0387986995Basic TheoryC

X-4. Multipliers of the periodic orbit of the van der Pol equation

In Example X-2-5, we looked at the van der Pol equation

where a is a positive number. Using Observation X-2-4, it was shown that every orbit of the system

is bounded for t > 0. It was also remarked that 0 is the only stationary point of system (X.4.2) and that the stationary point 0 is not stable as t - +oo. In fact, the

linear part of the right-hand side of (X.4.2) at y = 0 is Ay, where A = [ 01 and

[i]. Since trace[A]=f and det (A) = 1, the stationary point is an unstable

Y21-

node for e > 2, while 0 is an unstable spiral point for 0 < e < 2. Therefore, using the Poincare-Bendixson Theorem (cf. Theorem IX-4-1), we conclude that there exists at least one limit cycle. Now, Theorem X-3-1 implies that system (X.4.2) has exactly one limit cycle and all the other orbits except for the stationary point 6 approach this limit cycle asymptotically as t - +oo. In fact, since h(x) = e(x2 -1),

H(x) = e I 3 - x] , and g(x) = x, the seven conditions (i) - (vii) of Theorem X-3-1

3

are satisfied. In particular, the positive zero of H(x) is a = J > 1.

In this section, we prove the following theorem concerning the multipliers of the unique periodic solution x = x(t, e) of the van der Pol equation (X.4.1).

Theorem X-4-1. The multipliers of the periodic solution x(t, e) of (X.4.1) at 1 and p such that )pI < 1.

Proof.

Y

FIGURE 11.

5. THE VAN DER POL EQUATION FOR A SMALL PARAMETER 319

On each of two curves Cl and C2, let us denote y as a function of v by yl (v) and y2(v) respectively. Note that x2 > 1 on C1, but x2 < 1 on C2. Hence, yl(v)2 < y2(v)2. This implies that

fT

e (x(t)2 - 1)dt > 0,

where T is a period. Therefore, we can complete the proof by using the following lemma.

Lemma X-4-2. Assume that two functions f (x) and g(x) are continuously differentiable in R. Assume also that the differential equation

d 2 2 + f(x) dt + g(x) = 0

r1

has a nontrivial periodic solution x(t) of period 1 such that J0 f (x(t))dt > 0. Then, the multipliers of the periodic solution x(t) are 1 and p such that jpI < 1.

Remark X-4-3. If system (X.4.2) has more than one periodic orbit, then at least one of them must be orbitally unstable. Therefore, the proof of Theorem X-4-1 is another proof of the uniqueness of periodic orbit of (X.4.2).

X-5. The van der Pol equation for a small e > 0

In this section, we explain a method to locate the unique periodic orbit of dif- ferential equation (X.4.1) (or system (X.4.2)) for a small e > 0.

Set e = 0. Then, system (X.4.2) becomes

(X.5.1)

dt[y2J LyYi

Every orbit of (X.5.1) is a circle and of period 2n in t. We expect that the periodic orbit of (X.4.2) must be approximated by one of those circles as e -+ 0. The main problem is to find the radius of the circle which approximates the periodic orbit of

(X.4.2) for a sufficiently small e > 0.

Since the periodic orbit and its period are functions of e, we normalize the independent variable t by the change of independent variable

so that the period of the periodic orbit becomes 27r for every e > 0. Transformation (X.5.2) changes differential equation (X.4.1) to

x722 + e(1+eW)(x2-1)d + (1+&,d)2x = 0

that can be written in the form

22

5. THE VAN DER POL EQUATION FOR A SMALL PARAMETER 321

Now, let us consider an integral equation

as long as v is periodic in r of period 2ir. This integral equation can be solved by using successive approximations in such a way that the solution v(r, K, W, E) is a convergent power series in c:

where vm(r, K, w) are polynomials in (K, w) with coefficients periodic in r of period 2n. For any given positive numbers Ko and wo, there exists another positive number Eo(Ko,wo) such that series (X.5.11) is uniformly convergent for

This implies that S(K, w, c) and C(K, w, E) are also convergent power series in c:

where Sm(K,w) and C,,,(K,w) are polynomials in (K,w) with constant coefficients.

These two power series also converge uniformly for

Observation X-5-4. Inserting series (X.5.11) and (X.5.12) into system (X.5.10), we obtain

d2vi + vi = (1 - K2cos2(r))(-Ksin(r)) - 2wKcos(r)

dr2

-So(K,w) sin(r) - Co(K,w) cos(r)

=K3 cos2(r) sin(r) - K sin(r) - 2wK cos(r)

-So(K,w) sin(1T) - Co(K,w) cos(r)

4

- [2wK + Co(K, w)1 cos(r).

322 X. THE SECOND-ORDER DIFFERENTIAL EQUATION

Therefore, the system of equations S(K, w, e) = 0, C(K, w, e) = 0 has a solution K(e) = 2 + 0(e), w(e) = 0(e). The functions K(E) and w(e) are power series in e which converge if Iej is sufficiently small.

Observation X-5-5. Set

x(t, e) = v (T+ f1w(E) K(E), w(E), e/f

I

Then, x(t, e) is a periodic solution of (X.5.3) and

From the fact that

we obtain the following conclusion.

Conclusion X-5-6. The unique periodic orbit of (X.4.2) tends to the circle yl +

Y22 = 4 ase-.0+.

X-6. The van der Pol equation for a large parameter

In this section, we consider the van der Pol equation (X.4.1) for a large e. Let us write (X.4.1) in the form

where a = 1-

E

6. THE VAN DER POL EQUATION FOR A LARGE PARAMETER 323

Observation X-6-1. Note that, along an orbit of (X.6.2),

3 - wtl

Therefore, if 3 > 0 is sufficiently small, the slope dw, of any orbit is small at a

point (wl, w2) far away from the curve C : w2 = 3 - w!. This implies that every

orbit moves toward the curve C almost horizontally (cf. Figure 12).

Observation X-6-2. From Observation X-6-1 a rough picture of orbits of (X.6.2) is obtained (cf. Figure 13).

Actually, defining the curve Co by Figure 14, we prove the following theorem.

Theorem X-6-3. For a sufficiently small positive number J3, the unique periodic orbit of (X.6.2) is located in an open set V(/3) such that the closure of V(f3) contains the curve Co and shrinks to CD as 0 -+ 0+.

Proof of Theorem X-6-8. In eight steps, we construct an open set V(13) so that

(1) the closure of V(O) contains the curve Co,

(3)if an orbit of (X.6.2) enters in the open set V(!3) at r = ro, then the orbit stays in V(/3) for r > ro.

Step 1. Fixing a number a(3) > 2, we use the line segment

C1(a) = j wi, a(3)3 - a((3) I : 0 < wi <_ a(/3)}

as a part of the boundary 8V(i3) of V(/3) (cf. Figure 15).

Note that the slope of an orbit given by (X.6.3) is negative on C1 (O) and decreasing

to -oo as wt increases to a(f3). Also, is 0 on C1(f3).

dw2

Step 2. Let us consider a curve

CO) = j (wi, ws) : w2 = 3' - wl - Op(wi, 0), 1 < wl < a(fl)} ,

where

(i)µ(w1,3) is continuous for 1 < wl < a(0) and continuously differentiable for

1< wl < a($),

(ii)µ(w1, 0) > 0 for 1 < w1 < a(0), and (iii) µ(a(,3), 0) = 0.

On the curve C2(f3),

3

\ 31 - wl = -0µ(w1,13)

and, hence,

where dw2 denotes the slope of C2(0). The unique solution to problem (X.6.4) is

i

given by

as a part of the boundary OV(#) of V(f3) (cf. Figure 16). Note that on the curve

C2(,8),

w2 = -3 - Q a(f3)2 - 1 at w1 = 1.