# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

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r1 = r2

-X-—<g>—>Î-

r{ r{

= r2

Î r{= ß -

³ö

FIGURE 9.3.2 Schematic perturbation of rj = r2.

9.3 Almost Linear Systems

481

EXAMPLE

1

EXAMPLE

2

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 xT = (x, y), then ||x|| = (x2 + y2)1/2 = r. Similarly, if gT(x) = (gx(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

g\(x, y)/r ^ 0, g2(x, y)/r ^ 0 as r ^ 0. (6)

Determine whether the system

y) = (0 0.5) (y) + (-0.75xy- 0y>5y2) (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 â, y = r sin â. Then

gj(x, y) -x2 - xy -r2cos2 â - r2sin â cos â

r r r

= — r(cos2 â + sin â cos â) ^ 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, — = —Ø sin x— y y. (8)

dt dt

The critical points are (0, 0), (±n, 0), (±2n, 0),, so the origin is an isolated critical point of this system. Show that the system is almost linear near the origin.

482

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 g1(x, y) = 0 and g2(x, y) = —«2(sin x — x). From the Taylor series for sin x we know that sin x — x behaves like —x3 /3! = —(r3 cos3 6)/3! when x is small. Consequently, (sinx — 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

Ó = F(x, y), Ó= G(x, y).

(10)

The system (10) is almost linear in the neighborhood of a critical point (x0, y0) whenever the functions F and G have continuous partial derivatives up to order two. To show this, use Taylor expansions about the point (x0, y0) to write F(x, y) and G(x, y) in the form

F(x, y) = F(x0, y0) + Fx(x0, y0)(x — x0) + Fy(x0’ y0)(y — y0) + Ï1(x> Ó)>

G(x, y) = G(x0, y0) + Gx(x0, y0)(x — x0) + Gy(x0’ y0)(y — y0) + n2(x’ y)>

where n1(x, y)/[(x — x0)2 + (y — y0)2]1/2 ^ 0 as (x, y) ^ (x0, y0), and similarly for Ï2. Note that F (x0, y0) = G(x0, y0) = 0, and that dx/ dt = d (x — x0)/dt and dy/dt = d(y — y0)/dt. Then the system (10) reduces to

xx

_(Fx(x0, y0) Fy(x0, y0)

xx

— \x x0 )_ ( ~x^'-0’S0' [x x0

dt\y — yj \Gx(x0, y0) Gy(x0, Ó0)) \y — y0

+

Ï1 (x, y) n2 (x, y)

or, in vector notation,

du df 0

-ã: = -r (x0)u + 'n(x),

dt dx

(11)

(12)

where u = (x — x0, y — y0)T and ^ = (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,

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