The twist coefficient of periodic solutions of a time-dependent ...

Journal of Dynamics and Differential Equations, Vol. 4, No. 4, 1992

The Twist Coefficient of Periodic Solutions of a Time-Dependent Newton's Equation Rafael Ortega I

This paper gives sufficient conditions for the existence of periodic solutions of twist type of a time-dependent differential equation of the second order. The concept of periodic solution of twist type is defined in terms of the corresponding Birkhoff normal form and, in particular, implies that the solution is Lyapunov stable. Some applications to nonlocal problems are given. KEY WORDS: Periodic solutions; normal forms; twist theorem; Lyapunov stability. AMS SUBJECT CLASSIFICATIONS: Primary 34C25, 34D20o

1. I N T R O D U C T I O N Consider the scalar equation

x" + f(t, x ) = 0

(*)

where f is T-periodic in t and sufficiently smooth. Let go be a T-periodic solution. It is well-known that the stability properties of go are not determined by the linear variational equation. In fact the behavior of the solutions of (*) in a neighborhood of go will depend on the nonlinear terms of the Taylor expansion o f f The theory of normal forms combined with the Moser twist theorem provides a method of proving that go is Lyapunov stable in certain cases (Siegel and Moser, 1971, p. 226). We shall say that go is of twist type if the twist coefficient at go of the Poincar6 operator associated to (,) is not zero. This coefficient corresponds to the first nonlinear term in the Birkhoff normal form of an area-preserving map. l Departamento de Matemfitica Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada. Spain. 651 1040-7294/92/10004)651506.50/09 1992PlenumPublishingCorporation

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In this paper we prove the existence of periodic solutions of twist type under assumptions that depend on the derivatives of f up to the third order and, especially, on the signs of these derivatives. From the point of view of stability theory, these results can be seen as examples of "a method of the third approximation." It is important to remark that the assumptions are not of the kind of small parameters and that they do not imply that (.) is close to an autonomous equation. It follows from the Birkhoff fixed-point theorem and from KAM theory that, around a solution of twist type, there exist subharmonic solutions with minimal periods going to infinity as well as many quasi-periodic solutions (see Siegel and Moser, 1971; Arnold and Avez, 1968). Therefore some consequences on the existence of solutions of these classes can be derived from this paper. These results are applied to a pendulum of variable length and to a periodic problem of Ambrosetti-Prodi type. The existence of solutions of the periodic Ambrosetti-Prodi problem was analyzed by Fabry etal. (1986). The stability properties of these solutions were studied by Ortega (1989, 1990), assuming that (.) was perturbed by a linear friction. A partial extension to the nonfriction case of some of the results is obtained. The paper is organized as follows. In Section 2 we recall some facts on normal forms of area-preserving maps which are needed and state the main results. In Section 3 we combine the results of Section 2 with some of the techniques developed by Ortega (1989) to obtain the applications. Section 4 is divided into four parts and devoted to the proofs of the results in Section 2. Section 4.1 is concerned with the computation of the twist coefficient of an abstract area-preserving map. In Section 4.2 the derivatives of the third order of the Poincar6 map are found. Section 4.3 shows a useful property of the m o n o d r o m y matrices of Hill's equation. Finally, the proofs are completed in Section 4.4. Throughout the paper we use the following notation:

Ixl = Euclidean norm of a column vector x ~ R 2 R[O] = (cos 0 \ sin 0

- s i n 0~, cos 0 /

rotation of angle 0 e

Given g, h: g2 ~ R, I2 measure space, g So, at least one T-periodic solution if s = So, and no T-periodic solution if s < So. Massera's theorem and a truncation argument given by Ortega (1989) prove, in addition, that every solution of (3.1) is unbounded if s < So. This result is inspired by the paper of Ambrosetti and Prodi (1972). The original result was obtained for a Dirichlet problem and replaced the coercivity condition (3.2) by more restrictive assumptions on convexity and

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jumping of eigenvalues. These conditions can be translated to the periodic case as

g"(x)>O,

Vxe~

- ~ ~< g'( - oo ) < 0 < g'( + co ) ~< (2~/T) 2

(3.3) (3.4)

[Note that 0 and (2n/T) 2 are the first two eigenvalues of the operator - d 2 / d t 2 acting on T-periodic functions.] On the other hand, if (3.3) and (3.4) are assumed, the conclusion can be sharpened and (3.1) now has exactly two T-periodic solutions if s >So and exactly one if s = So. A proof can be obtained adapting the arguments of Ortega (1989, 1990). In the process one obtains some additional information on these periodic solutions. In fact, assuming that (3.3) and (3.4) hold and S>So, the two T-periodic solutions ~ol and q~2 are 1-elementary and the corresponding indexes satisfy

~)T(q)i)=(--1) i,

i=1,2

[Here 7T refers to the index of a T-periodic solution as given by Krasnoselskii (1968)]. Additional information is now obtained when inequalities more restrictive than (3.4) are assumed.

Proposition I.

Assume that g satisfies (3.3) and - ~ ~< g ' ( - ~ )

< 0 < g ' ( + ~ ) ~ So, ~ol is hyperbolic and ~oz is elliptic. The proof follows from the previous remarks together with the result below, which concerns Eq. (,).

Lemma 1. assume that

Let q~ be a 1-elementary T-periodic solution of ( , ) and

L(.,

)

Then q~ is elliptic (resp. hyperbolic) if and only tfTr(q~)= 1 (resp. - 1 ) . This lemma is inspired by Theorem 1.1 of Ortega (1989) and the proof is essentially the same. In order to apply Theorem 1 in combination with Proposition 1, we impose g"(x) > 0 and g"(x) 0 and D~ = diag(~, ~ - ' ). In addition, J0akF*(0) > 0

for each triplet

1 0. We now apply the previous discussion to G.

Proposition 5. Assume that F ' ( O ) = D ~ R [ - O ] D2 ~,

0 ~ (0, ~/2)

In addition, O,~F(0)=0 for each couple 1~~(Tt/2~)2 f~ y(s)2 ds, VyeHl(0,'c),

y(0)=0

or

y(r)=0

(4.10)

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This follows from the variational characterization of the first eigenvalue of the Sturm-Liouville problems y" + 2y = 0, y(0) = y'(r) = 0 [or y'(0) =

y(~)=0]. By contradiction, assume that for some r s (0, T), bg(z) = 0. Multiplying (4.7) by ~ and integrating over (0, r),

Io (b~(s)2 ds= fo a(s) q~i(s)2 ds ~ a~ fo ~i(s) 2 ds a contradiction with (4.10) since a2 < n/2r.

Lemma 3. Assume that (4.9) holds and y is the solution of (4.8) with p~C[O, T], p>.O on [0, T]. Moreover, if p(to) 0 for each t ~ [-to, T]. The function y is given by y(t)= -Sto G(t, s)p(s)ds, where G(t,s)=r O~s 0, 0 r nn, n = 0, 1, 2 ..... For each toe ~, let ~(t, to) denote the matrix solution of Y ' = A ( t ) Y , Y(to)=I, where A(t) = (_~m ~). The matrix ~(to + T, to) is a monodromy matrix of (4.7) and belongs to the class of matrices which are similar to R[ _+0]. The following result shows that r + T, to) can be chosen in a smaller class.

Proposition 7. In the previous assumptions there exist to and ~ > 0 such that either (i)

q~(to+ T, to)=D~R[O] D21 or (ii)

q~(to+ T, t o ) = D , R [ - O ] D~'

with D~ = diag(~, a - l ) . Remark 4. The first alternative (i) occurs when 2 is Krein-negative and (ii) occurs when 2 is Krein-positive, in the sense of the theory of linear Hamiltonian systems (see Ekeland, 1990). In the proof of the proposition we use the following elementary result.

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Lemma 4. Let M be a real 2 • 2 matrix with eigenvalues 2 and ~ and eigenvector w e C 2 - {0} such that M w = 2 w . Define P = ( R e w [ - I m w). Then M = PR[O] P-~. Proof. Let ~b(t) be a nontrivial (complex-valued) solution of (4.7) such that ~b(t + T) = 2qt(t), t ~ ~. Since t ~ L~b(t)[2 is real-valued and T-periodic, there exists to~N such that (d/dt)I~b(t)12=0 at t=to. Also, ~b(to) ~0. [-Otherwise (~(t)/O'(to)= )~(t) is a real-valued solution satisfying ;~(t+ T ) = 2Z(t), contradicting 2 r ~]. Define O ( t ) = O(t)/(~(to). Then O ( t o ) = l and (d/dt) lO(t)l 2 = 0 at t = t o . Setting ~ = ~ 1 + i ~ 2 , one has O l ( t o ) = l , O2(to)=0, O](to)=0. Now (O(to),O'(to)) t is an eigenvector of qb(to+T, to) and L e m m a 4 implies that q)(to+T, t o ) = P R [ O l P -1 with P = d i a g ( 1 , - ~ ; ( t o ) ) . If O~(to)0, the whole reasoning is repeated replacing 2 by J. and ~b by q~, and the second alternative holds. 2 Proposition 8. In the previous setting assume, in addition, that a 1 2. Then there exists to and ~ > 0 such that q~(to+ T, t o ) = D ~ R [ - O ] O~ ~ with 0 ~ (0, n/n].

2 First, we Proof. The case a = a] is trivial so that we assume a,~ a n. prove that the multipliers of (4.7) satisfy

2 = e i~

[0[ 0 such that 9 (to+ T, to)=D~R[ +_0] D~ 1, O~ (0, n/n]. Lemma 2 implies that the coefficients of the first row of ~b(to + T, to) are nonnegative so that the rotation R must be of angle - 0 .

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4.4. Conclusion of the Proofs Proof of Theorem 1. Let t o ~ ~ be given as in Proposition 8 (n = 3) and make the change of the independent variable z = t - t o . Then (,) is transformed into (d2/& 2) x + g(z, x) = 0

(4.12)

where g(y, x) = f ( t 0 + r, x) = a(to + ~) x + b(to + ~) x 2 + C(to + r) x 3 + .... Denote by Q the Poincar6 operator associated to (4.12). We have Q ' ( O ) = D , R [ - O ] D ~ 1, 0 e ( 0 , z/3]. Now,. according to Section4.2, OokQ*(O)=Jd ~, and since the coefficients of g are translates of a, b, and c, Proposition6 is applicable and d~ Thus JaokQ*(O)=J2d ijk= - d U~> 0 and, from Proposition 4,/?(Q, 0) > 0. Now Q and P, the Poincar6 map of (.), are symplectically conjugate, implying/~(Q, 0) =/~(P, 0), and the proof is finished. The proof of Theorem 2 is similar, Proposition 5 is used.

Proof of the Example of Instability. We use the following instability criterion, due to Levi-Civitfi (Siegel and Moser, 1971, p. 222): let F e C3(0, R 2) be such that F ( 0 ) = 0, where f2 is some open neighborhood of 0 in N 2. Assume that, in complex notation, F(z,~)=r

3)

as

z~0

where o92 + ~o + i = 0 and c r 0. Then z = 0 is unstable as a fixed point of F. The translation of these assumptions into real notation is f'(O)=R[+__2rc/3],

(011-022)f-2JO12F=/=O

We apply this result with F = P and assume that T = 2r~/3. (This is not restrictive, thanks to the change of the time-variable v=27rt/3T). Then P'(O) = R [ - 2 ~ / 3 ] and, since it commutes with J, (011 -- 022) P - 2JOI2P = R [ --2rc/3 ](J(d H - d 22) + 2d~2),

which is different from zero if and only if (2.4) holds.

REFERENCES Ambrosetti, A., and Prodi, G. (1972). On the inversion of some differentiablemappings with singularities between Banach spaces. Ann. Mat. Pura Appl. 93, 231-246. Arnold, V. I., and Avez, A. (1968). Ergodic Problems of Classical Mechanics, Benjamin, New York. Arrowsmith, D. K., and Place, C. M. (1990). An Introduction to Dynamical Systems, Cambridge University Press, Cambridge.

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Cesari, L. (1971). Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, Springer Verlag, Berlin. Ekeland, I. (1990). Convexity Methods in Hamiltonian Mechanics, Springer Verlag, Berlin. Fabry, C., Mawhin, J., and Nkashama, N. (1986). A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations. Bull. London Math. Soc. 18, 173-180. Fonda, A., Ramos, M., and Willem, M. (1989). Subharmonic solutions for second order differential equations. Preprint. toos, G. (1979). Bifurcation of Maps and Applications, North-Holland, Amsterdam. Krasnoselskii, M. (1968). Translations Along Trajectories of Differential Equations, American Mathematical Society, Providence, RI. Magnus, W., and Winkler, S. (1979). Hill's Equation, Dover, New York. Markus, L., and Meyer, K. R. (1980). Periodic orbits and solenoids in generic hamittonian dynamical systems. Am. J. Math. 102, 25-92. Moeckel, R. (1990). Generic bifurcations of the twist coefficient. Ergod. Theor. Dynam. Syst. 10, 185-195. Ortega, R. (1989). Stability and index of periodic solutions of an equation of Duffing type. Boll. Un. Mat. ItaL 3-B, 533-546. Ortega, R. (1990). Stability of a periodic problem of Ambrosetti-Prodi type. Diff. Integral Eqs. 3, 275-284. Siegel, C. L., and Moser, J. (1971). Lectures on Celestial Mechanics, Springer Verlag, Berlin. Wan, Y. (1978). Computation of the stability condition for the Hopf bifurcation of diffeomorphisms on ~2. Siam J. Appl. Math. 34, 167-175.