ASYMMETRIC OSCILLATORS AND TWIST MAPPINGS

ASYMMETRIC OSCILLATORS AND TWIST MAPPINGS RAFAEL ORTEGA

1. Introduction In this paper we consider some aspects of the dynamics of the differential equation x" + ax+-bx-=

l+p(t),

(1.1)

where x+ = max(x, 0),x~ = max( — x, 0), a and b are positive constants (a # b) and p(t) is a small 1-periodic function. This equation models the motion of a particle subjected to an asymmetric restoring force and appeared, after separation of variables, as a simplified version of the model of the suspension bridge of Lazer and McKenna [13]. In this model the asymmetry is due to the fact that the cables exert no restoring force when they are not tight. The same equation had been previously considered by Fucik [6] and Dancer [4] in their investigation of boundary value problems associated to equations with 'jumping nonlinearities'. After these works, the Dirichlet, Neumann and periodic problems for (1.1) have been the subject of several papers [7, 8,12,... ]. In this paper we study other properties of the solutions of this equation that are also relevant for the physical model and obtain results on boundedness and existence of certain recurrent solutions. We first prove that if p{t) is smooth and small enough then every solution of the equation is bounded. This result is in contrast with the well-known phenomenon of linear resonance that occurs in the case a = b = (2nn)2 for n e N. In such case unbounded solutions often exist even if p(t) is small. We notice that the smallness of p{t) is essential in our result, since otherwise unbounded solutions can also exist in the asymmetric case. After the boundedness result and assuming again that p is small, we find families of quasiperiodic and subharmonic solutions with large amplitude. This class of solutions resembles the structure of the closed orbits of the autonomous equation (p = 0) in a neighbourhood of infinity. The equation (1.1) is a special case of the class of equations x"+g(x)=M,

(1-2)

where/(0 is periodic and g(x) sgn JC -> + 00 as x -> + 00. As remarked in [17], it was Littlewood who proposed the consideration of equations of this form. Also in [17] Morris studied (1.2) when g(x) = 2x3 and /(/) is a piecewise continuous periodic function, and he obtained conclusions for such an equation that are similar to ours for (1.1). The result in [17] deals with a concrete nonlinearity and more recent papers [5,11,14,18] have enlarged the class of admissible nonlinearities. In all these papers g(x) has either a superlinear or a sublinear growth at infinity, a condition that

Received 21 December 1993; revised 5 September 1994. 1991 Mathematics Subject Classification 34C28. Research supported by DGCYT PB92-0953, M.E.C., Spain. J. London Math. Soc. (2) 53 (1996) 325-342

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RAFAEL ORTEGA

excludes (1.1). The method of proof in these papers is as follows. First it is proved, using the Twist Theorem of Moser, that the Poincare map associated to (1.2) has arbitrarily large invariant curves. This property already implies the boundedness of every solution of (1.2) and, at the same time, allows us to look at the Poincare map as a monotone twist mapping in certain annuli tending to infinity. The abstract theory of this class of mappings leads to the existence of different kinds of subharmonic and quasi-periodic solutions. The Twist Theorem is only valid for mappings with a considerable degree of smoothness (see [9]) and this fact restricts the method of proof just described to cases where the Poincare map is rather smooth. For this reason it is assumed in [5,11,14,17,18] that g(x) is at least of class C4. In our case g(x) is not even C1 and a direct application of the previous method does not seem possible. The Poincare map Px has the remarkable property of reducing the study of (1.2) to a problem in dynamics of planar maps, however it is not the only map with such a property. In [19] Picard considered an equation related to (1.2) and used the oscillatory properties of the solutions to define another map P2 with similar characteristics. This map can be described in the modern terminology of dynamical systems as the first return of the autonomous system t' = 1, x' = y, y' = — g(x)+f(t) with respect to the section {(t,x,y): x = 0, y > 0}. More recently Alekseev [1] and Jacobowitz [10] have used a related map to study certain Newton's equations. With respect to equation (1.1) the crucial difference between P1 and P2 is that P2 is smooth if the function p{t) is smooth. Thus we can compensate the lack of regularity of the nonlinearity with some extra regularity of the forcing term p{t). The proof of boundedness in this paper will be based in the application of the Twist Theorem to P2. As a consequence it will be shown that in a neighbourhood of infinity P2 is a twist mapping such that the theory of Aubry and Mather [16, 3] applies. This theory will produce subharmonic and weak quasi-periodic solutions with oscillatory properties determined by the rotation numbers of the corresponding Mather sets. It should be noticed that the use of P2 in the proof of boundedness leads to an approach that is similar to the approach of Levi in [14] for the superlinear case. In our approach the twist mapping is constructed for time and velocity, while in [14] it was constructed for time and energy. I thank the referee for informing me about [14] and for pointing out this similarity. The main results of the paper are stated and proved in Sections 4 and 5. Sections 2 and 3 deal with some technical results that are employed in the proofs of the main results. 2. The successor map

This section is inspired by [1]. Given co > 0 and peC(U/Z) consider the linear differential equation (2-1) A solution of (2.1) is said to be of constant sign if either x(t) ^ 0 V/e U or x(t) < 0 We IR and it is oscillatory if there exist sequences of real numbers {tn}neI, {t*}neI with tn, t* -*• ± oo as n -* + oo and such that x(/ n ) x(t*) < 0 for each « = 0, ± 1, ± 2 , PROPOSITION

2.1. Each solution o/(2.1) is either of constant sign or oscillatory.

Proof. We distinguish between the resonant and the non-resonant cases.

ASYMMETRIC OSCILLATORS AND TWIST MAPPINGS

327

Case (i): — eN In

and

Jo

p(t)sin[co(-1)]dt > 0 for some eU.

Each solution of (2.1) is written in the form 1 f1 x(t) = cx cos cot + c2 sin cot + — (1 +p(s)) sin [co(t — s)] ds with c1? c2 € R. When (i) holds it follows from this formula that x{(f>±n)

>±oo,

x( + 7i/co±n)

>>+oo as n

•oo

and so x(t) oscillates. Case (ii) n — £N In

or — G^J In

and

Jo

p{t) sin [co(d> — t)] dt = 0

V0ER.

Let x(t) be a solution that it is not of constant sign. We can select tx,t2sU such that x(tx) < 0 < x(t2). Since (ii) holds the solution can be expressed in the form x(t) = cx cos cot + c2sin cot+x(t) (with c1,c2eU), where/(0 is a 1-periodic solution of (2.1). For a given co > 0 it is always possible to find a sequence of integers {nk} with « fc -* + oo, such that e±nkCOi -* 1 as A:->oo. Then x(t1±nlc) -> x(t1),x(t2±nk) -* x(t2) and this shows that x(t) oscillates. Given (x,v)eU2 let x(t;x,v) denote the solution of (2.1) satisfying x(x;x,v) = 0, x'(x;x,v) = v. When v ^ 0 the solution x(t; T, y) cannot be of constant sign and the previous proposition implies that it is oscillatory. We denote by x' the first zero of x(t;x,v) to the right of T; that is, x' > x,x(x';x,v) = 0, x(t;x,v) # 0, W G ( T , T ' ) . We also use the notation v' = x\x'; x, v). We are interested in the mapping (T, V) -*• (xf, v') and, in order to specify the possible domains, we introduce the notation R+ = {(X,V)EU2:V>0}, +

S :R+

>R.[)RQ,

R_ = {(X,V)ER2:V(T>');

S~:R_

RO = {{T,V)EU*:V

>R+ U tf0,

(T,V)

= 0}; ^(T',^)-

(Sometimes we shall use the notation S+, S~ to emphasize the dependence on co). Following [1] the map S+ (and also S~) will be called the Successor Map corresponding to (2.1). The uniqueness of the initial value problem implies that S+, S~ are one-to-one and the periodicity of p(t) leads to (2.2)

In the rest of this section we obtain certain properties of the map S+ that will be used later. They are also valid for the map S~ with the obvious modifications. 2.2. Let Z + = {(T, V) e R+: v' = 0}. Then S+ is of class C1 on R+-Z+ and it is symplectic with respect to the differential form vdx A dv in the following sense PROPOSITION

v'dx' A dv' = vdx A dv +

Moreover, S is of class C on R+-I+

V(T, V) e R+

ifpeC^Ufflfor

- Z+. k>\.

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RAFAEL ORTEGA

REMARK. This result is similar to [1, Proposition 5] and to [10, Lemma 1]. However these papers deal with a nonlinear Newton's equation that admits the trivial solution and in such a case the singularity set £ + is empty.

Proof of Proposition 2.2. The regularity with respect to initial conditions implies that the solution x of (2.1) is of class C 1 with respect to the three arguments (t;x,v) and that dx/dx and dx/dv are the solutions of y" + co2y = 0 satisfying the initial conditions y(x) = — v,y'(x) = — (1 +p(x)) and y(x) = 0 , / ( T ) = 1, respectively. In consequence the wronskian has the value W{dx/dx, dx/dv) = —1>. These facts also follow from the formula (2.3) co

co

Now the implicit function theorem can be applied to solve with respect to x' the equation x(x'; x, v) = 0 as long as v' # 0. In consequence, x' is of class C 1 on R+ — Z + . The smoothness off' follows from the chain rule since x' is also C 1 with respect to (t;x,v). In the case peC*'1 a similar reasoning proves that S+ is of class Ck. To prove that S+ is symplectic we verify the identity

dx' dx' t/det

dx dv' dx

dv dv' ~dv~

= V.

By implicit differentiation

V

dv>

dx

-

,dx' dx ~dx+!h dx'

,dx'

V - z iv

dv

dx'

+A T )) ~dx~ + ~lh'

dx

(2.4)

Tv = u, dx

dv' piT)) -dv~dv~~' (i +

^Tv

(2.5)

These identities show that

dx' dx' l/det

dx ~dv~ . . . . dv' dv' dx ~dv~

= — rr l-z-t-zdx dv

leading to the previous formula. To state the next result we introduce some notation. In the plane with coordinates (T, V) consider the differential form Q = \v2 dx and denote by A p the class of C 1 Jordan arcs y: R -• IR2 satisfying

y(M) a R+-l+,

y(s+\) =

(where I + is given by Proposition 2.2).

VseU

ASYMMETRIC OSCILLATORS AND TWIST MAPPINGS PROPOSITION

2.3. For each yeAp

329 +

denote by / the transformed arc / = S oy.

Then

Proof. We proceed along the lines of [1, Proposition 5] and [10, Lemma 1]. In the three-dimensional space (x, P, t) consider the differential form rj = — Pdx + Hdt, where H = ±P2 + ±a>2;c2-(l +p(t))x. In IR3 define the curves T = {0} x y|[Oi 1}

and P = {0} x / | [ 0 X]

then )nOil]Q = Jr'/' J/io,i]^ = irV- We shall prove the identity jrrj = jrn. This is almost a consequence of the theorem of the Poincare-Cartan invariant [2, p. 235] but that theorem deals with closed curves, so we shall give the proof for our situation. For each point on F follow the orbit of the autonomous system x' = P, F = —co2x+ 1 +p(t), t' = 1 until reaching the curve F \ In this way we obtain the surface

S = {(x(t;T,v),x\t;r,v),

t): (r,v)ey,T ^ t ^ T'}.

Since x' is smooth on y and x' > r it follows that S1 has a piecewise smooth boundary composed by the arcs F, F', ^ 0 ,// x , where ji0 and ^ are orbits of the system that differ only by a translation of the vector (0,0,1). We apply Stokes theorem to the field F and, since rotF is tangent to S, we obtain j8srj = 0. Since j ^ rj = jM n the proof is finished. REMARK. The previous proposition could be deduced from Proposition 2.2 and Green's formula when S + = 0 . However this is not possible when E+ # 0 . As an example consider the map

S: R+ T' = T + 1 ,

2

>R_ U Ro,

(x,v)

>{x', v'),

1/2

v>\,

v' = - v

v' = -(V -\)

if

if

v^l.

Then 5 is smooth on R+ — Z with L = {v = 1}, and satisfies v'dx' A dv' = vdx A fifo V(T,y)ei? + -I. On the other hand the conclusion of Proposition 2.3 does not hold. As a consequence of the previous proposition we prove that S+ has the following intersection property. PROPOSITION

2.4. For each yeAp with y' = S+oy, we have /(IR) n y(M) # 0 .

Proof. Let II be the projection FI(T, V) = (f, v), f = x + Z, of R+ onto the cylinder R/Z x (0, oo). In view of (2.2) it is enough to prove that II o y and II o / have a nonempty intersection in the cylinder. By contradiction assume that Uoy{U) f| I I o / ( I R ) = 0 . Each of these curves divides the cylinder into two components and we denote by D and D' the corresponding bounded component. Since both curves are homotopic to

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RAFAEL ORTEGA

the circle {v = constant}, one of the bounded components must be strictly included in the other, say D + oo,peA will also be used. In practice the dependence of/with respect top will not be mentioned explicitly and the set A will be of one of the following types

A_hr =

{peC(n/Z):\\p\\Li|| c n 0,

« = 0,1,2,....

We now state the results of this section that will be employed in the rest of the paper. The first result shows that the singularity set I + is confined to a finite strip, the second proves that S+ satisfies a condition of twist type in a neighbourhood of infinity and the last result gives estimates on the derivatives. PROPOSITION

3.1.

There exists v > 0 (depending on \\p\\Li) such that

PROPOSITION

3.2.

There exists p > 0 (depending on co) such that if p(t) satisfies \p(t)\dt 0 (depending on \\p\\Li) such that

v' = k(x,v) — \/(v2 — a(x,vf)

for v ^ vv

(3.4)

Proo/. The change of variables y = x—jlx(\ + p(s))(sm[co(t—s)])/cods reduces (2.1) to the autonomous equation y" + co2y = 0. The conservation of energy implies

332

RAFAEL ORTEGA

that | y ' ( O 2 + &2y(x'f = \y'ir)2 + \w2y{x)2. Undoing the change of variables in the previous expression one obtains (v' — k(x, v))2 + max(v0, vl52/w) = v and applying (3.4) we obtain \v'\ ^ — m + V(v2 — m2) ^ v — 2m > 0. In consequence the set I + is included in

LEMMA

3.6. For each r > 0, we have

a(x,v) = 2/co + O(\\p\\Li + l/v),

k(z,v) = O(\\p\\Ll+\/v),

v

> + oo,

peA_Xr,

where a and k are given by Lemma 3.5. Proof. We prove the estimate on a. In fact a(x, v) = Ix + /2 + /3 + /4, where sin[a)(T' — s)]ds, /2 = z+n/ai

{sin [co(r' — s)] — sin [w(r + n/co — s)]} ds,

JT

rr+n/co

/3 =

/Y

sm[co(x + 7t/co—s)]ds,

/4 =

J X

p(s) sin [co(xr — s)] ds.

JX

It follows from Lemma 3.4 that IX,I2 = O(l/v), /4 = O(\\p\\Li). LEMMA 3.7. There exists p>0 such that if p satisfies (3.1) then there exists v2 > 0 (independent of p) such that

x' = x + n/co + (l/co) arcsin {(1 /v) o(x, v)} ifv^v2

(3.5)

(arcsin: [— 1,1] -• [—|7r,|7r]). Proof. The constant p is chosen in such a way that, for large v, we have a > 0. This choice is possible thanks to Lemma 3.6. Also for large v, Lemma 3.4 implies that 0 < co(x' — x) < In. the definition of x' and (2.3) imply that 1 1F sin [co(r' — T)] = — (1 + p(s)) sin [co(x' — s)]ds = — a ( x , v). J T

This formula and the previous inequalities show that co(x' — x) lies on the third quadrant, leading to (3.5). Proof of Proposition 3.2. From Lemmas 3.5 and 3.6 v' + v = k -\- a2 /(v + \/ (v2 — a2)) = 0(||/>||Li + l/v). From Lemmas 3.7 and 3.6 and the Taylor expansion of arcsin,

T' = T + - + -arcsinj — + 0 ( M ^ + co co

[cov

\ v

ir/J

H = T + £+ l

co co v

+0

Given a multi-index a e N2 with |a| ^ 1, a decomposition of a consists in a family of nontrivial multi-indexes f5x,...,f}r (with r ^ 1) and corresponding positive integers

ASYMMETRIC OSCILLATORS AND TWIST MAPPINGS

333

nv...,nr satisfying nl^1 + -•• +nrf3r = a. The trivial decomposition is given by r = 1, y?x = a,«! = 1. Let D{ot) denote the class of nontrivial decompositions of a. Given v > 0, let Mv be the half-plane Mv = {(T, V) : v ^ v}. LEMMA 3.8. Assume that ^eC°°(lR) and f€Cn(Mv) for some | / ( T , V)\ ^ m, V(T, V) e Mv. Then there exists a constant C > 0 such that

n^\,

with

(3.6) /or eac/i multi-index a w/f/i 1 ^ |a| ^ n. /n addition, if (p'(0) = 0, then

£

(3.7)

-

D(a)

constant C depends on m, n and \\(_m m)for

0 ^k

^n.)

The proof is easily obtained by induction using Leibniz's rule. Proof of Proposition 3.3. Given n ^ O w e say that the assumption (An) holds if peCn(U/Z) and then for each multi-index a with |a| ^ n we have

+ oo, peA_lp(\A^

(with r > 0).

We shall prove that if (An) holds then the same kind of estimates is also valid for the derivatives of order a with |a| = n+ 1. Since (Ao) is true, as shown by Proposition 3.2, the result follows by an induction argument with respect to n. Step 1. If (An) holds then for each a with 0 ^ |a| ^ n, we have (3.8) ,r.

(3.9)

For |a| = 0, (3.8) and (3.9) can be derived from Proposition 3.2. To prove (3.8) for |a| ^ 1 we apply (3.7) with q>{Q = cos
coFWJd'il/cov

+ £ ||dh {2/cov + coF} | | ^ . . . ||tfr{2/cov + coF} || J ] . Applying (An) we deduce that Wd^l/cov + coF}^ = O(\/vy*+1) with |y| ^ n and (3.8) follows. The proof of (3.9) is similar with (p{£) = sin 0, a2 > 0, define a* = (a l5 a 2 — 1). Then d«\v'dvz') = -(l/ft))r{sin[«(T'-T)]} and the previous argument still works; in fact, it is even simpler because d**(2/cov) = 0. Step 4. Estimate on dnx+lF. Again from (2.4) and (2.3), M'

/,

/ xv sin M r ' - T)]

v'— = (1+/?(T))—L_^ OX CO

_ , ,

,.

^ + UCOS[W(T'-T)].

If n = 0 the estimate on dr.F is obtained using Steps 1 and 2. If n ^ 1 then d? {v cos [aK^ - T)]} and the same kind of reasoning as in Step 3 ends the proof. (To estimate the first term in the right-hand side one has to combine Step 1 with Leibniz's rule and use the fact that p is of class C"). Step 5. Estimate on d*G with |a| = n+1. When a2 > 0 they are obtained by a similar procedure using the identity (derived from (2.5)) dv'/dv = To estimate &{P{T')} we use (3.6). To estimate 5"+1 G we use the identity (also derived from (2.5)) dv'/dz = (l+/>(T/))(dT7dT) + awsin[©(T'-T)]-(l +p(r))cos[co(z'-r)].

ASYMMETRIC OSCILLATORS AND TWIST MAPPINGS

335

If n = 0 it follows from Step 4 that (dz'/dz) = 1 + O(\\p\\co+ \/v). On the other hand, the formula for dv'/dz together with Step 1 leads to (dv'/dr) = (1 +p(T'))(dr'/dr)-1 + O(\\p\\co+ l/v) and these two estimates can be combined to conclude that (dv'/dr) = O(\\p\\co+ \/v). If n ^ 1 we differentiate to obtain d? + V = aTn{(l +p{z'))(dz'•/dz)} + covd?{sm[co(j''-T)]}-5?{(1 +p(z)) COS [CO(Z'-Z)]}. The bounds on the third and second terms follow from Leibniz's rule and Step 1. To bound the first term one combines Step 4 and Lemma 3.8. 4. Boundedness of solutions In this section we obtain a sufficient condition for the boundedness of every solution of (1.1). By a bounded solution we understand a solution x(t) such that THEOREM 4.1. Assume that a, b > 0 with a^b. Then there exists e > 0 such that every solution of (I.I) is bounded if p satisfies

peC\U/Z),

\pk)(t)\ 0 let z_ < t be the first zero to the left of t. Then 12 T m X(T_) = 0,X'(T_) ^ 0 and \x(z_)\ + a' ' |*'( -)I ^ a~ v. Applying the previous lemma 1/2 we obtain \x(t)\ + a~ \x'(t)\ ^ F where F depends only on a, p and v. The case x(t) < 0 is treated similarly. The proof of Theorem 4.1 will be based on the Invariant Curve Theorem. There are many variants of this theorem and we shall employ a version due to Herman that requires less regularity than the original version of Moser. In applying this theorem we need the concept of irrational number of finite type and some of its properties that we state first. A number a s U — Q is of constant y type if the quantity y defined by

y = M{q2\ 0 and 1

then a is of constant type if and only if sup{flf: / ^ 1} = C < oo. Moreover, in such a case (1/C) ^ y > (l/(C+2)) (see [9, p. 160]). LEMMA 4.4. Given a real interval [a,b] c [0,1] with b — a^e>0 there exists a e [a, b] — Q of constant type and such that the corresponding Markoff constant satisfies

Proof.

Let q ^ 2 be such that \/q < e/4 ^ \/(q—\).

For s o m e p e { \ , . . . , q }

the

interval [(p— \)/q, (p+ l)/q] must be included in [a, b]. The rational number/?/