RAFAEL ORTEGA

Departamento de Matem~tica Aplicada Facttltad de Ciencias, Universidad de Granada

18071 Granada, Spain

ON LITTLEWOOD'S PROBLEM FOR THE ASYMMETRIC

OSCILLATOR

Conferenza tenuta il giorno 15 Giugno 1998

In the papers [11, 12] Littlewood considered the equation

+ g ( x ) = f ( t ) ,

with g ( x ) - . +oo as x - _+o% and he found some unexpected exam- pies of unbounded solutions. His techniques were based on the me- chanical interpretation of the equation as a forced oscillator. After these papers, the name of Littlewood was associated to the study of the boundedness problem for the forced oscillator. Many results on this problem have been obtained when f is periodic and g has su- perhnear growth at infinity (see [10] and the references therein). In these notes I shall present some recent results on this problem for a class of oscillators that have linear growth in g. These oscillators look very similar to the linear harmonic oscillator but they display in- teresting nonlinear dynamics. To compare the linear and nonlinear situations it will be interesting to start with the harmonic oscillator and look at it from a geometrical perspective.

154 R. ORTEGA

1 T h e h a r m o n i c o s c i l l a t o r r e v i s i t e d

Consider the linear equa t ion

+ co2x = f ( t ) , (1.1)

where the f requency co is a posi t ive cons tan t and the external force

f is per iodic (with per iod 2rr). Let us review some e lementa ry facts about r esonance ( u n b o u n d e d / b o u n d e d solutions) and the F redho lm

Alternat ive (periodic solutions):

(i) If co E N and the Fourier coefficient fm is different f r o m zero t hen all solut ions are u n b o u n d e d . Here,

fw := 2"-~ f ( t ) e - U ~

(ii) If co e N and fw = 0 t hen all so lu t ions are 2rr-periodic.

(iii) If co r ~ then all so lu t ions are b o u n d e d

and there exists a un ique 2rr-periodic solution. The previous resul ts lead to a remarkable conclusion:

the solutions of(1.1) are bounded if and only if there exists at least one periodic solution.

Let us now forget these classical resul ts and try to explore numer i - cally the l inear oscillator. Probably the easiest way is to draw on the screen of the c o m p u t e r the i terates of the Poincar~ map. Given an

initial state so = (xo, vo) ~ R 2 we define the new state as

Sl = ( x l , v l ) , Xl = x (2 r r ; so ) , Vl = ~:(2rr;so),

where x( t ; so) is the solu t ion of (1.1) sat isfying x (0 ) = xo, :~(0) = vo. The Poincar~ m a p (also called the m o n o d r o m y operator) is def ined

by T : ~2 --. 1~2, so ~ s l .

It is easy to verify that T is an affine t r ans fo rma t ion that can be

c o m p u t e d in t e rms of fw.

ON LITTLEWOOD'S PROBLEM ETC. 15 5

Let us see, in each of the cases (i), (ii) and (i/i), the dynamics of T on the computer .

(i) co �9 N, fw 0. S3

$1

SO

All orbits go to infinity and T is a translation. The transla- tion vector v depends o n / w . (The stepsize for numerical inte- gration mus t be chosen small, otherwise the line on which the particle runs will appear slightly curved.)

(ii) co ~ ~q, jew = O. 5 0 = 51 = 52 = 53 = " " "

(iiiq)

All orbits are fixed points and T is the identity. (After many ite- rations with the computer it will seem that the particle moves very slowly along a curve.)

We shall dist inguish two subcases for (iii).

co E Q, co = p / q (the fraction is in reduced form with q > 1).

s'4

o f . p . Sl

$2

Most probably the reader is a very lucky person and will choose the point on the screen that corresponds to the initial condit ion of the periodic solution (the fixed point). An average person will find a periodic particle (of period q) that rotates around the fixed point. The Poincar~ map is now a root of unity; that is, T q = I, and all solutions are periodic with period 2rrq (su- bharmonic solutions of order q). After many iterates on the

1 5 6 R. ORTEGA

c o m p u t e r an ellipse will p robably appear in the screen. This curve is invariant u n d e r P and it has appeared due to numer i -

cal effects. Notice that the orbit lies on the ellipse bu t it is no t dense.

(iii-2) co ~ Q.

$ 4 �9 I

s3 of .P. :So

$2 51

It is poss ible that the reader will select again the po in t corre- s p o n d i n g to the per iodic solution. This means that she (or he) is

really lucky. Typically, orbits ro ta te a round the fixed point wi- thou t periodicity. After m a n y s teps we shou ld see an ellipse on

the screen. This ellipse, E, is an invariant curve of P, P ( E ) = E, and the orbit {Sn} is dense in E. In this case all so lu t ions of

(1.1) are quasi-periodic wi th f requencies COl = 1, 00 2 = 03. (A func t ion x : �9 ~ ~ is said to be quasi-periodic with fre- quencies 001, 002 . . . . . con if there exists a con t inuous func t ion F : ~ n ~ ~ , F = F ( 0 1 , . . . , On), that is 2~-per iodic in each 0i, and satisfies x ( t ) = F (001 t , . . . , 00nt) for each t).

F rom the geometr ica l po in t of view we can s u m up all the previous cases in two situations: ei ther there exists a family of s traight lines

that are invariant u n d e r P and foliate the plane, case (i), or there exists a family of ellipses in the same condit ions, cases (ii) and (iii). These ellipses are t ransla tes of v 2 / 2 + 002x2/2 = c o n s t a n t , the

energy levels of the a u t o n o m o u s oscillator.

2 The equation of an asymmetric oscillator

Let us consider a particle of mass m = 1 a t tached to two spr ings with Hooke 's cons tan t s 002 and 00 2. The second spr ing is only in one side

and has a s top at the origin. Under the action of an external forcing

ON LITTLEWOOD'S PROBLEM ETC. 15 7

f = f ( t ) , the equat ion of mot ion is

I(u~12 + u~2)x i fx_<O

2 + [ o o ~ x i f x >_ 0 = f ( t )

or, equivalently, 2 + a x + - b x - = f ( t ) ,

where a = oa 3, b = o0~ + co~. The above equat ion with a , b will be referred to as the equat ion of the asymmetr ic oscillator. When a = b we go back to the harmonic oscillator.

Piecewise linear oscillators appear in the engineering li terature (see [5]) and, in particular, the equat ion of the asymmetric oscilla- tor has been proposed by Lazer and McKenna in [9] as a simplified version of their model of suspension bridge.

3 The periodic Dancer-Fu~.ik spectrum

In the seventies Fu~ik and Dancer s tudied the equation of an asym- metric oscillator by purely mathematical reasons. They were inte- rested in nonlinear boundary value problems (Dirichlet, Neumann, Periodic, . . . ) and they found that the equat ion

+ a x + - b x - = f ( t ) , a > 0 , b > 0 (3.1)

was a paradigm for a large family of equations ("equations with jum- ping nonlinearities"). More informat ion on this point of view can be found in the book by Fu~ik [7]. Next we shall review some of the results in [2, 3, 4] on the periodic problem for (3.1). They can be seen as a sort of Fredholm Alternative for the asymmetric oscillator.

In the space of parameters (a, b) E ~2 define the family of curves

1 1 2 Cn . . . . n = 1, 2 . . . .

4"d + 4 F n '

These curves are similar to hyperbolae and the intersect ion of each of them with the diagonal a = b is the point (n 2, n2). These points are in the periodic spec t rum of the harmonic oscillator.

The role played by the set {n 2 : n = 1 ,2 , . . . } in the periodic linear problem is played by Y. = Un=l Cn in the asymmetr ic oscillator. Namely,

158 R. ORTEGA

(i) If (a, b) r Z then the equat ion (3.1) has a 2rr-periodic solution for each f ~ L ] (R/2rrZ). Notice that this solution is not always unique.

(ii) If (a ,b) E E then there exist functions f E LI(R/2rrZ) such that (3.1) has no 2rr-periodic solutions.

In [4] Dancer gave examples of piecewise constant functions f for which there are no periodic solutions. Later, in [9], Lazer and Mc- Kenna have found new examples of nonexistence for functions of the type f ( t ) = a c o s n t + f l s i n n t . Some sufficient condit ions for existence of periodic solution when (a, b) ~ Y were also given in [3]. They have been improved in a very recent paper by Fabry and Fonda [6].

4 C o e x i s t e n c e o f u n b o u n d e d a n d p e r i o d i c m o t i o n s

Our next task is to find under which condit ions are there unbounded solutions of (3.1). The Massera's theorem applied to our equat ion says that the existence of a bounded solution (in the future or in the past) implies the existence of a 2rr-periodic solution. In consequen- ce, if (a, b) ~ ~ and f is such that there are no periodic solutions, then all solutions are unbounded . By analogy with the linear case we can formulate the following question:

are all solutions of(3.1) bounded when there exists a perio-

dic solution?

For some time I thought that the answer should be yes, but my collea- gue Jos~ Miguel Alonso had the opposite impression. My conjecture was wrong. In a paper recently appeared, [1], Alonso and I proved that for any couple (a, b) ~ ~2 with

1 1 + ~ E Q, (4.1)

there exist functions f ~ L 1 (~ /2 r rg ) for which "large solutions" are unbounded . By a large solution we mean a solution with sufficiently large initial energy. When (4.1) holds but (a, b) ~t Z, the results of

ON LITTLEWOOD'S PROBLEM ETC. 159

Dancer imply the existence of a periodic solution. In consequence,

coexistence of u n b o u n d e d and periodic solut ions can occur. This is a genuine nonl inear p h e n o m e n o n . In a paper to appear [13], Liu Bin

has cons idered again the case (4.1) and has f ound sufficient condi- t ions of f for the b o u n d e d n e s s of all solutions. The resul ts in [13] and [1] are complementa ry . They provide condi t ions for the boun-

dedness or u n b o u n d e d n e s s that are ra ther sharp when the func t ion

f is smooth . (The resul t of Liu requires f E CS). The precise condi- t ions can be found in the original papers but it may be interes t ing to

show how they apply in a concrete case.

EXAMPLE 1 X + 4X + -- X- = A + Cos4 t , A E ~.

The couple (a, b) = (4, 1) is no t in Z and so we know the existence of a periodic so lu t ion for any A. Also, the n u m b e r 1/.,/-a + 1/x/~ = 3/2 is in Q and the resul ts in [13] and [1] apply. They lead to the

following assert ions:

�9 If IAI < 1/45 then all large solut ions are u n b o u n d e d

�9 If [AI > 1/45 t hen all solut ions are bounded .

The case IAI = 1/45 cannot be dec ided with the resul ts in these pa- pers. The reader may d e m a n d how the n u m b e r 1/45 appears. In the next sect ion I will try to jus t i fy it.

The previous example shows that the resonance set for the asym-

metr ic oscillator is not Z but a larger set that at least conta ins

1 1

This is dense in the space of pa rame te r s and the next ques t ion is, of course, what h a p p e n s in the complemen ta ry set. In a work in

p repara t ion I have p roved that if

1 1

and f E C4(~/2r rZ) is such that f2rt f ( t ) d t ~ O, t hen all solut ions of

(3.1) are bounded . Let us i l lustrate this resul t wi th another example.

160 R. ORTEGA

EXAMPLE 2 de + 5 X + -- X - = ~ + COS 4t, 2t E ~ .

The number 1/,,/-d + 1 / v'b = 1 + 1 / ~ is not rational and so there is a periodic solution for any h. When 2~ ~: 0 all solutions are bounded but for 2, = 0 1 do not know what happens.

5 T h e d y n a m i c s a t i n f i n i t y

We are now going to sketch the proofs of the results in [1] and [13]. As we shall see, these proofs give additional informat ion on the behaviour of large solutions of the asymmetr ic oscillator.

The first step will be to int roduce appropriate coordinates.

5.1 The a s y m m e t r i c polar coordinates

Let us first consider the "homogeneous" equation

+ a x + - b x - = O. (5.1)

In the phase space (x, ~:) the orbits can be const ructed by gluing two ellipses. These orbits are closed and have the minimal period

Tr 71

This means that the origin is an isochronous centre. The equat ion (5.1) is positively homogeneous and so the general

solution can be expressed as

x ( t ) = a C ( t + ~ 8 ) , a > O, fl ~ ~ ,

where C ( t ) is the solution of (5.1) satisfying the initial condit ions C(0) = 1, C(0) = 0. This funct ion can be de termined explicitly by the formulas

c ( - t ) = c(t),

cos 4-tit,

C ( t ) = l - ~ f ~ s in ,/-b ( t - - -

C ( t + T ) = C ( t ) ,

A)' ?T

O_<t_< - -

/1" T - - < t _ _ _ - . 2,/-d 2

ON LITTLEWOOD'S PROBLEM ETC. 161

The func t ion S(t ) = C(t) is also T-periodic and the conservat ion of energy leads to the ident i ty for S and C,

S ( t ) 2 + a C + ( t ) 2 + b C - ( t ) 2 = a , V t ~ . (5.2)

(It is convenient to th ink that the func t ion C is an "asymmetr ic co- sine". In fact, for a = b = 0.) 2 one has C(t) = cos tot, S(t ) =

- to sin to t.)

Let to = 2Tr/T be the f requency associated to (5.1). The funct ions C(O/to) and S(O/to) are 2rr-periodic with respect to 0 and allow us to define the change of variables

x = r C ( O ) , y = r S ( O ) ; r > O , 0 E S 1 .

The func t ion C is of class C 2 while S is of class C 1. These facts,

toge ther wi th (5.2), imply that the m a p p i n g (0, r ) - (x, y ) is a C 1 d i f f eomorph i sm f rom S 1 x (0, oo) onto R2 _ {0}. With this change

(5.1) becomes O = t o , ~ = 0 .

As a second s tep we shall apply the same change of variables to the forced equat ion and we shall s tudy the equa t ion for r ~ co.

5.2 The Poincar6 m a p at inf in i ty

In the new variables 0, r the equat ion (3.1) becomes

z r 0 \ t o ]

where y is a posit ive constant . Let us consider the Poincar~ map

P : (00, to) --- (01 , r l ) , where

O1 = O(2rr;Oo, ro), rl = r(2rr;Oo, ro).

From the new sys t em we can obtain the expans ions

01 = 00 + 2rrto + 1 ~ ( 0 0 ) + O ( 1 )

rl Y0 -- r + O ( 1 ) ,

162

with

R. ORTEGA

r = f ( t ) c + t a t .

The func t ion �9 plays an i m p o r t a n t role in the theory of the asym- metr ic oscillator. It was already employed in [3] and [6] to s tudy the existence of per iodic so lu t ions w h e n (a, b) E Y,.

We are now going to apply (4.1). In fact, we can f ind a f ract ion q/p in r educed fo rm such that

1 1 = 2 q "

Then, r = 2rrq/p and w = p/q. The l inear izat ion of T at infinity is a ro ta t ion of the f o r m

R " 01 = 0 o + 2 r r p- r l = t o q

This is a root of un i ty (Rq = I) and it will be s impler to s tudy Tq ins tead of T. The expans ion of Tq is

{Oq=O~176176 (5.3)

rq to- p'(Oo) + o(1) ,

where p is a discrete average of ~, namely

q k=0

For example 1 this func t ion can be computed , its exact value is

u - 4 h + ~-~ cos 30 = - 3

Our final s tep will be the s tudy of the dynamics of (5.3).

5.3 The discrete m a p p i n g

The behaviour of the m a p p i n g (5.3) d e p e n d s on the oscil latory pro- per t ies of the func t ion p. We shall cons ider two cases:

ON LITTLEWOOD'S PROBLEM ETC. 163

�9 /.I has a finite n u m b e r of zeros 01 , . . . , On with p'(Oi) ~ O, i = 1 , . . . , n

�9 p(O) > O, VO ~ �9 [or p(O) < O, VO ~ ~].

In the first s i tuat ion one can find a large ball a round the origin B and

n narrow sectors Ki a round 0 = 0i such that K i n B is posit ively inva-

riant [resp. negatively invariant] ff O'(0i) is posit ive [resp. negative].

Moreover, the orbits in K i n B go to infinity in the fu ture or in the past depend ing of the type of invariance of the sector. The rest of orbits outs ide the ball B travel be tween two consecut ive sectors and

so they are u n b o u n d e d in the fu ture and in the past .

When the func t ion p has a definite sign (second case) the map T q has a twist at infinity. This m eans that the derivative

=

~ro r 6

has a definite sign for large v0. This fact sugges ts the use of Moser 's Invariant Curve Theorem. Actually Liu uses a variant of this t heo rem that can be found in [15] and shows the existence of invariant curves su r round ing infinity. This implies the b o u n d e d n e s s of all orbits. As in the case of the ha rmonic oscillator, these curves p roduce quasi- periodic solut ions with two frequencies. However, in contras t to the linear case, these invariant curves do not foliate the plane and iso- la ted subharmonic solut ions or Mather sets can appear (see [14] for more details).

To conclude we go back to example 1. In this case the func t ion

/J has 6 nondegene ra t e zeros when [hi < 1/45 and does no t vanish when Ihl > 1/45. The reader is invi ted to explore numerica l ly the ma pp ing (5.3) for different p ' s and to compare the p ic tures with the linear s i tuat ion that we d i scussed in the first sect ion of these notes.

R i f e r i m e n t i b i b l i o g r a f i c i

[1] J.M. ALONSO, R. ORTEGA, Roots o f unity and unbounded motions o f an asymmetric oscillator. J. Differential Equations, 143 (1998) 201-220.

164 R. ORTEGA

[2]

[3]

E.N. DANCER, On the Dirichlet problem for weakly non-linear elliptic partial differential equations, Proc. Roy. Soc. Edinburgh 76A (1977) 283-300.

E.N. DANCER, Boundary value problems for weakly nonlinear or- dinary differential equations, Bull. Aust. Math. Soc. 15 (1976) 321-328.

[4] E.N. DANCER, Appendix to the previous reference, unpublished.

[5] J.P. DEN HARTOG, Mechanical Vibrations, Dover, New York, 1985.

[6] C. FABRY, A. FONDA, Nonlinear resonance in asymmetric oscillators, J. Differential Equations, 147 (1998) 58-78.

[7]

[8]

[9]

[lO]

[11]

[12]

[13]

[141

[15]

S. FU~IK, Solvability of nonlinear equations and boundary value problems, Reidel, Dordrecht, 1980.

A.C. LAZER AND J.P. MCKENNA, A semi-Fredholm principle for pe- riodically forced systems with homogeneous nonlinearity, Proc. Amer. Math. Soc. 106 (1989) 119-125.

A.C. LAZER AND J.P. MCKENNA, Large-amplitude periodic oscilla- tions in suspension bridges: some new connections with nonlinear analysis, SIAM Rev. 32 (1990) 537-578.

M. LEVI, Quasiperiodic motions in superquadratic time-periodic potentials, Comm. Math. Phys. 143 (1991) 43-83.

J.E. LITTLEWOOD, Unbounded solutions of~P + g ( y ) = p(t) , J. London Math. Soc. 41 (1966) 491-496.

J.E. LITTLEWOOD, Unbounded solutions of an equation ~ + g ( y ) = p(t) , with p(t) periodic and bounded and g . ( y ) / y ~ oo as y -. _0% J. London Math. Soc. 41 (1966) 497-507.

B. LIu, Boundedness in asymmetric oscillations, J. Math. An. App. 231 (1999) 355-373.

R. ORTEGA, Asymmetric oscillators and twist mappings, J. London Math. Soc. 53 (1996) 325-342.

R. ORTEGA, Boundedness in a piecewise linear oscillator and a variant of the small twist theorem, Proc. London Math. Soc., 79 (1999) 381-413.