Non-linear oscillations of third order systems

Proceedings of the
V IIIu
International
Conference on
Nonlinear
Oscillations
PRAG UE 1978
la tl itu t* of Thcrmom cchaoic*
Czechoslovak Acftdcmy
of
Scicact*
NON-LIN EAR O S C IL LATIO N S
OF THIR D ORDER SYSTEMS
Nguyen Van Dao
National Center o f Scientific Research
of Vietnam, Vien khoa hoc Vietnam - Hanoi
ABSTRACT In this work different kinds of non-linear oscillations :
f re e o s c i l l a t io n , s e lf- e x c ite d o s c il la t io n , f o r c e d o s c i l la t io n and p ara
m e tric one o f the dynamical system d e sc rib ed by t h ir d orde r d i f f e re n t ia l
equation (1) in the critical case are examined systematically by asymp
totic method. '
Introduction
A lo t o f mechanical and physical problems lead to study or o s c i l
la tio n in the 3ystem governed by the equation of third order
X + ax + bx + cx s > einvt f (x,if3c) . (1)
ns-N n
The following definition is accepted
a) If the characteristic equation + a 7Ỉ + b A + c = 0 has a nega
tive root A 1s.( and a pair of complex roots with negative real part
* 2 = - rị + i Q t ^ 2 = -rj-in ,then we have the non-crỉtical case.
b) If the characteristic equation has a negative root and a pair of

imaginary roots (f]sO) then we have the critical case.In addition,if Vr pQ/q
here p,q are integers,then we have the critical resonant case.
The non-critical case has been investigated in many publications
(see,for example [1-6] ) but the critical case has not,to the author#s
knowledge,been examined hitherto.In this work we atudy the non-linear
oscillations of the third order system (l) in the critical case.We shall
find the family of two parameters particular solution of equation (1)
which has strong stability property.To assess the validity of the theory
a series of experiments on the analog computer have been made.The theore
tical results show a good agreement with the analog computer ones.
§1. Non-linear Oscillations in third order autonomous Systems
I»et us consider the following equation
x# + f X + n 2x + ị a 2x s £R(x,x,x) • (1 .1)
We shall find a partial 'two parameters solution of this equation in form
X a G.COS <p + £ u . |( a , <p) + £2u ^ ( a , (Ọ) + . . . , ( 1 . 2 )
- 517 -
<p«nt *4/ .Here a find ^ rrc f u rr .ti o ’ip 3 fit Isf.yirp, tbc fo llo vin r, eq ua tio ns
£Y~ = £ A ^ ( a ) 4 €2A^(») + C^in) + e?B^(a) f . . . (1.3)
S u b s t i tu t in g (1.>),(!.3) in to Í1.1) Zi\r\ com p^rinc the term s co r-
tal.nỉn r £ rrit1 r ircp , co~vp ’.ve jrt
___
(1.4)
ca'(42+Q2)' ’ “ 1 ?n,-’( ỉ ?+ a 2 )
' - _ ^ 1 Tj 1 D — -‘ 1

— J

I
■|= " n ( |a+n*) ’ ?ai,(^ +
iG m i h rr- + , n f t -'rr .

u 1 m ? " . . ,2 ? ? , v 1n p 0 0 3
A (1-n-Ml +mn ) n.2 (1-T,? ) ( | ?H tn V )
w he re r ,h , u„_ , V , r r t h e r o f f f i c i m t r . o f ĩ>i»r.iế>r CVỈV n r:‘or:- :
*~T ’ m ’ In till ^
R(acoscf, - i la s in < f , - il'«'CCi<p)s > H h Pinn <f ) , ( Ì . ^ )
u. » — (U- oosrrcP H V. ::1nn<P).
I rr. in ^ * n
Fo r th e P u ffi n; c ire R a ~ f i y'^ ,ve rpve
ậ ạ - r.? * 3 E /3 ^ Í,)
dt = 2 f ’ fli ? a p 1 ’ p = — -ji— .
r 4 ( 1 + n 2 )
I f fi > 0 the 'imp] i t uric of OGCÍ J.lrt.1 on in c re - if £ ir. tlxne.Tf 6 < n th(r
"jppljtuf’e Jec ^er.sea.
For the Van der Tol c ise ii =. (1- x ^ )A th e eq u a tio n s of the first
approximations are
t 3 «
X * a c o s <p -*■

3“

R—( 300 3 ? ^ - i s i n ? ( p ) , (1*7)
32( I n
4 a
-

— £ ( « . £ ) - .Q -

- —7“ (1 - T ) . ( 1 . p )
dt 2 ( I2 + a‘) « dt V(|“ +Q2) 4
The stationory amplitude is p- s 2.The oncillation d1f.0rnms on ftiialoi; -

computer are presented on fi£. ! for ( a A s1 , 6 s ' .1.
$2. N o n -lin ea r O sc illa tio n s in t'lir d o r d e r non-cT.utononous Systems
Now v:e study the osci 11/itio n s of the fjyiitem governed by equation
X -»• fc * + n ? x + Ị a ?X s t.\ (x f5:,x) + iP c i m ft , (2 .1 )
n = s f €6~ .
The p a r t i'l l pe rio d ic so lu tio n w ith p e rjo d 27C /S of equ atio n (2. 1 )
is found in the form
X « r.co s ( * t+ tị») + gu-ti*, + , i t ) + e2u ^ (a , tị/ , S t ) , (2.2)
v/he>*e u .(a,4» t i t ) ir e period ic fu n ction s w ith period 21T rela tiv e ly
end 'St rnd n , Ỷ fire determ ined from th e e q u atio n s
~ * e A1 ( , 4/ ) + £2 A2 (p, 4>) + , e 3, (p., 4») + £ 2fl2 (a , <|> (2.3 J
Ô.V substituting (2.2), (2. 3) into (2.1) and comparing the coeffici-
on tr of £ '.rid f l i n cp , COS cp we o b tain a f t e r sim ple c a ] c u l G . t i . 0 n 3 :
r ^ s i n f - i c o s ỷ ) - » r n d L r 21 t B H i a * < 0 0 « l l z i r , 1± í ĩ 21 J ( 2 . 4)
? ff ( i + 1 ) 2 1 a ( i + v )
u _ lllm -'n<r2m v1m- mifr1m 4 * r2m
I Hi 2 >> " o 0 o o 1m— o 9 9 o o •
* ( 1-1 ) ( | V r ) f (l-n rM ị^ + n r r )
here r i 1 fr 0., u . tV- re i-o u rie r c o e f f i c ie n t s :
I I 21 I m 1171 00
n u , 8 - x - f r x = y ( r 1 n c o s n < p + r ? n s i n n < f ) , (2.5
n s c
I ^ s 2 _ ^ ( u 1;noos>n<? + v ^ n i n n u p ) .
-'he stn tto n rw y s o lu tio n o f e quation s (2 . 3) in th e f i r s t approxima
tio n Ì3
f? 2 .2
7 * r ii + rh - - •
519 -
+ ir21) ] > , > 0 . (2.6)
For the Duffing case Rz-px^ v/e have the fol]ov;in£ equation of

resonance curve
The stability condition of the stationary solution ia of form
2 I 4 2 -4- / ft^ 4 * „ 3£p
V = 1 + ± £ a2 ± %/ i - jL a4 ,Pr — * p =

0 ~r~ ' (2-7;
a 2 r V * n 2 a ( fc2+ a ) r M iW )
and the neccescry condition fo r the s t a b i li ty of solution is 0.
Figure 2 is presented for fr1U ,fl*1, p*x-10 ^ and (curve 1) ,
p*= 1 ,2 5 .10“k (curve 2 ).The fat line s correspond to the stable state of
o s cilla tio n .
For the Van der Pol case R ss(1-x ^)x the equation of resonance
curve is
2 ea(A -n ± eỉ
V = 1 +

- J -
This relation is plotted in fig.3 for f £s0.1 cind J*0 (point
A*1 , v 2= 1) » Ja4/81 (curve 1), Ja4/27 (curve 2 ),J s8/ 27 (curve 3 ).
The unstable branches are presented by hatching.
ự
ị -u -1 )2
Js
c)
( | 2+ a2)p2
4 a 2 I 2
(2 . 8 )
- 520 -
\
The process of frequency entrainment has been in v es tig ate d and

presented gra p h ica lly in f i g . 4 .This figure shows the in ter a ctio n b et
ween the forced and s e lf- e x c ite d o s c il la ti o n s in the n e a r resonance
zone.For exam ple, observing the o s c il la t io n diagram v/hen increasing the
excitdrv. frequency Í we see f ir s t the b e at.In the in te rval
the brat dioaope^.rs :\nd there i s only the harmonic o s c ill a tio n v/ith
e'v citin ~ fre quen c y . In c re a s in g more y we se e th a t th e beat ap pea rs
r.~ain.As i t i s Sfcf.n in the figure 4 the freauency entrainment zone ỈS
n rrro p r th e r thr.t in th e second o rd e r system Ịlừ Ị .T his phenomenon
war: observed on nnalo''-computer too.
fig. 4
r
3.Non-autonomous Systems (continued)
Parametric Ocnillotion of third order non-linear System
? h e f o l l o w i n g e q u a t io n
i X - Q2X + l a ' x + e [ k x 3+ h 7 ? + R (x ,s ,x ) - CXC03tft]sO , (3.1)
£A * n 2(1- f|2 ) , r ỵ st/ĩa hns been investigated.The partial two parameters
so lu tio n o f (3 .1 ) is found ỉn th e s e r i e s
X = COof | t + i | o + e u , {n9 | t ) + «2u2 (n, 4>,|t) + . . . (3.2)
- 521 -
ẫ t : 7 ^ [ 8 (k- ỉ- o c o . ^ - §fl:,in?+H n.3 , (%3)
& ■ Ĩ T ĩ r ^ - [ í (í 2 ^ 2jA“ 4 f ,« l:+ Q -V )r .^ r ,ir .2* - g | c o c ? Ỷ +R ,],
H1 » <K0C0!5cf> + | l <E0::in4>> , r?2» ự < v * 0= * > - < V si” *F> •
HQ s H(acor. «f , - «^o.Rin<p ^ .^C03<f ).
Tr. ri 5 the a'Tip'J' tilde curve '»£: pi" tic.*? for the n,*.iu 2 0 , is il « 1 j
c«O.C5 1 ks-0.1 iinri h sC (curve 1 ) , ‘'l * • r> (c.irvG 2) /’.rid
h * c .
1
* * o o o *
(curve 3) 1 here k s e ’ /n , h s eh /a; , c* C c/si .■
v/h*re a, Ỷ satisfy

The influence
oulo^b 'Tri.ct.ion
1 r.i;~nx on O'-«’%rv“i?t?,ic
i ỉ . i ; i J J. .'.uv:.V/V * */ -J u .L u ' u I J J. v> 1 « u 11 u 1
It’ti c n was stu d ie d , in th i c c tco th e reso nance curve hiư c 1 rre rorr»
S e e r : l £ j . b f o r t * f l s 1 , h s 0 . l 5 , k s - 0 , 1 ,c = 0 . ' o .'VV? ! j* a 2 ,5.
(curv e l),h*s 5. 10*; (curve 2)ty*sZ\/t! •
a
0.3
ft 2
Ỡ./
f i g . 6
Ỡ.M
I.oz
72
- 522 -