1951] HASKELL B. CURRY 205
Now Fm+1 is itself such an F. Hence, by induction, we have, for suitable gh ,
Fm — % + <7l/ + Q2S2 + ■ • • + '•
This suggests a change of variable to y — f(x) (which is possible since f'(X) ^ 0). If
x = u(y) is the inverse function, and
$m(y) = Fm[u(y)], My) = 9k[u(y)],
then
= W + iiy + $2 y2 + • • • + im-l 2/m_1. (B)
The conditions which must be satisfied by <3?„ are
*(0) = X,
$'(0) = $"(0) = • • • = $ra-1(0) = 0.
The first of these is automatically satisfied.
Now a function $m(y) satisfying these conditions is given immediately by the inverse
Taylor expansion of X = u(y — y). In fact, if we set $m(y) equal to the sum of the
first m terms of this expansion, viz.:
fc = 0
then
x = *"(y)+LJf
and hence
*m(y) = X-
satisfies the above conditions.
This method avoids the necessity of slapping down Bodewig's formula (14) or of
motivating it by tedious experimenting with small values of m.
BOUNDARIES FOR THE LIMIT CYCLE OF VAN DER POL'S EQUATION*
By R. GOMORY and D. E. RICHMOND (Williams College)
1. Introduction. In non-linear mechanics much interest centers on the Van der Pol
(VDP) equation
d zc - / 2 doc
Te + »(x ~ " a
J + M* - 1) 37 + * - 0 (!)
*Received Sept. 11, 1950.
206 NOTES [Vol. IX, No. 2
or its equivalent in the phase plane
ft = V> ft = "* + "(1 ~ X*)y- (2)
It is well known that (2) possesses a unique trajectory which represents a limit
cycle in the sense of Poincare. Using a different plane, La Salle1 has located this limit
cycle between two boundary curves in a very ingenious manner which however seems
artificial and difficult to motivate. The present paper sets forth a simple and natural
method for constructing outer and inner boundaries. The method admits of unlimited
improvement but even its simplest application gives results superior to La Salle's in
that the limit cycle is localized somewhat more sharply.
In the phase plane all trajectories other than the limit cycle spiral into it from the
inside or the outside. The curves are described clockwise with increasing t. We use these
facts to enclose the limit cycle between an outer boundary B0 and an inner boundary B{ .
Introducing r2 = x2 + y2, we transform (2) to
dr . 2 2 dx . .
rJt = ^l-x)y, Tt=y. (3)
Eliminating t, one obtains for the trajectories
r fx = ~ X**y"
Since the field of (4) is symmetrical in the origin, it is sufficient to discuss solutions in
the upper half-plane (y ^ 0).
If C is a curve, r = F(x), y > 0, which intersects the x-axis only at ( — a, 0) and
(a, 0) and if at every point the value of dr/dx for C is greater than or equal to that of
the VDP solution through that point, all YDP curves intersect G from above to below.
Then C together with its image in the origin forms an outer boundary B„ .
The construction of an inner boundary B{ requires the substitution of less than for
greater than in the above statement.
2. The outer boundary. To construct an outer boundary B0 , write (4) in the form
r | = m( 1 - x2)(r2 - Xy/2, y > 0, (4')
On an x-interval within which 1 — x2 is positive, replace x under the radical by
(z2)min , its least value on the interval. The curves defined by the solutions of
r £ = m(1 - *2)[r2 - (*2)min]1/2, y > 0, (5)
have at every point of the interval a value of dr/dx greater than or equal to that of the
corresponding VDP curve.
Similarly, on an x-interval within which 1 — x2 is negative, replace x2 under the
radical by (x2)m„ , its largest value on the interval. The curves defined by
'J. La Salle, Relaxation o scillations, Q. Appl. Math. 7,1-19 (1949). If La Salle's co-ordinates are (x, u),
the relation to ours is given by y/n + u = x — (x3/3). His t is n times ours.
1951] R. GOMORY AND D. E. RICHMOND 207
r | = „(1 - x2) [r2 - (z2)mJI/2, y > 0, (6)
have at every point of the interval a value of dr/dx greater than or equal to that of the
corresponding VDP curve.
It remains to join together solutions of (5) and (6), valid over different intervals, to
generate a continuous curve which will serve as C, the portion of B0 in the upper half-
plane.
This boundary, at least for large n, may be expected to lie rather close to the limit
cycle since over much of the short x range, x2 is small relative to r2 so that the error
made by the proposed substitution is not serious. In fact, by using a large number of
intervals a very accurate outer boundary may be constructed. But even the simplest
outer boundary, using three intervals, is surprisingly good. We proceed to the details
for this case.
Fig. 1. Sketch of Boundaries.
Let C intersect the x-axis at —a and a. The intervals to be used are [—a, —1],
[ — 1, 1] and [1, a]. We start at —a and work across to a in the upper half plane. The
equations become
[-a, -1] M(1 - z2)(r2 - a2)1'2,
[-1,1] rg = M(l-*2)(r2-02)I/2,
[1 ,a] rfx = „(1 - x2)(r2 ~ a2)1'2.
Each of these equations is separable and possesses a general and a singular solution.
We use the singular solution r = a for [—a, —1] since it is the only solution through
208 NOTES [Vol. IX, No. 2
( — a, 0) which is real in the interval. For the other intervals, we use the general solutions
which take the form
(r2 - c2)"2 = »(x ~f) + k.
It is easy to choose the constants k to secure continuity.
The principal interest is in the amplitude a which determines the size of the en-
closure. To find a, we take definite integrals successively over the intervals [—a, —1],
[ — 1, 1] and [1, a], finding in each case the relation between r at the left side and r at
the right side of the interval. The results are
r(— 1) — a — 0,
r(l) — r(—1) = (7)
Combining
[r2(l) - a2]1/2 = | (a3 - 3a + 2).
a3 - 3a + 2 - 4(1 + ' = 0
Thus even in the simplest case there is a slight improvement over La Salle's result,
which in this form is
a3 - 3a + 2 - 4(1 + |^) = 0.
We quote some numerical results for n = 3. La Salle's boundary gives a = 2.21+.
Our result is a = 2.18~. If seven intervals are used (joining at — 31/2, —1.6, —1.4, —1,
1 and 3I/2), one obtains a = 2.10".
3. The inner boundary. To construct an inner boundary Bi , use
r ^ = M(1 - x2)(r2 - z2)1/2, y > 0, (4')
as before, but replace x2 under the radical by (x2)mM if 1 — x2 is positive within the
x-interval, by (x2)mia if 1 — x2 is negative there.
Using three intervals [ — c, — 1], [— 1, 1] and [1, 6], we obtain the equation
r|=M( l-/)(r2-r,
applicable to all three intervals. One is tempted to set c = b and reflect in the origin-
There is however a difficulty. To the left of x = —1, dr/dx is negative but —dr/dx g 1
for x ^ — c. This condition restricts the choice of c.
Since r(— dr/dx) = n(x2 — l)(r2 — 1)1/2, y ^ 0, and since on the x-axis, r = c, it
follows that
c > M(c2 - 1)3/2.
1951] R. GOMORY AND D. E. RICHMOND 209
The largest c corresponds to the equality. If
d = »(d2 - 1)3/2
defines d for a given n, we therefore choose c ^ d and follow the curve along the upper
half-plane to its right-hand intersection at b. Reflection of this curve in the origin leaves
two gaps, one between —b and —c and one between c and b. If b > c we may use seg-
ments of the cc-axis to complete the inner boundary. Since the YDP curves are described
clockwise with increasing t, these curves cross the added segments in the required
direction.
The relation between — c and b is given by integrating between these limits and is
(6» _ i)i" _ (c« _ i)V = | + c - (9)
We know that c g d. The largest inner boundary using three intervals arises from the
choice c = d. However, for the sake of simplicity, we may obtain an inner boundary
valid for all n by choosing c = 1. Then (9) becomes
b3 - 3b — 2 + - (62 — 1)I/2 = 0,
which is a slight improvement over La Salle's result
b3 - 36 - 2 + — = 0.
i"
For n = 3, for example, La Salle's result is b = 1.77. Ours is b = 1.81 with c = 1
and b = 1.87+ with c = d — 1.248. If additional points of division are placed at —.5,
.5 and 31/2, b = 1.94 is obtained.
4. Conclusion. In conclusion, it is clear that the simplicity of the calculations makes
it relatively easy to obtain indefinitely better boundaries by increasing the number of
intervals. Over these intervals, the form of the solution is (with one exception) always
the same, different (x)2mCLX and (a:)2 in being inserted. It is therefore sufficient to be armed
with a table of square roots and cubes to find a or b by trial solution. With a moderate
amount of work, it is fortunately possible to supplement the method of perturbations
which is useful for n <5C 1 and the known results for ju —> °o by giving a good account of
the limit cycle in the intermediate range of //.
It is also obvious that the method of this paper can be applied if x2 — 1 in Eq. (1)
is replaced by other suitable functions f(x). Further generalizations are possible but
will not be discussed.