N U M E R I C A L S T U D Y FOR P A R A M E T R I C E X C I T A T I O N
O F D I F F E R E N T I A L E Q U A T I O N N E A R A 4 - R E S O N A N C E
M. BELHAQ
Lab. M6canique, Facult6 des 6ciencas I , BP 5 3 6 6 Maarif, Casablanca, Morocco.
(Received 24 October 89; accepted for print 28 February 1990)
INTRODUCTION
In this paper, we will study the Polnc, ar6-Hopf, bifur~tiOh of .a class of Llenard forced differential equations in IR 2 , which coefficients vary periodically in time:
{<=y
= F ( x . y , I J ) + O ( x , t ) , I J • IR 2
( i )
where F and G are supposed sufficiently regular with r~pect to their arguments, angwhere g is T-periodlc in time. Many mechanical systems are governed by such equations (see [ I O] ).
For an autonomous I R 2 differential equation, the Polncari~-Hopf blfure~lon has already been studied by Poincar~ [9] and Hopf [ 7 ] , and we know a umque periodical and asymptotically stable solution ( Polncar~ limit cycle) can be found.
For periodical coefficient systems ( 1 ), research of stationary solutions is make by using the stroboscopical method [8] which associates a Poincar6 map :
: (x,y) ~ (g (x,y,T), f (x,y,T)) ( 2 )
where g( . ) and f( . ) are the values at (t + T) of solution of ( 1 ) initialized by ( x ( t ) , y ( t ) ) , T being the coefficients' period. We can therefore study the properties of the differential system ( I ) by using lnvariant solutions of ~ . A fixed point p of ( 2 ) (corresponding to a T-periodical solution of ( 1 )) is said to be resonant for I~-I~ if D ~ I ~ , p) has complex eigenvalues X ( I ~ ) a n d - ~ ( l ~ ) such that ~,k (1~) = 1, k • IN. In the p -parameter space E, such e point is noted Pk.
Analytical results ( see [ 1 ]) concerning the study.of, r.asonant.cases for k = 1,2,3 show that a unique k-periodic points and one closed lnvariant curve can exist around Pk.
199
For resonant cases with k >_ 5, Arnold [ I] demonstrates, in the i=- parameter space E, the existence of horn shaped regions ,J,',',',',',',','~r k whose boundaries are the saddle-node bifurcation locus of k-periodic points, the existence of a "resonance circle" ( Fig. I ) in phase portrait of J,~, for values I~ • ,,,hf" k , can also be proved.
For strong resonant case k = `i, known results are inpomplete : inde#d, It is Known sin.ce Iooss and Joseph [ 1 1 ] that for k ='t, in i~-parameter space E, there is a horn, ~ 4, in which there exists a pair of ,t-periodic points of ,.~'. The boundaries ~ 1 and " ~ 2 of this horn, are the saddle-node dlfurcation locus of 4-periodic points produced at P4 •
W a n [2] demonstrates that In the region of E where the bifurcation does not create a
`i-periodic points, a closed invariant c u r v e generally appear ( Fig. 2).
There is no ex]stenca theorem to orove the coexlstence of 4-oeriedlc ooints and a closed"
invariant curve in horn =¢'~f~. Such possibility is described in a possible bifurcation scenario proposed by Arnold [ I ].
The aim of this paper Is a quantitative numerical study of such a coexistence for a parametrically Lienard forced equation
}=y
= -(cx + px)y - ~ ( I + hcos((~t))~ * C2 X2
C3)
where I~ : (c = exp (-etlII) ,~,~1}), h • IR, I~" .it, ~ ,. 2, c 2 = I, (c, ~ 0 ) are the system's natural parameters for h=O.
To do so, a strategy In calculating lnvarlant solutions for I~ • ~ ls developed. Indeed, presence of several solutions around P4 needs a particular numerical treatment if we want the lteratlve algorithms to converge.
This explains perhaps w h y no results are to be found in precedent studies.
t'2
CONSTRUCTI VE ALGORITHM5
In the parameter space ( c, gO ) ,the locus defined by c = I ( (z =0) represents the Poincar(~,- Hopf bifurcation curve ~ ' of fixed point p ( 0 . 0 ) of map ~ (Fig.4).
Construction of P,~ resonant Doint
This point is found by solving by Newton's method the following system:
g(x,y,(~,(~O) - x = 0 f(x,y,(~,(~)) - y = 0 d e t ( D - q ' ) - 1 = 0
2 ~ tr (D 3-) - 2 c o s - - = 0
4
where the first two equations are those of the fixed point of represents the condition
I~,I =I that is
ag )f _~g ~ f - I = 0
)x ,)y ~y ~x
and the fourth represents the condition 2~
arg(~.) = - - 4 that is
( 4 )
, the third equation
~f 2 ~
~'g + - 2 c o s 0
ax ay 4
Resolution of ( 4 ) by Newton's method requires, at each iteration, the knowledge of 24 functions which take part in newton's formulas. These functions are calculated by integration between to=O and t= ~, using Runge-Kutta method, of the 24 equations resulting from derivation of ( 3 ) with respect to x O, YO, c~ a n d ~ [3]
To start newton's method in resolution of ( 4), we notice that x = O , y=O, c = 1 and eo =.5 is a solution of the unperturbed problem.
We thus obtain, for h=. 1, resonant point coordinates (Fig.4) : c=1 , ~0=.499896.
Construction of saddle-node locus of 4-periodic points
This locus is found by solving by Newton's method with unknown values (x,y, ~) on the following system,,
g4(x.y,~) - x --- O,
f4(x,y,oI) - y = O, ( 5 )
I - t r ( D , . ~ 4) + det(D ,.~'4) : 0
where g4, f4 define the 4-iterate ~ of ,.~'. the f i r s t two equatii~s are those of the fixed
point of ,.,~Ir'.
The third one characterizes an eigenvalue ~,4 of the 4-periodic point such that
~Pq.4 : I ( 6 )
that is
~g4 81f4 ~g4 bf4 J g 4 ~If4
I - + - - + - - - - - - 0
a X ~ y ~ X b Y ~ y a x
Resolution of (5) by Newton's method requires, at each iteration., the knowledge of 18 functions which take part in newton's formulas, these functions are calculated by integration between t O = 0 and t=4 7[ , using Runge-Kutta method, differential system obtained by derivating (3) with respect toxo,Y O, ~ (See [3]).
]he convergence of Newton's method In the resolution of (5) requires the knowledge of Initial conditions for x and y, for a fixed 4-order point and for ~. These are obtained by using the following process..
- W e used as initial values of x and y those given by a Bogelloubov-Mitropolsky [6] type formal approximation for II =0.
Let us remember that an asymptotioal meth~¢onsists in searching approximative solutions of a differential system
d2x
) + ¢ g((~t,x, ) ( 7 )
+ % x =~: r ( ( ~ t , x ; d t (It d t 2
where ¢ is a small parameter, f and g are ~ t periodic function ( 2 ~ period), by w r i t i n g
2 r ( ~ )2 (~ -- ( +z~(~
o k
where ~ is the difference between the squares of the proper frequency end the foroed frequency, an(l
x = a c o s ~ + c u l ( a , (~t, ¥ ) + ... ( 8 )
where u 1 .... are o t and ¥ periodic functions ( 2 ~[ period). Values of a and V can be determined using the differential system
da = ¢ Al~a,e)" + d t
where
de
= c B ( a , e ) +
d t t "
e = ¥ - - - m t r k
and A 1 , A 2 .... B 1, B 1 ... are periodic functiol~5 ( 2 ~ period).
For 1~ = 0, r = 1 and k = 4, equation ( 3 ) can be written
2 2 2 _~)2
h = oh, C 2 = EC 2 , A ~ = E A(~, Ot = E Or, (O ° = ( 2
÷ E Z ~
dx 2 (01)2 ~ 2 : dx
+ ~ X = c ( - h ( ~ ) XCOS01t + C2 x2) + {: (-401X - e d t ) ) , d t 2
By developing this esymptotical method, we obtain the following formal approximations x = a c o s ( - ~ : e ) , u i
01 that exist only when
y=~
2 66oi 01 2 h201 4
a - - - *
2 2
80C 2 5120 C 2 8 c 2 a " 8 c 2 a - u I - - - cos (
2 2 4
3 ~
(IO)
201t+20) ha 5~t ha 3~t
+--48 cos ( 4 +e) *~-~ cos ( ~ - - e )
(9)
C 2 (~ 2 h201 2
% > (¥) - 38--~
The choice of x and y, given by ( I O) and ¢orrast~M!:l~g; ~ the first term of development ( 8 ) , is sufficient to obtain convergence of Newton's method to resolve
g4(x ,y ) - X = 0
(11) f4(x,y) - y = 0
T '..>'_
4 1 . ,
oJ I i
~ j . ~4 ~ . t,~,
- by incrementing I1 , we can follow the evolution of the fixed point of the 4-periodic points in the phase representation, as well as its elgenvalues up to p =. 2.
- By lncrementlngo~ or c ( ~ = .5) sufficiently to obtain a 4-periodic point's elgenvalue close to 1. We therefore obtain the initial values x, y and a to resolve Newton's method of system (5).
-With these values of (x,y, ( x ) Newton method's to solve ( 5 ) rapidly converges to (x 1 , y l (Z t ) for eel = .5 (Fig. 3a),therefore the point (cl=exp(-octx) ,=tO ) on c u r v e ~ 1.
To find another point of curve ~ 2 , we solve ( I I) by Newton's method with initial
conditions x 1 , y l for c = c 1 and by incrementing ~ between ~)1 and ~l) 2.
Evolution of the eigenvalue ~1 shows that it Increase to a certain value, and then decreases to come back to close to 1, while the other eigenvalue remains inferior to 1.
-
