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Fx(x0, Ó0) Fy(x0, Ó0Ê |U1 Gx(x0, Ó0) Gy(x0, Ó0W U.
where u1 = x — x0 and u2 = y — y0. Equation (13) provides a simple and general method for finding the linear system corresponding to an almost linear system near a given critical point.
9.3 Almost Linear Systems
Use Eq. (13) to find the linear system corresponding to the pendulum equations (8) near the origin; near the critical point (n, 0).
In this case
F(x, j) = j, G (x, j) = wW sin x - ó j; (14)
since these functions are differentiable as many times as necessary, the system (8) is almost linear near each critical point. The derivatives of F and G are
Fx = 0, Fy = 1, Gx = w2 cos x, Gy =-ó. (15)
Thus, at the origin the corresponding linear system is d (x\ ( 0 Ë (x
iii 2 ii
dt\j) \-wf -ó)
which agrees with Eq. (9).
Similarly, evaluating the partial derivatives in Eq. (15) at (n, 0), we obtain
d M(0 Ë (u
dt \v) \à>2 -ó) \v
where u = x - n, v = j. 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 r1 and r2 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: r1 and r2 pure imaginary, and r1 and r2 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 r1 and r2, 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 other 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 Linear System Almost Linear System
Type Stability Type Stability
r1 > r2 > 0 N Unstable N Unstable
r1 < r2 < 0 N Asymptotically N Asymptotically
r2 < 0 < r1 SP Unstable SP Unstable
r1 = r2 > 0 PN or IN Unstable N or SpP Unstable
r1 = r2 < 0 PN or IN Asymptotically N or SpP Asymptotically
rj, r2 = ê ± ip
ê > 0 SpP Unstable SpP Unstable
ê < 0 SpP Asymptotically SpP Asymptotically
rj = 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.