Download (direct link):
r1 = r2
x r 1 = X + iM
O r2' = X - iM FIGURE 9.3.2 Schematic perturbation of r1 = r2.
9.3 Almost Linear Systems
First, let us consider what it means for a nonlinear system (3) to be close to a linear system (1). Accordingly, suppose that
X = Ax + g(x), (4)
and that x = 0 is an isolated critical point of the system (4). This means that there is some circle about the origin within which there are no other critical points. In addition, we assume that det A = 0, so that x = 0 is also an isolated critical point of the linear system x; = Ax. For the nonlinear system (4) to be close to the linear system x; = Ax we must assume that g(x) is small. More precisely, we assume that the components of g have continuous first partial derivatives and satisfy the limit condition
llg(x)||/||x|| ^ 0 as x ^ 0; (5)
that is, ||g|| is small in comparison to ||x|| itself near the origin. Such a system is called an almost linear system in the neighborhood of the critical point x = 0.
It may be helpful to express the condition (5) in scalar form. If we let x T = (x, y), then ||x|| = (x2 + y2)1/2 = r. Similarly, ifgT(x) = (g^x, y), g2(x, y)), then ||g(x)|| = [g^x, y) + g;(x, y) ]1/2. Then it follows that condition (5) is satisfied if and only if
g1(x, y)/r ^ 0, g2(x, y)/r ^ 0 as r ^ 0. (6)
Determine whether the system
y) V0 0.5) \y) + (,0.75xy 0.25y2) (7)
is almost linear in the neighborhood of the origin.
Observe that the system (7) is of the form (4), that (0, 0) is a critical point, and that
det A = 0. It is not hard to show that the other critical points of Eqs. (7) are (0, 2),
(1, 0), and (0.5, 0.5); consequently, the origin is an isolated critical point. In checking the condition (6) it is convenient to introduce polar coordinates by letting x = r cos Q, y = r sin Q. Then
g1 (x, y) - x2 - xy - r2 cos2 Q - r2 sin Q cos Q
r r r
= r(cos2 Q + sin Q cos Q) ^ 0
as r ^ 0. In a similar way you can show that g2(x, y)/r ^ 0 as r ^ 0. Hence the
system (7) is almost linear near the origin.
The motion of a pendulum is described by the system [see Eq. (13) of Section 9.2]
dx dy 2
= y, = co sin x y y. (8)
dt dt W
The critical points are (0, 0), (±^, 0), (±2^, 0),..., so the origin is an isolated critical point of this system. Show that the system is almost linear near the origin.
Chapter 9. Nonlinear Differential Equations and Stability
To compare Eqs. (8) with Eq. (4) we must rewrite the former so that the linear and nonlinear terms are clearly identified. If we write sin x = x + (sin x x) and substitute this expression in the second of Eqs. (8), we obtain the equivalent system
On comparing Eq. (9) with Eq. (4) we see that gi (x, y) = 0 and g (x, y) = m2(sin x x). From the Taylor series for sin x we know that sin x x behaves like x3/3! = (r3 cos3 d)/3! when x is small. Consequently, (sin x x)/r ^ 0 as r ^ 0. Thus the conditions (6) are satisfied and the system (9) is almost linear near the origin.
Let us now return to the general nonlinear system (3), which we write in the scalar form
y = F(x, y), y= G(x, y).
The system (10) is almost linear in the neighborhood of a critical point (xo, yo) whenever the functions F and G have continuous partial derivatives up to order two. To show this, use Taylor expansions about the point (xo, yo) to write F(x, y) and G(x, y) in the form
F(x, y) = F(xo, yo) + Fx(xo, yo)(x - xo) + Fy(x0 yo)(y - yo) + Vx> y) G(x, y) = G(xo, yo) + Gx(xo, yo)(x - xo) + Gy(xo> yo)(y - yo) + n2(x y)
where n1(x, y)/[(x - xo)2 + (y - yo)2]1/2 ^ o as (x, y) ^ (xo, yo), and similarly for ^2. Note that F(xo, yo) = G(xo, yo) = o, and that dx/dt = d(x - x^/dt and dy/dt = d(y - yo)/dt. Then the system (1o) reduces to
(x - xo ) = dt\y - yo)
or, in vector notation,
Fx(xo yo) Fy(xo yoA (x - xo
Gx(xo yo) Gv(xo yoV V - yo
n1 (x, y) n2 (x, y)
du df n ,
~r. = ~T (x0)u + M*), (12)
where u = (x x0, y y)r and m = (n1,n2)T.
The significance of this result is twofold. First, if the functions F and G are twice differentiable, then the system (10) is almost linear and it is unnecessary to resort to the limiting process used in Examples 1 and 2. Second, the linear system that approximates the nonlinear system (10) near (x0, y0) is given by the linear part of Eqs. (11) or (12), namely,
Fx(x0> y0) Fy(x0> y0y
Gx(x0, y0) Gy(x0, y0),
where U1 = x x0 and u2 = y yj. 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, y) y, G(x, y) -m2 sin x yy; (H)
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