NATIONAL BUREAU OF STANDARDS REPORT

ASYMPTOTIC SOLUTION OF VAN DER POL»S EQUATION

fcy

A. A. Dorodnicyn [Dorodnitsyn]

Translated from Lhe Russian by Curtis Do Benster

Edited by

Wolfgang Ro Wasow

2U89

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS REPORT

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1101-10-5100 April 30 9 1953 21*89

ASYMPTOTIC SOLUTION OF VAN DER POL'S EQUATION'"

by

A. A, Dorodnicyn [Dorodnitsyn]

Translated from the Russian by' Curtis U® Benster

Edited by

Wolfgang Ro Wasow

National Bureau of Standards^ Los Angeles

‘This translation was sponsored in part by the Office of

Naval Research^ USN„

The publication, rep, Approved for public release by the

unless permission is C Director of the National Institute of

25,d.c. Suchperm Standards and Technology (NIST)

cally prepared If thr on October 9, 2015

i part, Is prohibited

ndards, Washington

ort has been speclfllort for Its own use.

ASYMPTOTIC SOLUTION OF VAN DER FOL'S EQUATION

1

Ao A. Dorodnicyn [Lorodnitsyn]

Moscow

1. Statement of the problem. In this article is considered the

solution of Van der Pol's equation

(1.1) - V(1 - X

2

)ff x - 0

dt '

for large values of the parameter

In the phase plane xp 5 equation (1.1) is transformed into the

form

(1.2) pp’ - v(l - x

2

)p + x « 0 (p =

where the prime sign denotes differentiation with respect to x«

The solution of this equation has the character schematically

represented in Fig. 1 (for the limit cycle).

It is known that in domain I and in domain III the solution of

equation (1.2) tends to the solutions of the "shortened" equations

(1.3) pp’ " vU - x?)p s 0

(1.U) -v(l - x^)p + x - 0

respectively.

1Prikladnaia matematika i mekhanika. Vol. XI * 19k7 9 pp. 313-328.

2

P

The domains in which these two limit solutions are suitable are

not 5 however3 contiguous s and therefore these solutions cannot be

linked* We do not know how to choose the constant of integration in

equation (1*3) so that for the analytic continuation of the solution

into domain III this solution shall pass into that which tends to the

solution of the second equation,, (lJi)o

3

In the present paper we introduce two '* connecting 18 domains II and

I

V

a for which are established singular asymptotic solutions of equation

(l.^)^ different from the solutions of the 19 shortened*1 equations ( 1 0 3 )

and (loU) e Domains I5 lls III 3 and IV overlap each other, the possibility being thus obtained of finding the complete solution for the

entire cycle accurate to quantities of any order of smallness relative

to Vo

2o Solution for domain I. Denoting by and a~ the values of x

for which p

= 0 (for the limit cycle a-^ s = a 5 where a is the amplitude of the steady-state auto-oscillations) } we will define two parts

of domain I thus;

It will obviously be sufficient to consider the solution in one

of the parts of domain I s say the first e We seek the solution in the

form

n=o

Substituting (2d) in (1„2) and equating the coefficients of like

powers of V9 we obtain the recurrent system of equations;

-l+e<x<a1 -e } p > 0 ^ € > 0 ;

a2 + £<x<l=6 ^ p<0 ^ t > 0 o

00

( 2 . 1)

n

k-1

( 2 . 2 )

9
.
h

whose solution is elementary-® Thus for the first two functions we

have

(2.3) f (x) - C + X - “X^

0

V 3

fx

(x)

L1

x

2 - 1

log 1

X

X,

1 (2x + x^)

2

+ 3(x2 - k)

log

U(x? - 3)

(2.U)

2X + Xn Xi arc tg ~ arc tg

v3(xf - U) v5(? U)J

Here,, in (2®1j.) 5 by x-^ is denoted the real (positive) root of the

equation f (x) « c + x 0 9 and it is assumed that c >2/3 (which

holds^ for example,, for the limit cycle) ®

The functions fn(x) have a singularity in the neighborhood of the

point x = x-^o From system (2„2) it is not hard to discover the character of these singularities ? namely %

(2.5) f

n

(x)

[log(x - xj]n-1

(x - x-j-)

n®”l

Hence it follows that series (2®1) preserves its asymptotic character up to values of x satisfying the condition 0(x-^ - x) > O(log ) ®

In particular,, series (2®1) is an asymptotic series for

x = x^ - Here p will be of the order of unity® We shall utilize

this in what is to follow®

We exhibit the expansions of the first three functions f

n

(x) in

the neighborhood of x-^s
'
5

f

0 (x) * - (x^ ~ l)(x - x

1 )»x

1 (x - Xx

) 2 - “(x - X^-

fjCx)

X-, ^T"i

iog r^j +s

1

o 9 (x “ X-j j (x" - 1)^

(2.6)

Xi(x2-U) 2

X

1 “ 3x^ - 1

:(x - x.) +

s f 2 \ 3 1

6(x

1 - 1Y 9(x

1 - 1)

X - Xl)

f

2

( x) ZFP 108

l

1 “ I

1

1 + g(l + xj)

X “ x.i

<L / 2 \ 1 x

l^ xi “ ^

(1 + g)x? x x

1

(x| + 1)

^ 2 £^

(x? - 1)3 *

*r og

t

=x

i)

+

where

x

(2o7) g

1

x.2 - 1

V5(x^ - 2) , 3(x^ - 1) arc tg - | log

X

1VX1 V3(xJ - 2) X

1 - 3

We will not present here proof of the convergence of series (2ol)o

This proof is obtained from a consideration of solution (l e 2) by the

method of successive approximations:

Pn = *(1 “ )

0 9 0

S 2 "n+1

vC X

X

p.n

which converge in domain I, After this,, estimating the difference

pn+ i - p , we become convinced that it has the order V

(^n+l.

^ -whence

indeed follows the convergence of expansion (2ol) (at least the asymptotic convergence)

•
I
6

3° Solution for domain II. Domain II is the neighborhood of the

point p

= 0 3 x = a-^<, x = ”a

2

° For definiteness we will consider the

part of domain II corresponding to the neighborhood of the point p

= 0 5 x = a-^. We will introduce the variable q

~ -yp and seek x as a function of q, Equation (l e 2) is written in the form:

(3.1)

dx _ ^ q dq ’ V

2 q(x2 - 1) - x

The solution of this equation we will represent in the form:

co (3-2) x 7Cn

(q) v

" 2n

n=0

Substitution of this expression in equation (3°i) gives a recurrent system of equations for the determination of the functions ^(q) „

We obtain

Xo

s0 . X

Gt-i o 1

[q(a2 - 1) - * q » [q(a

l “ !) “ aih2 (1 - ^q&pa

(3.3)

n

[q(a

! ^ ~ altyn+l

^

Xk&i+l-k ” q

k—

1

(oo+

f$ + y~ n + 1 9 l^cc^n 9 O^/S^n 3 O^y^n)

The solution of this system is elementary*. For the first two

functions }£n

(q) we iia^e
,
7

ft/*) q + -^~log 1

2 4- 1

q

a, - 1L a-, - 1

X2 (q)

(a‘ - 1)'

,2 u f

*1 + 1

i (a. - 1) q q +

a

l^ al “

r 2 , al

+ 1

+ 2a

l^

2(a2 - l)q2

log[l“ q(a?“ l)/a1 ] 3at + 1

1“ q( aq- 1)/

a

1 2(a£-l)

P The functions )(n

(q) have singularities for q-»a^/(a^ -= 1) and for

q-»-co. Let us discover the character of these singularities. From

formulas (3»k) it is evident that, for Iq"5* a

]_/( a

q “ 1) the function Xq

has a singularity of the form log(i = u) 9 and ^ a singularity of the

form (1 ~ u) log(l - u) 3 where u - q(a^ - l)/aq» We then easily

obtain from system (3*3) that generally

(3.U) n

LOgU - uj

1 - u

n-1

Hence it follows that series (3 • 2

)

preserves its asymptotic charac

ter up to values of q satisfying the condition

0 q > 0

Analogously for large negative values of q we obtain Xt ~ Hs n

^2 ~ q and generally ~ q « Thus for negative values of q series

(3.2) preserves its asymptotic character up to values of q bounded by

the inequality 0(q) < 0(v ). In particular, asymptotic convergence

obtains for q

*= - i;(p - 1)

.
-V"
8

We will not adduce proof of the convergence of series (3»i)

either o It is not difficult to obtain this proof from the solution

of equation (3.1) by the method of successive approximations , setting

x

Q = ^

fdd _ 1. dq ' y

2

o

For linking the solutions obtained for domains I and II, the

constant must be determined in accordance with the assigned value

of the constant c in (2.3) or, what is the same things in accordance

with a given value of x-^. Since series (2d) converges asymptotically

up to the values of x for which p = 0(1) , and since it is for these

values that series (2.3) converges, we obtain the possibility of linking solutions (2.1) and (3.2). Putting p ~ 1 (q = ~v) in (2.1) and

(3.2) , we obtain two equations in the two unknowns x' and a-, t

00 00

(3.5) 1 = f

n

(x

:;_ )v“2n

n=o n-o

The solution reduces to finding x" from the first of equations

(3.5)

5 and next—by the x" found—finding, from the second equation of

(3.5) s which figures in the expressions for the functions

yn

(q)

.

Substituting for the functions fn

(x' ) their expressions (2.6), we

will find x' by the method of iterations. Wanting to obtain x’" with a

definite accuracy, we stop the process of iterations when subsequent

iterations do not alter the quantities of a given order of smallness

relative to V. Thus, for example, the functions cited in (2.6) are

sufficient for the computation of x" accurate to a quantity of the

order log2

v/v^. Accurate to quantities of the order of l/v ^ , performing three iterations, we obtain
'
9

x = X-, 1

V xn

xn

1)‘

(3.6)

1

x-, L(i - i) 2

l0g X

1

(X1

1)

v3 °v;

o

(3.9)

(3.7)

Proceeding in exactly the same way with the second equation of

3 we obtain for the expression

fa

(x:2 - l ) 2 2

V

2x-. ? —-s—slog(x^

(x^l) 2 1

1 ) 0(i) o

U« Solution for domain IIIo Domain III is defined by the interval of variation of the variables

a-^*=€>x>i+e ^ p<0 ^ e > 0 j

-ag + e < x < “3. - s 5 " p > 0 3 e > 0

Domain III has this essential significance for relaxation oscillations ^ that when the oscillatory system falls into this region^ it at

once passes ^ with a high degree of accuracy3 to steady-state autooscillations o We will dwell in detail on the obtaining of the solution

' in this region*,

First we will find a particular solution satisfying the condition

p-» 0 for x~> 0O 3 denoting this solution by P(x) <, This is precisely the
'
10

solution for whose expansion the solution of the second shortened

Van der Pol equation,, (IJ4.),, is the principal term* We therefore put

(Lul)

1^ P(x) ^ —- + Tf(x)

iKx: - 1 )

(we are considering that part of domain III for which p < 0)„ Then

for the function lT(x) we obtain the equation

(U°2) ff

8 (x) ^

(x

2 ~ lY + x

2 + 1

X x(x2 “ 1).

Tf(x)

x

2 » 1

v(x2 “ 1)‘

Regarding the right side of equation (lu2) as the free term5 we

reduce equation (H°2) to an integral equation;

(U.3) lx)

v3

F(x) + V

: x

=l

p

V

<:'k(x) er

V k ^W*d£

where

,oo w 2x - 1 V"'k(x) F(x; » v " e /k(!

'x (C - D

3

(U-U)

k(x) * -xf4 - x^ + log x + ^

2 ?

and k(l) = 0 S k(x) > 0 for x > 15 and k

! (x) - (x~ - 1) /x > 0 o

Moreover9 it is easily seen that F(x) = 0(1) „ Indeed, integrating (UoU) once by parts 5 for which we multiply and divide the integrand

by k

! (£) s we obtain

Editor’s note; Formulas (3°8) and (3°9) do not appear in the

original paper

„

11

CO FU) - I e-V

2 k(|) 6£t12|l^

d^

= 0(1)

(x -1)

X

and,, in addition,, since F(x) > 0 and the integral term in the last

expression is positive,, we have

(U.S) F(x) £ V-hf

(x

2 - 1)

U

Next performing an integration by parts in equation (h«3)

,

we

shall have reduced it to a non-linear integral equation^

fT(x) = TMF(x) + ^ £_irf2 (x)

Oi-6)

X

^ X

2 - l_y

2 k(x) / e-V

2 k(^)^2

+ _ ^2^2^^ T

~^e

00

Lastly,, replacing the sought function by the formula

Ow7) 'n'(x) - -—L—^x)

X~ - 1

we shall have for u>(x) the integral equation

(U*8) oj(x) = if(x) + Vo;^(x) + V

^ V

J x^

2 '-' .3i£in!ev

2k(x)

00

where

(U.9) f(x) . S-~F(x) .

12

We solve this equation by the method of successive approximations,

putting

4rf(x)

V-* (kolD)

u

n+l

(x)

1 ,, Z, s 3(x2 -1) 2

y

2k(x) f -»

2k(£L 2,^, - X + x + ^ / e 4a>n

(|?)d{; <, J

co

Denoting max |f(x) | by M and max |« | by ^ in the interval

1.+ e < x ^ oo, we shall have the estimate

LO )

2 _ M

1

“ “3

v3

© • • e

o z M rt 2 „3n2 | V

2 k(x)(x2 - l)

2

f -V

2

ktf), ,

J

^n+l p

+ v n

+ ^ £n max e —— / e 5 d£

' x

_ J/oo

2 2 2 Since £ /(£ “1) is a monotonically decreasing function <, we have

V

2k(x)Cx? "' 2

00

v

2k(n

x X

V

2k(x) (x

2 ~ l)2

.00

X X

'v k(4) k'(l)—2^—jd? < “

(£

2 - 1)

2 v

2

Accordingly

(mi) o M o x ^ q q2 ffi l

_ -j

, •••

, s n+1 " J3

+ 2*^n

Let us consider the series of relations

Y-, = M

1 v3*

y <_ 1l + 2vl2

n+l

v

.3 VI

n

13

This series is formed during the solution, by the method of iterations , of the equation Y = M/y 3 - 2UY2

, and converges , if this equation has real roots, to the lesser root. From the condition of the

reality of the roots, we obtain that the process of iterations con2

verges as long as 8M/V" < 1,

Since

f(x) <

(x2

4- l) X 1

2x(x2 - l) 3 < 8(x - l) 3

3

the condition 8M/v

2 < 1 can be represented in the form x - 1 > V

2 /3

<

2

On fulfillment of the condition 8M/V” < 1, we shall have

Y < n

2v‘ <

2M

and consequently, so much the more will < 2M/v , From the boundedness of ft promptly follows the convergence of the method of successive

approximations. Indeed, from (U.13) we shall have

max|a>n+ 2 - < U^Y max \co^ Cl) ~i

and consequently the series

converges if UvY < 1, i,e,, also for 8¥i/v Z < 1, and thus the successive

approximations converge uniformly to the solution of equation (U„8),

“2/3 given fulfillment of the condition x - 1 > V
'
Lastly,, noting that - oJ^ - 0(v ^n+3)

) f( x) ancj

each o^(x) are developable in asymptotic series in Vs x^e conclude

that the functions u>(x) , which means lf(x) too 5 are expansible in

asymptotic series in powers of 1/V

,

This asymptotic expansion is most simply obtained directly from

differential equation (1.2 ) 9 setting

oo (U.12)

which gives

P(x) « - - \ P

n

(x)y“'

2n

n-o

P

0 M - 2

X

X - 1

* 1

P-i(x)

p

p* o o _____ _ x{x + 1) x

2 - 1

"

(7TI?

(U.13) 0090000 * ooooo

n

P

n+l

(x) -

2 , X “l

P* (x)P (x) k

v n-k'

k=o

One is easily convinced that the functions P (x) for x-» 1 have a n singularity of the form. l/(x - l)^ n+^

5 and consequently series (Iul2)

preserves its asymptotic character under the condition 0(x - 1) > 0(y

We note that for an approach to the boundary of convergence P(x) will

-1/3 have the order of V

' .

Let us pass on to finding the solution in accordance with the

assigned initial conditions.

In the preceding we had the possibility of obtaining the solution up to values of p falling short of attaining the line

p

s -I)] by a quantity of the order of log V/v . In particu

lar, we could obtain the solution for
mm
15

1 - al

V a?*1

where x differs from by a quantity of the ordgr of log v/v .

Thus we have to construct in domain III a solution satisfying the

condition pc = pr for x - x «, where

o o s

X,

» 0(i) > V(x2 - 1) V

2

We will seek the solution in the form

(U.Hi) p(x) = P(x) + cr(x) i

\ then for o(x) we obtain the equation

(Uol5) P(x) o' (x) + [P(x) + y(x

2 - l)]cr(x) ^ ~(j(x)cr’(x)

which,, analogously to what was done for f?(x) 3 we reduce to the integral

equation

<**) - n^r°-^K) v f

r -c., 2 x 2 -V m(x)

+ 2^x7'- "

°oe

(U.16)

+

V3 ~V2m(x)

2n(x)

.x y

2

m(^)^^o2

(i) dt n

where we have used the notation

or - cr(x ) =

o o

„c_ V2

(U.17) m(x) =

S n(x) = -vP(x) 9
'

,
16

From expansion (U.12) it follows that

(U.18) m(x) = 0(1) >0 , n(x) - 0(1) > 0

In exactly the same way5 from the initial conditions for p one

may consider c = 0(1) (or even less). Mow put

(U.19)

2

cr(x) = 2n(x)e” V m^s(x) |

we obtain for s(x) the equation

(U.20)

s(x) = ~0(x) + vs

2 (x)e~

V

n

2 (x) e-v

2

m( ? ) n(g)(S

2 _ 1)s2(£ )d£

X

where

0(x)

n(xx )c - c fv

o2n

? (x)

We will obtain the solution of this equation^ again by the method

of successive approximations^ setting

sl(x) = h#(x)

(U.21) o o o o o

sn+1(x) = + ye'^a<' y '' s^(y) + v

3

n

2

(x) J

e-v2m(^)n(S)(|2 -l)s2 ^)d^

Having denoted by 2 nthe maximum of the modulus of sti (x) in the

n'

interval (1 + e 5 x ) a by M the maximum of the modulus of (Zf(x), and
'

.
17

taken into account the fact that n(x) is a decreasing function of x,

we obtain the estimate

ElM

An+X ^2 + 2v

From this, by a method completely analogous to the preceding, we

shall prove the convergence of the successive approximations under the

2 condition 8M/v < 1. Here we have |s(x)| < 2M/y ,

Turning to the expression for 0(x), one can write the last ine»~

quality in form

(k°22) |s(x)| < p C — , or \&(x) < o V n(x)

This result can be formulated as follows, Whatever be the initial

values or origin of motion of the oscillatory system, on falling into

domain III its motion approaches the limit cycle with an accuracy to

_2 2 quantities of the order of V "exp(=p m), where m > 0 is of the order

of unity.

Since the basic solution P(x) can be obtained accurate to quantities of the order of 1/V n

, where n is any number, the order of smallness of the correction d(x) to this solution is less than any remainder

term in the series for P(x) and therefore it makes no sense to seek an

asymptotic expression for <y(x) e

5, Solution In domain IV, This domain is defined as follows?

l-g<x<i+g 3 p<0 s . e> 0 |

“l-€<x:<“i+e s p>0 , @ > 0

18

We have already seen that for an approach to the boundary of

domain III 5 which is defined by the condition (x - 1) > the

—l/3 order of p(x) approaches 0 {\) ~ 0

We therefore introduce the following change of variables;

(5.D P

= -v-^AHu) 3 u - y2/3 (x - 1)

(we are considering the first part of domain IV). In the new variables

equation (1.2) will take the form

(5.2) € “ - 1 - v" 2/3(u

2

Q - u)

We will seek the solution in the form of the series

00 (5.3) Q(u) Qn

U)v‘ -2/3 n

n-o

Substitution of (5*3) in equation (5«2) gives for the functions

Q (u) the recurrent system;

dQ

(a) Q

-5-2 - 2uQ +1-0 o du o

, . . dQi

Qi 2

(b) Qo-ar ~ r

u Q0 - u

(S ’ U)

C.)

%

Idu

^n+1 ^n+1 2^

?o1T - — ' U Qn

o

OOO0OO

n

dQn+l-k

Qk"W“

k-1

19

As regards the initial conditions for Qn

(u) 3 they are determined

subsequently by the condition of the linkage of the solution with the

solution in domain III 0

Let us find the solution of equation (5°U 5 a) „ Setting Q

du

o " dr5 after the substitution and integration we obtain in succession

(5 . 5 )

,2 d u

dt

+ 1 - 0

dr

du

dr u + r - 0

(we take the constant of integration as equal to zero 3 in view of the

arbitrariness in the choice of the variable r) <, Equation ( 5 . 5 ) is the

Riqgati equation „ It reduces to the equation

(5.6)

d

2

,v dr2

rv « o u

1 <±V

'

v dr'.

the general solution of which is

(5.7) v - 1/rl c1

K

1/3 (2/3 df3/2

) + c

2

IlA (2/3 r

3/2

)J 2 1/3

du. To link it with the solution F(x) ,, we must require that Q0 = ^ —> 0

as u oo, i«e. 5 from ( 5 * 5 ) we obtain that u oo as r =» oo 0 This

condition is satisfied only by the solution

(5.8) v = c

lV%1/3 (2/3 n?/2

)

Utilizing the familiar relations for the Bessel functions

k'(x) = K -.(x) + K (x)n/x K (x) = K (x)

we obtain for u the solution
.
20

(5o?) 2 ) u = V?k2/3 (2/'3 r3/2 )/K

1/3 (2/3 r3/2

)

For negative values of ^ formula (5«9) is more convenientlyrepresented in the form * sf)

U =Vf-{j.2/3

(2/3rj/2

)

( 5 . 10 ) J 2A£2/3 r

3/2 ) i/{J V3(2/3 r

3/

2

) * J_

1/3

(2/3 r/2

)}

2 According to (5«5) 5 the quantity Qo - u - ^ and,, using asymptotic expansions for the Bessel functions $ the asymptotic expansion

for Q(u) is easily obtained,, We have

(5 oil) 11 -/F(i + r

-3/2

u -

r

2Vr

4* o o o 2u

The more complete asymptotic expansion is more simply obtained

directly from ($ek 9 a) „ We have

(5.12) Q0 (u) - & -

jJj

* s-gl3 ” -

For negative values of the denominator in expression (5 „ 10

)

has sjeroa ® Denote by oc the least root of the equation

Jl/3

(2/3 r{/2

) + J„

1/3 (2/3 ^3/2

) . 0

2 ) 2/3' read K

2fy

Editor’s notes For K‘
'
21

Then u-» -oo as a(T-, < ac) « Moreover,, according to (5»6) ? u « or u = v”1”-, and therefore u has for r1

- oca. simple

pole with residue equal to unity., and accordingly

u = ^ + a holomorphic function

Hence ^ = a + - + • • ® 5 and consequently as u-> -oo we have

p i

Q s u + tt»+“+ ,,o ° o u

A more detailed computation gives

p

/r^-j-vN ( \ 2 1 OL 1 1 OU 7 OL 0 . 13 ) «,(*>-« + - +3 -

5 --5 --7 + -

?

+ _

?

1 2 5 1 -oo-"-

U a

7

Let us pass on to the determination of Q-^(u), The general solution

of (5«U ? b) will be

(5 oil;) Q^u) =

uj

„u

c + / A(u) [u* u

Q

du

oJ J

A(u) = exp

p

u

1 bu

Q

2

o oy

For linking with the solution P(x) we have to require that the

quantity V ( u) be bounded for u - 0(v e

)»

Utilizing asymptotic expansion (5«12) s we easily obtain for the

constant of integration c the value

du o p

00

/

A(u) 2 u U -Q1 Q L oj

Hence finally
'
22

00

(5. 15) Q

x

(u) A(u) u^ |du u

For Q-^ we have the expansion

for uTM» oo

(5.16)

ior U“> “Co

n f \ 13 -u OL fl b

0 \ 1

Ql

(u) - fP * b

o - -j -

+ yj-j

1 2. \ 1

+ ?

°J 7

(5.17)

+ o • | log|u| (l - + -2%

3

'v 3u3 Su5

where

oo (S.18) b

o

" Av"CoT

A(u) u 1 u"

5 2

-CO 3 Q

o

u 1 / 2 ah

3 7^i + ^o

log

t

+J

J

The solutions for the rest of the functions will have the form

oo ^ n

(5.19) Qn+1(u) = A(u)

'u

„ dQn+l“k 2^

Q

k^_ - u Qn

'k-1

du

Q^o

where the constants of integration are determined by the condition of

p / p the boundedness of the quantities V ' Q n

( u) f°r u - 0(v e

) „

The asymptotic expressions for the functions Qn

(u) will bes

du

for u-» +oo

23

Q

2 (u)

fr)

for U“> “oo

Q

2

(u) = |u * + & * 0

(5.21) 2 2b

u +

Generally9 the principal singularity of the functions Q (u) for

These results show that series 0?„3) preserves its asymptotic

i.e., for values of x satisfying the condition 0(x - 1) < 0(1) , and

thus the domains in which solutions (5>»3) and (Ud2) are suitable

overlap. As we shall see 5 the same thing holds for solutions (5°3)

and (2d).

6. Linking the solutions for domains I and IV. Let us return

to the first part of domain I. We shall have to link solution (2d)

with solution (5 .3), in which by u should be understood the quantity

u-£- -oo s beginning with n = 2, s will be Qn ^ u

character up to values of u bounded by the condition 0(u) < 0(y

2^) 5

-V2/^(l + x)

p

= + V

and one sets
o£
2h

We note at the outset that since p > 0 for x = =1 (u = 0) $ the

constant c in formula (2„3) must be greater than 2/3j (for c = 2/3

we shall have f

o (-l) = 0). Set c = 2/3 + y*, The order of magnitude

of ycan be at once determined; since p(-l) - 0(v

^^

) s yy will also

be of the order of p

^

and consequently y

- 0 ( )T4// ^

,

Up to what negative values of x is expansion (2,1) applicable?

Let us now discover this B

Consider the case x^hen c = 2/3 <> Increasing the constant c leads

only to improving the convergenceo For c - 2/3 we have

f

Q

(x) = |+ x - = i(x + 1)

2

(2 - x)

xdx

f

o

1

X + 1 | log(x + 1) + |

log o

From system of equations (2 0 2) it is now easy to find that in the

neighborhood of x = -1 the principal singularity of the functions f^(x)

has the form fn(x) ~ ( x + 1)

2

3 and accordingly series (2d)

preserves its asymptotic character up to values of x satisfying the

“1/3 condition 0(x + 1) > 0(y '”) [which corresponds to the values ~u > 0(1)] 3

and thus the domains in which expansions (5°3) and (2d) are suitable

overlap.

In particular ^ the asymptotic convergence of expansions (2,3) and

(3=3) is guaranteed for x = -1 + V~^J

) 3 and thus the constant of

integration c can be determined by equating., for x = d + V

values of p obtained from formulas (5°3) and (2d.)®

the
v
25

( 6 . 1 )

CO CO -1/3 V v-2“/3 V-v-l/3) - V yv\(-l + v-V3} n=o n=o

We will not exhibit here the quite cumbersome but elementarycalculations. Using the expansions of the functions Q (u) for large

negative values of u [formulas (5 <> 13 )* (£.17)* (5.21)] we obtain for

the left side of equation (6.1) the expression

p(_l + y V3) - - i + cev”1/3- v' 2/ 3 + b

Q

V“

( 6 . 2 ) + 2- _ 2 v V3

3 9

2 log y A, 1

9 v

1 + 3V

On the other hand,, the expansions of the functions f (x) in the

neighborhood of x * -1 give for the right side of equation (6.1) the

expressions

P(“l +

2

3

log

(6.3)

2j*/3

O O O

(the cited terms of the expansions are sufficient for the determination

of y accurate to quantities of the order of y~®/6

) o

Equating expressions (6.2) and (6.3)* we obtain the equation for

the determination of 3/. Thus accurate to quantities of the order of

V we obtain
%

.
26

(6oIt) T'- ayh/3 b log v "Tv ' i log OtiT8/3

)

7

«

Determination of the amplitude of the steady-state autooscillations . After the determination of the constant c - 2/3 +

the root x-^ of the equation f

Q

(x^) - 0 is easily computed. The solution of this cubic equation can be represented in the form

^ . x1

2 2

27* + o o o

and substitution of the value of ^

gives

(7 ol) x

1

« 2 + fv

4*73 - + ~ 6 - 1 -

|

log ijv2 + 0(v-6/3

) * V J

,3) after which equation (3°9) allows'5

' the amplitude of the autooscillations to be founds the computations yielding

2 + ^v h' /3 - ~

27 . 2

(7 o2) + “(3b

0 - 1 + 2 log 2 - 8 log 3)V

2 + 0(V“8'/3

)

8. Determination of the period of the auto-oscillations » The

period of the auto-oscillations is computed by the formula,

a dx

pT£7

•

3) Editor's notes Apparently equation (3»7) is meant
'

.
27

We will here limit ourselves to an indication only of the method

of calculating the period,, without adducing the calculations 0

We subdivide the whole interval of integration into five parts 5 corresponding to the different domains %

1) from -a to -x^ over domain II (this part of the integral we

will designate as ) ; here x^ is the value of x obtained by formula

(3o2) for a value of q equals for instance^ to (1 -

y"U/3

) a/(

a

3

== 1)°

2) from -Xg to “(1 + V

'L/3

) over domain III (this part of the

integral we will designate by the letter T^)j

3) from •=(! + v ) to -(1 = v~1/3

) over domain IV (this part

of the integral we will designate by the letter T^);

U) from “(1 = V ) to x"

5

' over domain I (this part of the integral is designated by the letter T-Jj here x is determined by formula

(3.6);

5) from x" to a over domain II (this part of the integral is

designated by the letter Tjpj

The complete period T will then equal.

(8.2) T - 2(T1

+ Tg + + T

3 + T^) .

In each of the integrals we substitute for p the respective expansions (in domain II replacing the variable of integration by q and in

domain IV by u) . Using the estimates of the singularities of the functions set forth for each region,, we determine without difficulty the

number of terms of the developments needed to determine the period T

to the assigned accuracy,, after which the calculation reduces to the

computation of the integrals.

28

We will set forth the results of the calculations accurate to

•1 quantities of the order of y inclusively?

^ o

P

-l+V-3/3

(8.3) Tj_ = ~ * >"2/3

+ ^

- (l “ | log iV'

1

+ 0(^/3)

(8J4) T" = f ^

i2£Ji

+ |/l + log 3 + |

log

2) v”1

+ J o

(8*5) T

1

- T

1

+ T

1

» v"2/3

+ I + flog 3 - + 0 (v^/3

) 9 9

(8.6) « i + “ log | V

1

+ (v"^3

)

(8.7) T**

^ i°£JL + 0(v“^/3 log V)

3 ~ 2 log 2 )v - v1/3 + 1 + /> .

y-l/3

+ 7_ r2/3

3 2 10

(808)

18 y - (>0 + E

+ rloe 3V1

+ V3

'

iVl/3 _ H.-2/3

(8.9)

[ » //3 - i + |oc.+ ji |v -V j - ±L V 4 3 I 4 I 10

1 l0Ul + j v"

1

+ 0 ( V -V3)

where

m

(8.10) d It , 1

2 l0g OL " b

o “ U

+

00

u V

5°

u

2u +< du + Qo - i -f-W

29

Accordingly for the complete period lire obtain

T « (3 - 2 log 2)v + 3cxv

X/3 - ^

9 V

( 8 . 11)

+ n log 2 - log 3 - i + ” 2d j y”1

+ 0(y“k/3

)

¥e exhibit the numerical.. values of the coefficients of the basic

formulas:

oc = 2.338107 , b

Q

- 0.1723 , d = O.U889

p(0) « |v+ 2

„

338107 V”

1'/3 -

|

iSIUi _ l e 0980v“1

+ 0(v~3/3

)

a « 2 + 0.77 9369 V"14 /3 - ~y - 0.8762 v" 2 + 0(v”8/3

)

T 1.613706v + 7cOl432y"1/3 - p -

, 7 V

+ Q.0087^"1

+ Q(y“^/3

)

.

We have performed here the asymptotic solution of the simplest

equation of non-linear oscillations, so that it would be possible to

carry through the computations to the end. But the device used here,

of introducing "connecting" regions, is not limited to this particular

case only, being applicable to a more general equation

“f - ^

(f> l) * 0(x) 0

under certain limitations, to be placed on the function if) (x 5 p) . The

connecting domains will be neighborhoods of the lines (x, p) = 0.
.

'

'
30

If at a nodal point of the line ^ (x,, p) =0 the expansion of the

function tp in a Taylor's series commences with the term

a(x - x

Q )p ,

where a is a numerical coefficient^ then the basic solution for domain

9° Example . In conclusion we will present an example of the

solution of the equation for V ~ 10 „ For this case the asymptotic

formulas exhibited give p(0) = 7°5UO a a = 2.0138,, T = 18„831o In

Fig. 2 is depicted the graph of the function p(x) 3 two terms of the

expansion having been taken for each domains

for domain I

IV will remain without modification,, since,, setting p

= ±V

^

3

Q in the

neighborhood of the nodal point ±(x - x = u? we shall obtain for

the principal term of the expansion the equation

Q Q

1 “ auQ + 0(x ) = 0

0 0 o o*

reducing to ($oh 9 a) by the simple substitution

Q0 = [20

2

(x

o )/a] 1//3

Q^ , u' - [a2

/U0(xo

)

for domain II

for domain III
i

-
31

-i po« +

vVi(x)

for domain IV

P v“'1/3Q

0

(u) + yQ1(u) o

The values of the functions Q (u) and Q (u) are set forth in

vU\-»v^ -o. .. _ J..„ i Table 1.

Table 1.

Table of the Functions Q

q

(u) and Q^(u)

u Q

o

(u) Q

1

(u) u Q

0

( u) Qx

(u)

-6.0 38.171*7 “73.031*3 0.0 1.018? 0.1869

”5 °o 27 .11*36 -1*2.581*8 0.2 0 .81*21* 0.211*9

-l*.o 18.0985 -22.1123 0.1* 0.7018 0.2310

-3.8 16.5269 -19.0382 0.6 0.5901* 0.2398

”3.6 15.03U2 ”16.2668 0.8 0.5023 0.21*1*5

-3«1* 13.6203 “13.7819 1.0 0.1*331 0.21*71

-3.2 12.281*8 “11.5673 i 0 nii O 0.3778 0.21*85

-3.0 11.0276 -9.6070 1 © Lj- 0.3331* 0.21*92

-2.8 9*81*81* -7 .881*7 1.6 0.2971* 0.21*95

-2o6 8.7U69 ”6.381*1* 1.8 0.2678 0.21*96

-2 .1* 7.7225 -5 .0898 2.0 0 0 21*32 0.21*97

-2,2 6.771*9 “3.981*9 2.2 0.2225 0.21*98

-2 .0 5.9032 -3.0532 2.1* 0.201*9 0.21*99

-1,8 5.1068 -2.2790 2.6 0.1897 0.21*99

-1.6 1*. 381*5 ”1.61*58 2.8 0.1766 0.2500

-1.1* 3.7351 ”1.1379 3.0 0.1652 0.2500

-1.2 3.1568 “0.7393 3.2 0.1551 0 .2500

-1.0 2.61*76 -0.1*31*1* 3.1* 0.11*61 \ 0.2500

“0 0 8 2.201*7 -0.2080 3.6 0.1381 0.2500

-0 .6 1.821*9 “0.01*57 3.8 0.1310 0.2500

-o.l* 1.501*1 +0 .0661* l*.o 0.121*5 0.2500

“0 0 2 1.2372 0.21*03

0.0 1.0187 0.1869
-

-
32

In Fig. 2 the computational points corresponding to the formulas

of domains I and III are depicted as circlets , and those corresponding

to the formulas for domains II and IV, as crosses.

Fig . 2

The scale for p in the section from -a to -1 has been increased

to ten times that of the reamining part of the curve.

Received at the editorial office

April 17, 19b7

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