ASYMPTOTIC SOLUTION OF VAN DER POL'S EQUATION

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• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP82-00039R000200020046-8
• Release Decision: RIPPUB
• Original Classification: R
• Document Page Count: 17
• Document Creation Date: December 22, 2016
• Document Release Date: May 15, 2012
• Sequence Number: 46
• Publication Date: May 15, 1952
• Content Type: REPORT

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A"YNIPTOTIC SOLUTION OF VAN DER POL' S EQUATION
'ri.klad.naya I''iatemata.ka i Mokhanika,
Voiumo XI, No 3 (4ay/Jun U7), pages 313-328
1~1o S COW: I''1ay/Jun 19L17.
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'`STAT
 I
. ,,
2? The Soluto~n the Donia .1.. Designating by al and a2 the values of x for
which p 0 (fox the 1irnit cycle we have al a2 = a , where a is the ampli-
tude of stationary self-excited oscillation), we define two parts of the domain
In the xp-phase plane the er~uation (1.1) is transformed in the form following:
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A$YMPma~IO so ur ioi o vi DDP PoL' S Q,UATION
A. ,A. Dorodnitsyf. (Moscow)
Priklad Mat i' Mekh Vol XI,
No 3 (May/3i L'7) , pp313-28.
1.' Settin u hg Prblemn. In the present article we consider the solution
of van der Po1~s equation
2x/t2 - n(1 - x2) , ax/at k x ? 0 (1.1)
for large values of the parameter n.
pp1 - n(l .-X2)p + x = o (p = dx/dt}
where the prime designates differentiation with respect to x.
The soluution of this equation possesses the character schematically represent-
ed in fi re 1 (for the limit cycle) F- wre l i~ +e
It is well known that in the domain I and in the domain I +I the solution of
equation (1,2) tends correspondingly to the solutions of abbreviated eqa-
Lions following:
pp - n(l - x2)p 0 (1.3)
n( 1-'.2)x-0 (1.L)
However, the domains in which these two limit solutions' are applicable do not
come in contact with each other and. therefore it is impossible to join these
solutions. It is not known how to select the constant of integration in equa-
tion (1,3) in order that during analytical extension of the solution into do-
main III this solution would pass over into that solution which tends to the
solution of the second equation (1.3).
In the present work we introduce two htconnectin& domains II and IV for which
we establish particular asymptotic solutions of equation 1.2) that differ
~
from the solutions of the n abbrev'iat ed" equations (1.3) and (l.Lr) The domains
I, II, III, IV intersect and thus we can find a complete solution for the en-
tire cycle with an accuracy up to magnitudes Of any order of smallness relative
to n.
I so that we have;
_ l+eLxLal- e e~O andp~0
- a2 $eLXL+1 e PG0 and e~0
Obviously it is sufficient to consider the solution in one of the parts of
domain I, for example in the first part. We seek the solution in the form;
--?~-1
p - n ;$x). n-2m (2?i)
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equat~,ng aoe~,cts with td,entice,l QWexn,,
.2) Sn d
ats
.
(h
1) into ~1
.
Lls ~,Tlg (2,
Sbtitut
; ,nl rot' He po~rc\
n Ws obtabi a recurrence system o f e~uation$
c:h
4
t w.ctr (2 u2~
~
singularities, namely: (x -x )J. I (x ' x: )' n~ ~~ pA-'c~~~' `
~
2x + Xl ?' arc a~~ --- - -- i
arctan ......--
I.
... t. IIM N ~.
~-1 .. ~
.11'..11 r I , (3x12 - i12) n y 1.2
2 ? , ~Ix1 ( )
designates the real (positive) root of the eQuation
.
.3 :: and ~.t a.s assumeci. that a , 2/ 3
the limit cycle, for example).
(~,-hich ho~.ds ~'or
in the neighborhood of the point
The functions x) Possess a singularity
ot se `
fn( m 2.2) it is easy to clarify the nature of nthe
x :, x , Pram the syste ( n-l
n-1 ~" hire ~s ()
2,
n xe.~r
v-~~
be ~ nat the parameter unutt
t to s
ea
n
here are m
(note; the nis asymptotic chara ter up
it
s
~, ,
Hence it follows that
t those values of x the atisesfyries (2,1) preserves
ing the conc~itilQn_ 0(x1 - x) ~ Q(1og `n/ n )
, o sot? , -,n
e ~." "
here are the parameter ttnuUJ (h ~ r~
note; the n % ~Arlet~r
/n).
In pe,rticular the sex 11 ~ u 11~~
ies (2.1) is an asymptotic series for X " i11. be used
, ~,nformation w
thin case p will be of the order of unity. This
In
later on
Expanding the first three functiGns fn(x) in the neighbor~.ood of xl we
w x 'x x1 ).~ f
Y
2 _ 1) (X xl) ~ x1(x 1) 2 _ ( ."
fi(x) = '~Xl
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.~ f ~ r
fifl , ,, fof wi M x. 1'-Z
T4k
l
a for the first two unction we have
Thu
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(x~
2'
1) " x
-
n
a
meant to be
e
~
J
. , ~ 2m (notes capital x,
+ ++chi!t, (3?.
m( ) the Greek letter )
m~0
;
SubstitUti0 of this expresaion into the eQcation (3?1) gives the recurren
system - fuactions gym(q) ? We get
stem of the equations for the aeterminata.on o the chi
y
The tion of this system is elementary. For the "first
sale
)/ciJ _3z__4_ /%
,... -~-
a- and
~ a~ Let us clarify' se singularities for q . f(a12 1) for
The functions x~( q) posse ~ h?se t~irgularit~cs, from the formulas
y' the . nature of ~ a
t for q. a1/(a1 1) the function Xl possesses f ahe
(3 g it is o?n the form log l ) nd and. possesses a singular Y
a Hence from the #ystem
singularity of th e u _ a,(a2 1)/
l
form (1 - u) ...log(]. " u), when
eneral
t i
n g
) it's ea~Y to obtain tha
x - \;:c!::1 '
m
-
I
Hence it follows that the series (3;2) preserves its asymptotic character up to
.
n
Uti
o
the con
lues of q that satisfy
va
0 ( a11( al2 ., 1) q.) , 0 (log n/ n2) (note; nnn is the parameter
- -
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This
the Gonvergenca of the 'series ~ 2.l).
We wi1~ conduct haxa a woof of thti~on of the o~,uti,an of (1.2) by the me
proof is abtained_fxam a considexa
n
ti
t
a
. f fexence (2
the Vii,
main I. After which, evaluating
whiwhich converges in the do differenco possesses the order ljn ~,nced that the m we are ry conv l the expansion (21) (at least the asymp-
~nce it :must be convezg?nt ~ name Y?
II .is the neighborhood of the
totic difference) ? 3, _1 S p Toints p pol. = 0, ~? f. ? a the D_o_ma-n I~ .
The domain
Let us consider for the sa~ce of, defini mess
22 to the ne~,ghborhaod of f point p d
_ ng function of q.
x as a
that p part of x- x- a ~
s into II correspandi and
Let us introduce tha vaxiable ~np , ? frm;ind.
in the follow:~ng ~
x ~ ~
~
o
ma
thod of ouecess~.ve appxox
2), ?,..?, p mil~, n(I, .. x~)
t n(l - ,
p
ie~xitten
+
M
s r The equation (l.2)
1
dx j d.q
~ ;
Let us represent the sol.ution of this equation in the followinY,, form
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in ~, M ~
to end in
S mi1a1Y fax large negative valU of o we obt a e ~;r~.e~ (~ ~) ~x serves
naxal X N qm ? ,TJius for uegat~.ve values ofted:'b the ~.nequa~.ity
o
its aaymptmt3.c character upr to the as values yr~ of pt4t~c licomnvergerica holds for o. '~ -n
~
q) 4: p(n') , In particula ,
(p= 1).
1 w~.ll not. be carried out here..
~ the 0ey?gence of the sexi es (3,) the e ua,ti on (3.1) by the
The ;proof o
This proof is easily to obtain from the solution of o
method of ` successive approximations, setting,
r
m
ed or the domains I and II it is neces?
In order to loin the s?lutions obtain of the constant
( ,1) constant a1 relative to a given value. ? Since the sec
sary to determin the elative to the value of _xl L Since
and rie?
c in (2.3) or, what is the same, ies
n values
we of x can lfoxoin which p solutions (2.1)
o gever ee asy~ptatically n p to converges, the
same values the series (2?3) ~ (3;2) p w 1 (q _ rn) , we obtain two eoua~
and (3,2) , Setting in (2.1) ~ and al
,
tions with two unknowns an
-2m , x* : Xm(- n) , n
l n f (x) ..2m (3.5)
m
rno
we rind. x* and ,`;
,e first. sauation of (3?5)
lu ~ i.nd a relative to the found -...~ ; a
tion is handled thus: from th '
The so
en from the second . equation of (3.5) we f ;F
V
then
which a enters 1nto the expcessions or the functions Xm( ~.) ?
f
.
for the f1J.nctions f (x) their e xessions (2,6) We will find
i q(
Subs () +O() 3(1t+
2- 4--.
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0o
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Let us xeturn to the
etion of the olutions for the LomaifS I and IV? o.; then
b, Con un
first .,dart o.~ the domain x', We must join the solution of (2,1) with the sa
eh for the ).ttex a it is necessary' to take the quantity
of (5,3), in whi
x) and set p +
n /3(1 .
.
-tant c
First o all we note that since p 0 or x (u : 0) then the const
in formula (2.3) must be greater than 2/3 (for c _ 2/3 we will have o 0
IF
note: tht's under. fled "gn stands for ganmma) . The _ 4 _ ~ ~ - l~3 then
Let us sat e 2/3 order
be determi. ed at once; since p(_1) .? Q(i/n ), n3 ;
_ 0 (l / ~a
of magnitude of g can
1 n113 and consequently we will have g
n will be of the order of ,n
Let us now clarify up to what negative values of x the esion (2.1) is applica-
,
ble. .. .:. constant c leads
g the
Let us coneider, tY~~e' case where c = `2/3 . Increasin i
only to improving the onvergence. For c =2/3 we have;
.. ~ ,. ~
fo(x) ,, 2/3#x-x f3 - (1j3)(x 1)(2 - x)
x. f
.. J xdx0 - _lf (x 1) - (2 3),1og(x 1) + (2/3)01 x/(-2) ?
f (x) .. ~ f ~
1 0
z1
From the system of equations (2.2) it x s easy now to fin that in the neighborhood of
x .. 1 the main a~.mg'ulari ty o f the functions f (x) has the form f( c) N(~"l)series 2.1) preserves it m.,symptotic character up to values
and consequently the. ( 1)?~, 0(n-~1-'), (which corresponds to the
of x sati sfyi.ng the condition Q(x
values u 0(1) ) and thus the domains in which the expensions (.3) and (2-1~
are applicable' intersect.
en_
is convergence of th expansiofs (2.3) and (5.3) is
meg
ra
In particular the asymp'~~
cured fox x - _ 1 f l/n ( ''~ - ~~~ ~ 3) _ 1 ~ and ~ tn1hu~3 the the constvaluesant of of ~. p
t ion -
c ' c an be determined, by equating for x
obtained from the formulas (5.3) and (2.1)
y=a
We shall not cand~ct here fairly laxge_scale computations but merely elemtary ones.
.~.
Using the expansions of the functions Q (u) for large negative values o we
5 for the left side. of the eauat~on (6.1
( fo rmulas ( 5 . 13) , ( ? obtain
1r') , ( . 21) )
.5
the expression; _ ~.
I w ,?
(+) + ?'-( 3 "
c h S 1
/'I ? ~ of the f1ctJons f (x) ~i:the ne~.ghborhoad of
On the other side, the expansions o
x .: _ 1 give for the right side of the equation (6.1 the exj ressian
p(-
(I+;; +j(it
he expansions are sufficient for the 7terminatian of
. ..3
(The
with terms of t ~. up to the magnitude of the order of 1/n )
wth ~.~ accu.ra y
Icy
Equating r and L6.3) we obtain the eguati.on for t~ determination
the aitreasions (6.2) of 1jn 13 we obtain;
u magnitude of the order
of , Thus with accuracy p to the
_ 11.~h/ 3 _ Ulo n 9n 4 (b0 - 1 - tog ~) n o(l/n8/3). . (6.)
3 f`'
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 mr
k fi ~,i 4a
70 Determination o thA~1i~ of the ea Wit to Seg. '.- ite__ O____- tat ,
After the cleterrnination of the constant c = 2/3 + its easy to compute the
root x of the equation (xl,) ? 0 The solution of this cubic equation can
~.
be represented in the fo1&owing form; a 2 2 2 3
and substitution of the value of gives;
..'
n
after which the eg:uation (3.9) permits one to find the amplitude of self-excited
osci11ationa. Computations. give;
L. / + 1 i L ~- j-+ 4 ( ?b A +2 fyyI - M W YI J) h I 4WD C / ( 1
1 .` V y
of the Period of Self-Excited. Oscillations, The period of self-
,
8, Determinationn .....J.,._.~
excited oscil.latio,ns is computed according to the following formula;
a
T w 2 dx f p(x) (note: this T is not the same as that
-a -used previously for maul) (8.1)
Let us limit ourselves merely to indicatig the method of computing the period
without actual conduction of calculations
Let us divide the entire interval of integration into five parts corresponding
to the various ;domains;
i) from pa to -x22 according to the domain 'II ('this part of the integral let
us designate by r~ ) ; here x2 is the value of x obtained ccording to formula
( 3, 2) fox values `of q equal, for example, to (1-ri 11 of ( a~-1)
r
to the domain III (this part of the
from -x to -(l i/fVj) . according L
2) fr
integral is designated. by the letter T3),
3)
4)
5)
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from -( i.. n'13 ) to -(1 -. l/nV3) according to the domain IV ( this part of
integral is designated by the letter T~ )
i - n_1/J) to xaccording to the riomain I ( this part of the
om -(
at
integral is designated by the letter T1 ); here x is determined, according
to forula (3.6).
from x'c to a according to the domain IT ( this part of the integral is designs;
ated by the letter T92 ),
The total period T then will equal T n (T1 .~ T'2.+ T"2 . T3 } TL1, ) (8,
In each of the intervals we substitute in lace of p the corresponding
expansions (in the domain II replacing the variable of integration by q and
in the domain IV by u ). Using the evaluations of the singu&.rities of the
functions carried out for each domain we easily determine the necessary number
of terms of the expansions for obtaining the period T with given accuracy,
after which the computations reduce to a computation of the integrals.
Let us conduct the results of computations with an accuracy up to the quantities
of the order 1/n inclusively;
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where
~,
M- ,
o I ;ft'( ( /O)
c - -
:: - 5 4 - o -2- + Q L4 L+
nsequently for the total period. of oscillation we obtain,
Co . _1
_ to n /9n 4 (3" log2 - 1083 r 1/6 b - 'fl
~ 8
22
pl/n ). ~
~,21ab2)n 4 3a/nu/3
ical values of the coefficients of ..main foxn~ulas;
Let us determine the numbr
_ ba 0.1723 d ? 5*
l~k/ a2.33810? ~ /3
,U 2.33 7l
L - C 1 (, i/ _ _) 01 1 2/ /3)
ii i11
2 ht (x)
d2~/ dt nU(x, /d) +
etion (x ,p) ;~) .The cennactin~ ~.omai-ns Will be the
r certain limitat~.ons on the f`un u
ction point of the line
f
n
o
nQighboxhoads o~ the lines ~(x, p) G If in ~ the in ja Taylor seiies begins with the
't
terrn a~x ~ the ~)p expa where nsion a of is the a psi cal: function then the main solution for
term numeri ilboxhood of the Pk
the domain IV remains without variation, since setting in the`neig.
juncture point
( r/3 (
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X13 io
solution of the simplest equation of non-
We have carried out here.: an asymptotic
But themethodused bare of inthat we might carried out to the end the calculat ous.
li.ne~.x osaa.llat'~.ons in ordex troducin~ rrconnecting~r aome,itls is not li.m~.tec~ to j
. be erg , loy`efor more general equation of the t;ypet
tl~i s partial case and can ~ ~
?~ J ,C rr
1u I~
U
}
..) v1..P-I-- 0 C v'
 11
h?
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-- --- - - - - ----
we obtin far the main term of the expanon thQ pqua,ti.on
1
u 'x ? 0 (note; nan here is a Latin letter)
which reduces to (5-+a) by the 'simple substitution
a 7 J (xo )
An Bxam le. In conclusion we give an example of soiuti'ons of the equatiox fox
9.
Ii 10. Pox' this case the asymptotic formulas deduced give
M a 2,0138(!'ts Latin) , T 18.831 . In fil;Ure 2 is the
r of the functa.on p(x) , in which two terms of the expansion for each domeih
g.
were taken in the computations;
for domain I; p(x)rnfo(x) .' l(x)/n
f o r rdo main II: x ^ a + X1(q)/n2 (not e "X" ii chi ).
ID .'P0(x)/n P1(x)/n3
N n~~ ~
for domain IV: p %(u) / ,
the functions %(u) and Q1(u) are given in table 1.
The yarlues of
s onding to the formulas of domains I and III figure ? the computed pain
In f is correp
as cihcular paonts and the points corresponding to the formulas of the
are drawn as
domains II and fl are represented by crosses.
The scale for p on the interval from ?a to ..1 is magnified. ten times in corn-
.. portion of the curve,
parison with the scale of the remaining p
Submitted 17 April 1927
for domain 111
(nglish_iafEage summary of the Original)
The paper presents the solution to equation a 2x/dt 2 ? n( 1- x2 ) dx/dt + x
The entire cycle of the oscillation in the coordinate system p,x (where &/t)
is divided into four parts (domains I, II, III, IV in figure l). An symptotic
series. in teams of powers of 1/n is established for each domain. For domain I the
epaflSiOfl 1 holds where the functions fm(x) are determined from the recurrent
holds for domain II , functions xm( q) (where X
system (2.2) The seise (3.~)
_nP) 1 ein determined f ronl system (3.U)
stands for chi ,. and :.. a ~' ? The seies (L.1Z}a
is vala.d for domain III Junctions P(x) bexnfdetermined from A1) and the seies
r m _ 2 _
hot. for domain IV, where functions Q(u) (where u n x 1
(S 3)
n I3 are determi.n?d from the system ( 5.1) ?
The intervals of the variable x for Which the different asymptotic series area
valid cover each other making it possible to sonnet the solutions & tained for
different d.omains (that is. to determine the constantsof 'integration entering
?
into the expressions for functions fm(x) and Xm(q) )
fi The asymptotic expressions for the amplitude of steady oscillation (formula
.. 'n f thriod of oscillation ( formula ( 8.1.2) may now be found.
ad.ore pe. 7?2 w) 3
i.j ; {) C*X-
t 2' _ end- (() - :'
.r _
X~
X2.
C
0C:-my) +-Xt
try
c+-~ t .M r
12, (
'ii
l
(+t
t
.v)[ r~v-~~ - .
J
(w-yo,L -'- G. c
`),/r I t () hi
X 1 ~ v: Ta(() ~-}
N/) m 7 ( a,~M. l t"
.,w
~c} +z&'
(?/ "'
K z
~. s Latin)
CH ) 1::D)
-. (-v)(~+.:!)
1 A" ~ '. 4 x'11 y
)
(7_l
Z~i-xl
[ Wte , _
(3)(M2)
.c) )C >1
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Gf I!~__~I ~~
? U. 0
3,V
Uif
0i2_q9 -a
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