axis (x = 0 and y > 0). At such a point it follows from Eqs. (8) that dx/dt > 0. Thus the point (x, y) on the trajectory is moving to the right, so the direction of motion on the spirals is clockwise.