Periodic solutions of a system of coupled oscillators with one-sided superlinear retraction forces

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1 Periodic solutions of a system of coupled oscillators with one-sided superlinear retraction forces Alessandro Fonda and Andrea Sfecci Abstract We generalize the phase-plane method approach introduced in [7] to the case of higher dimensions. To this aim, the phase-space is assumed to be decomposable as a product of planes, where the corresponding components of the solutions can be controlled by means of suitable plane curves. We then apply our general result to the periodic problem associated to a system of coupled oscillators, with retraction forces having a linear growth, or with one-sided superlinear nonlinearities. Introduction The aim of this paper is to extend to higher dimensions the existence results obtained in [7] for the periodic problem associated to some planar systems of ordinary differential equations. We want to prove the existence of T -periodic solutions for a system like u = F (t, u), () where F : R R 2h R 2h is a continuous function which is T -periodic in its first variable. As a particular case, we have in mind a system of coupled oscillators of the type x + φ (t, x ) = e (t, x,..., x h ) x 2 + φ 2(t, x 2 ) = e 2 (t, x,..., x h ). (2). x h + φ h(t, x h ) = e h (t, x,..., x h ) Some existence theorems for general systems of this type have been provided by the use of functional analytical methods, typically using bifurcation theory, or degree theory. We refer to [, 8] and the references therein, for some classical results in this direction. Moreover, in the variational setting, when () is a Hamiltonian system, there is a large literature on this type of problems; see, e.g., [, 3] and the references therein. In [7], we studied the case h =, by the use of phase-plane methods. Such kind of methods are frequently applied to planar systems, but very rarely used in higher dimensions, due to the difficulty to control the solutions in the phase-space. The aim of the present paper is to provide a setting where it is possible to have such a control, at least when the coupling forces e i (t, ) have a sublinear growth at infinity. On the other hand, we are able to deal with many different situations involving the growth of the functions φ i (t, ). Like in [7], we can deal with functions having a linear growth, assuming either nonresonance at infinity, or a Landesman Lazer type of situation, or even with one-sided superlinear nonlinearities. Notice that our result is not of perturbative type, like e.g. the ones in [3, 9, 2], and many others, in the sense that we do not require the functions e i (t, ) to depend on a small parameter.

2 We will mainly concentrate on the situation of one-sided superlinear retraction forces, since, in our opinion, it has not yet been sufficiently studied in the literature, for higher dimensional systems like (2). We recall that, in the case of the periodic problem for a second order scalar equation, one-sided superlinear growth has been first considered in the pioneering papers by Mawhin Ward [0] and Fabry Habets [5], while a particular higher dimensional situation has been studied by Arioli and Ruf in [2], by the use of a variational method. Let us explain the main result of this paper, recalling first the approach we have used in [7]. The idea there was inspired from [5], where a particular curve γ was constructed in order to control the solutions in the phase-plane. We have then used such a curve to have the necessary estimates in order to apply the Poincaré-Bohl fixed point theorem to the Poincaré map associated to (), thus obtaining the existence of a T -periodic solution. Passing to higher dimensions, one could try to generalize this approach introducing some kind of manifolds in order to have the same type of control. This seems a very delicate problem, and it is not clear to us how such manifolds could be defined. As an alternative approach, we separate the phase-space as the product of h planes, and on each of them we construct a curve γ i, which controls the solutions in that particular plane. Assuming the coupling forces e i (t, ) to have an appropriate sublinear growth at infinity, the behaviour of a large amplitude solution x(t) = (x (t),..., x h (t)) of (2) will be approximately the same as if the oscillators were uncoupled, so that each component x i (t) of the solution will be controlled by the corresponding curve γ i. Let us briefly describe how the paper is organized. In Section 2 we provide our general setting for a system in R 2h, with h. We state and prove our main existence result, thus generalizing what we have done in [7] in the case h =. In Section 3 we particularize our assumptions in view of the applications we have in mind. In Section 4, we deal with a system with nonlinearities having either linear growth, or one-sided superlinear growth. In Section 5 we show how our existence result applies for a system of coupled oscillators. A few words about the notations, to be used on each phase-plane. We denote by, the Euclidean scalar product in R 2, and by the corresponding norm. The open ball, centered at the origin, with radius R > 0, is B 2 R = {v R2 : v < R}, and by S we denote the set {v R 2 : v = }. The cone determined by two angles θ < θ 2 is defined as Θ(θ, θ 2 ) = {v R 2 : v = ρe iθ, ρ 0, θ [θ, θ 2 ]}. (It will be sometimes convenient to use the complex notation for the points in R 2.) The closed segment joining two points v and v 2 is denoted by [v, v 2 ]. Finally, we use the standard notation ( ) 0 J =. 0 2 The main result In this section we are going to introduce a generalization of the main result obtain in [7] to a particular system in R 2h with h. Consider an open set A, containing the origin, with the following shape A = U U h, where, for i {,..., h}, the set U i R 2 is open and bounded. In the following we will write any point u R 2h in the coordinates (u,..., u h ), where u i R 2. In the same way, all the functions ϕ with image in R 2h will be written in the components (ϕ,..., ϕ h ). 2

3 We start by stating the following result, reminiscent of the Poincaré Bohl theorem, whose proof is a standard application of Brouwer degree theory (see, e.g., [4]). Theorem 2. Let ϕ : A R 2h be a continuous function such that, for every i {,..., h}, the following property holds: ϕ i (u,..., u h ) µu i, for every µ > and for every u U U i U i U i+ U h. Then ϕ has a fixed point in A. We consider the equation u = F (t, u), (3) where u = (u,..., u h ) R 2h and F = (f,..., f h ), being f i : R R 2h R 2 continuous functions which are T -periodic in the first variable. Let us recall the definition of clockwise rotating regular spiral in a plane, which can be found in [7]. Definition 2.2 A clockwise rotating regular spiral is a continuous and injective curve satisfying the following properties: γ : [0, + [ R 2,. there exists an unlimited strictly increasing sequence 0 = σ 0 < σ < σ 2 < < σ k < σ k+ < such that the restriction of γ to every closed interval [σ k, σ k+ ] is continuously differentiable, and such that J γ(s), γ(s) > 0, for every s [σ k, σ k+ ] ; (4) 2. the curve grows in norm to infinity: 3. the curve rotates clockwise infinitely many times: + 0 lim γ(s) = + ; (5) s + J γ(s), γ(s) γ(s) 2 ds = +. (6) A similar definition can be given for a counter-clockwise rotating regular spiral, by changing the inequality in (4), and requiring the integral in (6) to be equal to. Without loss of generality, we will assume that the spiral has the following parametrization: γ(s) = γ(s) (cos s, sin s). Now we want to introduce an admissibility condition for such spirals, extending the definition in [7] to the case of a higher dimensional space. 3

4 Definition 2.3 A clockwise rotating regular spiral γ is said to be i-admissible for system (3), with i {,..., h}, if, when restricted to any subinterval [σ k, σ k+ ], it satisfies J γ(s), f i (t, u,..., u i, γ(s), u i+,..., u h ) < 0, (7) for every t [0, T ], s [σ k, σ k+ ], and u j R 2 with j i. (The sequence {σ k } k is the one introduced in Definition 2.2.) Moreover, given a subset U of R 2, the spiral is said to be i-admissible in U for system (3) if (7) is satisfied whenever γ(s) U. Starting from a spiral γ i which is i-admissible for (3), it is convenient to define, for every n N, the set Ω i n: it is the open region delimited by the Jordan curve Γ i n obtained by glueing together the piece of curve γ i going from γ i (2πn) to γ i (2π(n + )), and the segment joining the two endpoints: Γ i n = { γ i (s) : s [2πn, 2π(n + )] } [ γ i (2πn), γ i (2π(n + )) ]. Notice moreover that, by the injectivity, one has Let us now state our main result. γ i (s) < γ i (s + 2π) for every s > 0. (8) Theorem 2.4 Suppose that the following assumptions hold, for every i {,..., h}. (H i ) There exists a clockwise rotating regular spiral γ i : [0, + [ R 2 which is i-admissible for (3). (H2 i ) There exists R i > 0 such that, for any solution u : [0, T ] R 2h of (3), satisfying one has that, either u i (T ) < u i (0), or u i (t) R i, for every t [0, T ], T 0 Ju i (t), u i(t) u i (t) 2 dt / 2πN. (H3 i ) There exist C i > 0 and θ i < θi 2 such that Jf i (t, u), u i C i ( u i 2 + ), for every t [0, T ] and u R 2h with u i Θ(θ, i θ2) i. Then, a T -periodic solution of equation (3) exists. Proof. Take R max{, R,..., R h } such that Ω i 0 BR 2 for every i. Let m be a positive integer such that B 2 R Ω i m for every i, and let n be an integer such that, for every i, n > (Ci + )T θ i 2 θi. (9) We can find a R > R such that Ω i m + n+ BR 2 for every i. In the same way we can find an integer m 2 > m + n + such that B 2 R Ω i m 2 for every i, and a constant R 2 > R such that Ω i m 2 + n+ BR 2 2 for every i. 4

5 Define, for any r > 0, B r = (Br 2 ) h = Br 2 Br 2 R 2h. }{{} h times Consider a sequence (F n ) n = (f n,..., f n h ) n of locally Lipschitz continuous functions converging to F uniformly on [0, T ] B R2. For any i, by (7), as long as, for some s, ũ = (u,..., u i, γ i (s), u i+,..., u h ) belongs to B R2, then, for n large enough, moreover, by (H3 i ), for n sufficiently large, J γ i (s), f n i (t, ũ) < 0, for every t [0, T ] ; (0) Jf n i (t, v), v i v i 2 C i +, () for every t [0, T ] and v B R2 whose i-th component is such that v i Θ(θ i, θi 2 ) (B2 R 2 \ B 2 R ). The solution to the Cauchy problem associated to the equation u = F n (t, u) (2) is unique for every n, and, if u n is a solution of (2) satisfying u n i (0) R for every i, then, for sufficiently large n, u n i (t) < R 2, for every t [0, T ] and for every i (3) (i.e. u n (t) B R2 for every t [0, T ]). Indeed, for such n, assuming by contradiction that max{ u n i (t) : t [0, T ], i =,..., h} R 2 (i.e. u n (t) / B R2 for at least one t [0, T ]), there exists an index j and t, t 2 [0, T ] with t < t 2, such that: u n j (t ) = R u n j (t 2 ) = R 2 R < u n j (t) < R 2 for every t ]t, t 2 [, (4) u n i (t) R 2 for every t [0, t 2 ] and every i j. (5) Then, for t varying from t to t 2, by (0) the component u n j of the solution would be driven by the curve γ j to make at least n+ clockwise revolutions around the origin, thus crossing at least n times the cone Θ(θ j, θj 2 ), in the clockwise sense. Writing the solution in polar coordinates from () we have that, if θ j ϑn j (t) θj 2, then u i (t) = ρ i (t)(cos(ϑ i (t)), sin(ϑ i (t)) ), (6) (ϑ n j ) (t) = Jf n j (t, un (t)), u n j (t) u n j (t) 2 C j +. So, the time to cross the cone Θ(θ j, θj 2 ) in the clockwise sense is at least (θj 2 θj )/(Cj + ), and then, by (9), the time to cross it n times should be greater than T. Hence, t 2 t > T, which is impossible. The Poincaré map associated to (2) is then well defined on B R. Let us now see that Theorem 2. can be applied for every n large enough, up to a subsequence, taking as A the set B R, in order to find a periodic solution to the equation (2). 5

6 Assume by contradiction that, for every n large enough, there exist µ n > and ū n B R with ū n = (ū n,..., ū n i n,..., ū n h ) B2 R B 2 R B 2 R (7) for a suitable i n, such that the solution u n of (2) with u n (0) = ū n satisfies u n i n (T ) = µ n u n i n (0). We claim that, for n large enough, it has to be R < u n i n (t) < R 2, for every t [0, T ]. (8) Indeed, we already proved above that max{ u n i (t) : t [0, T ], i =,..., h} < R 2. Assume by contradiction that min{ u n i n (t) : t [0, T ]} R. Then, since u n i n (T ) > R, there would be ˆt, ˆt 2 in [0, T ], with ˆt < ˆt 2, such that and u n i n (ˆt ) = R, u n i n (ˆt 2 ) = R, R < u n i n (t) < R, for every t ]ˆt, ˆt 2 [. Then, for t varying from ˆt to ˆt 2, by (0) the component u n i n of the solution would be driven by the curve γ in to make at least n + clockwise revolutions around the origin, thus crossing at least n times the cone Θ(θ in, θin 2 ), in the clockwise sense. Arguing as above, we see that ˆt 2 ˆt > T, which is impossible. By (8), necessarily it has to be < µ n < R 2 R, so, up to subsequences, we can assume that: [ i n ι, µ n µ, R ] 2 R and ū n ū B R. Moreover, since (F n ) n converges to F uniformly in [0, T ] B R2, there is a constant M > 0 such that F n (t, u) M, for every n N, t [0, T ] and u B R2. By (3), u n (t) B R2 for every t [0, T ], so (u n ) n is bounded in C ([0, T ]) and, by the Ascoli Arzelà Theorem, there is a continuous function u : [0, T ] R 2h such that, up to a subsequence, u n u uniformly. Passing to the limit in we obtain u n (t) = ū n + u(t) = ū + t 0 t 0 F n (τ, u n (τ)) dτ, F (τ, u(τ)) dτ, so that u is a solution to the equation (3) with initial value u(0) = ū B R. By (8), R u ι (t) R 2, for every t [0, T ], (9) and u ι (T ) = µu ι (0). Hence, u ι (T ) u ι (0) and, using polar coordinates as in (6), there is an integer k such that ϑ ι (T ) = ϑ ι (0) 2πk. 6

7 By the angular velocity formula (ϑ ι ) (t) = Ju ι(t), u ι (t) u ι (t) 2, as a consequence of (H2 ι ) it has to be k. Taking into account (9) and the fact that Ω ι 0 BR 2, let m Z be such that γ ι ( ϑ ι (0) + 2π( m )) < u ι (0) γ ι ( ϑ ι (0) + 2π m). (Recall that γ ι is parametrized in clockwise polar coordinates.) Then, by the admissibility of the curve γ ι and (9), since B 2 R contains Ω ι 0, it has to be So, using (8), u ι (t) < γ ι ( ϑ ι (t) + 2π m), for every t ]0, T ]. u ι (T ) < γ ι ( ϑ ι (T ) + 2π m) = γ ι ( ϑ ι (0) + 2π( m + k)) γ ι ( ϑ ι (0) + 2π( m )) < u ι (0), and we get a contradiction with the fact that u ι (T ) u ι (0). So, up to a subsequence, for every ū n B R (with associated, as in (7), an index i n such that ū n i n = R ), the solution u n of (2) with u n (0) = ū n is such that u n i n (T ) µū n i n, for every µ >. We can then apply Theorem 2. to find a T -periodic solution v n (t) of (2), for n large enough, up to a subsequence, starting from a point v n B R. Using the Ascoli Arzelà Theorem again, we find that, up to a subsequence, (v n ) n converges to a T -periodic solution of equation (3). 3 Some applicative conditions In this section we introduce three other hypotheses which are useful to obtain (H i ), (H2 i ) and (H3 i ). These conditions are simply the generalizations to the 2h-dimensional case of the conditions (H4), (H5) and (H6) introduced in [7]. (H4 i ) There exist R > 0 and η > 0 such that, for every v R 2h v i R Jf i (t, v), v i η v i 2, for every t [0, T ]. (H5 i ) There exists a continuous function χ : [0, + [ ]0, + [ such that f i (t, v), v i χ( v i ), for every t [0, T ] and v R 2h, (20) and + 0 r dr χ(r) = +. (H6 i ) There exist some values w,..., w m S and two positive functions ψ, ψ 2 : S \ {w,..., w m } ]0, + ], not identically equal to +, with the following properties: 7

8 (i) in each open arc of the domain these functions are either continuous with all values in R, or identically equal to + ; (ii) one has ψ (w) lim inf α + lim sup α + Jfi (t, u,..., u i, αw, u i+,..., u h ), w α Jfi (t, u,..., u i, αw, u i+,..., u h ), w ψ 2 (w), α (2) uniformly for t [0, T ], (u,..., u i, u i+,..., u h ) R 2h 2 and w in any compact subset of S \ {w,..., w n }; (iii) moreover, [ 2π 0 dθ 2π ] ψ 2 (e iθ ), dθ 0 ψ (e iθ ) where N 0 denotes the set of positive integers. { } T N : N N 0 = Ø, (22) Notice that, in (22), we use the convection that + = 0, and implicitly assume that the integrals have finite values. In [7], the following two propositions have been proved. Proposition 3. If (H4 i ) and (H5 i ) hold, then (H i ) is satisfied. Proposition 3.2 If (H4 i ) and (H6 i ) hold, then (H2 i ) and (H3 i ) are satisfied. We will see below how condition (H5 i ) can be weakened when dealing with different regions in the phase-plane. 4 Applications In this section we are going to prove the existence of a T -periodic solution to the following system: Ju = g (t, u ) + r (t, u,..., u h ) Ju 2 = g 2(t, u 2 ) + r 2 (t, u,..., u h ). Ju h = g h(t, u h ) + r h (t, u,..., u h ). (23) We assume that, for every i {,..., h}, writing u i = (x i, y i ), the i-th equation of the system has the following form: { y i = g i,(t, x i ) + r i, (t, x, y,..., x h, y h ) x i = g, (24) i,2(t, y i ) + r i,2 (t, x, y,..., x h, y h ) where the functions r i,j : R R 2h R are continuous, and T -periodic in their first variable. Moreover, we assume that there exist functions p i,j : R 2 R such that p i,j (u i ) r i,j (t, u,..., u h ) p i,j (u i ), with lim = 0. u i + u i 8

9 The functions g i,j : R R R are continuous and T -periodic in their first variable. We assume that, for j, k {, 2} and i {,..., h}, there are some constants µ i j,k, νi j,k ]0, + ] such that µ i j, lim inf ξ + ν i j, lim inf ξ g i,j (t, ξ) ξ g i,j (t, ξ) ξ With the usual convention that + = 0, let τk i = π 2 µ i,k µi 2,k for i {,..., h} and k {, 2}. lim sup ξ + lim sup ξ + + ν,k i µi 2,k g i,j (t, ξ) ξ g i,j (t, ξ) ξ ν,k i νi 2,k + µ i j,2, (25) ν i j,2. (26) µ i,k νi 2,k, (27) Theorem 4. Assume that all the constants in (25) and (26) are finite, and { } T [τ2, i τ] i N : N N 0 = Ø, for every i {,..., h}. (28) Then, system (23) has a T -periodic solution. The same is true if, for one or more index i, one of the constants µ i,2, νi,2, µi 2,2, νi 2,2 is equal to +, the three others being finite. The procedure to verify that (H i ), (H2 i ) and (H3 i ) hold for every i is indipendent of the index i, so in the following we will consider the case i = and, to simplify the notations, we will often write u = (x, y) instead of (x, y ), and λ = (x 2, y 2,..., x h, y h ). Hence we have Jf (t, x, y, λ) = g (t, x, y) + r (t, x, y, λ). In the proof we will need the following refinement of Proposition 3., where we focus our attention on the admissibility of a regular spiral on a particular subset U of R 2. Lemma 4.2 Given a positive constant η, a point P 0 R 2, and a continuous function χ : [0, + [ ]0, + [ such that + r dr χ(r) = +, 0 it is possible to build a clockwise rotating regular spiral γ, passing through P 0, such that s γ(s) is strictly increasing, which is -admissible for system (23) in any set U where. Jf (t, x, y, λ), (x, y) η(x 2 + y 2 ), ( ) 2. f (t, x, y, λ), (x, y) χ x 2 + y 2, for every t [0, T ], (x, y) U and λ R 2h 2. Proof. Let P 0 = r 0 e is 0. We define the curve γ : [0, + [ R 2 as γ(s) = r(s)(cos s, sin s), 9

10 where r(s) is the solution of the Cauchy problem ṙ = 2 η χ(r) r, r(s 0 ) = r 0. Notice that γ(s 0 ) = P 0. Since this curve is smooth, the sequence (σ k ) k, in this case, is arbitrary. Clearly, (4) and (6) hold, since s is the angle in clockwise polar coordinates. We see that r(s) is strictly increasing, and remains bounded for s bounded. Moreover, r(s) + for s +, so that condition (5) is satisfied, as well. We compute, J γ(s), f (t, γ(s), λ) = ṙ(s) r(s) J γ(s), f (t, γ(s), λ) + γ(s), f (t, γ(s), λ). Using the assumptions, we have that, for γ(s) U, J γ(s), f (t, γ(s), λ) η ṙ(s)r(s) + χ(r(s)) < 0, for every t [0, T ] and λ R 2h 2, thus completing the proof. Proof of Theorem 4.. It is easy to see that the functions ψ and ψ 2 which are involved in (H6 ) are µ, cos2 θ + µ 2, sin2 θ, if θ ] 0, π [ 2, ν, cos2 θ + µ 2, sin2 θ, ψ (e iθ ) = if θ ] π 2, π[, ν, cos2 θ + ν 2, sin2 θ, if θ ] π, 3π 2 [, and µ, cos2 θ + ν2, sin2 θ, if θ ] 3π 2, 2π[, µ,2 cos2 θ + µ 2,2 sin2 θ, if θ ] 0, π [ 2, ν,2 cos2 θ + µ 2,2 sin2 θ, ψ 2 (e iθ ) = if θ ] π 2, π[, ν,2 cos2 θ + ν 2,2 sin2 θ, if θ ] π, 3π 2 [, µ,2 cos2 θ + ν2,2 sin2 θ, if θ ] 3π 2, 2π[. Being all the constants in (25) ans (26) strictly positive, (H4 ) holds. Solving the integral it is easy to see that also (H6 ) holds (see [6] for computations). So, by Proposition 3.2, conditions (H2 ) and (H3 ) hold. If all the constants µ,2, ν,2, µ 2,2, and ν 2,2 are finite, the nonlinearity has an at most linear growth, so (H5 ) holds with χ(r) = ar 2 + b for some suitable constants a and b. By Proposition 3., condition (H ) is satisfied, and the proof is completed in this case, by Theorem 2.4. We now consider the case in which one of these constants is equal to +. For example, we assume µ,2 = +. 0

11 In order to build an admissible spiral γ in this case, we will glue together pieces of curves belonging to some regions of the plane. By construction the curve will pass through some points P α, whose distance from the origin gradually increases, giving to it the shape of a regular spiral. In what follows, we will sometimes use Lemma 4.2, whose condition is satisfied by (H4 ), so that we will only need to find a suitable function χ in order to apply it. Let ɛ > 0 be fixed, in such a way that ɛ < 8 min { µ,, ν,, µ 2,, ν 2,}. Then, there exists R > 0 such that, for every (x, y) for which x R and y R, r i,j (t, x, y, λ) p,j (x, y) ɛ( x + y ), j =, 2. We can assume R > 0 large enough to have x R 0 < µ, x g, (t, x), x R ν,2 x g, (t, x) ν, x < 0, y R 0 < µ 2, y g,2 (t, y) µ 2,2 y, y R ν 2,2 y g,2 (t, y) ν 2, y < 0, slightly modifying these constants, if necessary, without affecting (28). Moreover, we have the existence of a constant C > 0 such that x R g, (t, x) C, y R g,2 (t, y) C. We consider five different regions in the phase-plane (see Figure ): W = ], R] R, NE = [R, + [ [R, + [, E = [R, + [ [ R, R], ESE = [R, + [ ], R] {(x, y) : x y}, SSE = [R, + [ ], R] {(x, y) : x y}. The regular spiral γ will be constructed by glueing together pieces of curves belonging to each of these regions. Region W. We note that, in this region, giving us, for every (x, y) W, g,2 (t, y) C + max{µ 2,, µ 2,2, ν 2,, ν 2,2} y, g, (t, x) C + max{ν,2, ν,} x, f (t, x, y, λ), (x, y) = g,2 (t, y)x g, (t, x)y + r,2 (t, x, y, λ)x r, (t, x, y, λ)y C (x 2 + y 2 ) + C 2,

12 Figure : The regions in the phase-plane. for some suitable constants C and C 2. Fix a point P 0 = (R, y 0 ) with y 0 < R. By Lemma 4.2, taking U = W, we can build the spiral γ which passes through P 0 = γ(s 0 ). There exists s > s 0 such that γ([s 0, s ]) W and γ(s ) = P = (R, y ) with y > R. The spiral γ in W consists of the branch of γ which goes from P 0 to P, and it is abmissible in W by construction. Region NE. We have, for every (x, y) NE, f (t, x, y, λ), (x, y) = g,2 (t, y)x g, (t, x)y + r,2 (t, x, y, λ)x r, (t, x, y, λ)y µ 2,2 xy ɛ(x + y) 2 M (x 2 + y 2 ) for a suitable constant M. Similarly as what has been done in the region W, applying Lemma 4.2 with U = NE, we can construct γ going from P to a point P 2 = (x 2, R) with x 2 > R. Region E. In this region, we construct the spiral γ as a line y = mx where 0 < m < is sufficiently small. We recall that, here, g,2 (t, y) C, g, (t, x) µ,x, r,i (t, x, y, λ) ɛ( x + y ) 2ɛx, i =, 2. Hence, we have J γ (s), f (t, γ (s), λ) = (m, ), (x, y ) = m(g,2 (t, y) + r,2 (t, x, y, λ)) (g, (t, x) + r, (t, x, y, λ)) mc + 2mɛx µ,x + 2ɛx mc (µ, 4ɛ)x mc µ, 2 R which is negative choosing m < µ, R/2C. In this way we build a branch of the spiral γ which goes from P 2 to a point P 3 = (x 3, R), with x 3 > x 2 > R. 2

13 Region ESE. In this region the spiral γ simply coincides with the line y = 2(x x 3 ) R. Let P 4 = (x 4, x 4 ) be the intersection between this line and the line y = x. We recall that, here, g,2 (t, y) ν 2, y < 0, g, (t, x) µ, x, r,i (t, x, y, λ) ɛ( x + y ) 2ɛ x, i =, 2. So we have J γ (s), f (t, γ (s), λ) = (2, ), (x, y ) = 2(g,2 (t, y) + r,2 (t, x, y, λ)) (g, (t, x) + r, (t, x, y, λ)) 2(0 + 2ɛ x) (µ, x 2ɛ x) = (µ, 6ɛ) x < 0. Region SSE. In this region the following inequalities hold: g,2 (t, y) ν 2, y < 0, g, (t, x) µ, x, r,i (t, x, y, λ) ɛ(x y) 2ɛ y, i =, 2. At first, we note that for a solution of (23), with the above notations, as long as (x(t), y(t)) belongs to this region, we have that x (t) = g,2 (t, y(t)) + r,2 (t, x(t), y(t), λ) (ν 2, 2ɛ) y(t) < 0. (29) We have to build the spiral γ starting from the point P 4 = (x 4, x 4 ). Call SSE good the region SSE {x x 4 } and SSE bad the region SSE {x > x 4 } (see Figure 2). Figure 2: The construction of the curve in the region SSE. Note that, for every (x, y) SSE good, f (t, x, y, λ), (x, y) = g,2 (t, y)x g, (t, x)y + r,2 (t, x, y, λ)x r, (t, x, y, λ)y 0 M y + ɛ(x y) 2 M 2 (x 2 + y 2 ) + M, 3

14 for some suitable constants M and M 2. Setting U = SSE good, it is possible to apply Lemma 4.2 to obtain a spiral γ which links P 4 to a point P 6 = (R, y 6 ), with y 6 < R, passing through SSE. By construction, this spiral is -admissible only in SSE good. Nothing tells us that γ does not enter in the region SSE bad, but there exists a point P 5 = (x 5, y 5 ) with x 5 = x 4 and y 5 y 4 (possibly P 5 = P 4 ) on the curve γ after which γ is contained in SSE good. Using (29), we choose the spiral γ to be made of the vertical line linking P 4 and P 5 and of the branch of γ linking P 5 and P 6. With such a procedure we have constructed the first lap of the spiral γ. In the same way we can obtain the other ones. Such a spiral is -admissible in the whole plane by construction. So, (H ) holds, and the proof is completed in this case, too. 5 An example: coupled oscillators As a particular case of (23), we have the following system of coupled oscillators x x Here we assume that, for every i, and that the function φ i satisfies + φ (t, x ) = e (t, x,..., x h ) 2 + φ 2(t, x 2 ) = e 2 (t, x,..., x h ). x h + φ h(t, x h ) = e h (t, x,..., x h ) p i (x i ) e i (t, x,..., x h ) p i (x i ), with lim = 0, x i + x i µ i lim inf ξ + ν i lim inf ξ φ i (t, ξ) ξ φ i (t, ξ) ξ lim sup ξ + lim sup ξ φ i (t, ξ) ξ φ i (t, ξ) ξ. (30) µ i 2, (3) ν i 2, (32) for some suitable constants in ]0, + ]. With the usual convention that + = 0, let τ i = π µ i + π ν i, τ2 i = π µ i 2 + π, (33) ν2 i for i {,..., h}. As an immediate consequence of Theorem 2.4, we have the following. Corollary 5. Assume that all the constants in (3) and (32) are finite, and { } T [τ2, i τ] i N : N N 0 = Ø, for every i {,..., h}. (34) Then, system (30) has a T -periodic solution. The same is true if, for one or more index i, one of the constants µ i 2 and νi 2 is equal to +, the other being finite. 4

15 Let us show an example of a situation which permits us to apply the previous corollary. We will use the following notations: for every ξ R we write ξ + = max{ξ, 0}, ξ = max{ ξ, 0} and for every x = (x, x 2,..., x h ) R h we write x + = (x +, x+ 2,..., x+ h ), x = (x, x 2,..., x h ), exp(x) = (ex, e x 2,..., e x h ). Consider the equation in R h x B(t) arctan( x ) x + exp(x + ) = a(t), (35) where x is a norm in R h, a : R R h is a T -periodic continuous function, and B(t) = diag(b (t),..., b h (t)) is a diagonal matrix where each b i : R R is continuous and T -periodic. We assume that there exist some positive integers N i, and a constant δ > 0, such that 2π (λ N i + δ) < b i (t) < 2π (λ N i + δ), where λ k = (2πk/T ) 2 is the k-th eigenvalue for the T -periodic problem. We can see that this is a particular case of system (30), with and we find the values φ i (t, x i ) = π 2 b i(t) x i + exp(x + i ), ( e i (t, x,..., x h ) = arctan( x ) π ) b i (t) x i + a i (t), 2 µ i 2 = +, ν i = 4 (λ N i + δ), ν i 2 = 4 (λ N i + δ), for every i. Thus, choosing µ i large enough, we have T N i + < τ i 2 = 2π λni + δ < τ i = 2π λni + δ + π µ i < T N i, for every i. Corollary 5. can thus be applied, so that equation (35) has a T -periodic solution. Remark 5.2 We have focused our attention on one particular situation where our conditions (H i ), (H2 i ) and (H3 i ) hold, for every i. As we have shown in [7], many other different cases can be treated with the same approach, like, e.g., nonlinearities controlled by positively homogeneous Hamiltonian functions, Landesman Lazer situations at resonance, and nonlinearities with a singularity. Our theorem permits to mix together all these situations. For example, one could think about a system in R 6 with a one-sided superlinearity in the first couple of variables, a resonance case with a Landesman-Lazer condition in the second one, and a singularity in the last one. Remark 5.3 One can extend our results to the case when the phase-space R 2h is replaced by a space of the type R 2h+k, for some k, introducing some hypotheses on the last k coordinates. For example, one could think of some kind of dissipative situation, so that a variant of Theorem 2. will be appliable. For briefness, we will not enter into details here. 5

16 References [] A. A. Andronov, A. A. Vitt and S. E. Khaikin, Theory of oscillators, reprint of the 966 translation, Dover Publications, New York, 987. [2] G. Arioli and B. Ruf, Periodic solutions for a system of forced and nonlinearly coupled oscillators with applications to electrical circuits, Dynam. Systems Appl. 4 (995), [3] C. Chicone, Periodic solutions of coupled oscillators near resonance, SIAM J. Math. Anal. 26 (995), [4] T. R. Ding, Some fixed point theorems and periodically perturbed nondissipative systems, Chinese Ann. Math. 2 (98), [5] C. Fabry and P. Habets, Periodic solutions of second order differential equations with superlinear asymmetric nonlinearities, Arch. Math. 60 (993), [6] A. Fonda and J. Mawhin, Planar differential systems at resonance, Adv. Differential Equations (2006), 33. [7] A. Fonda and A. Sfecci, A general method for the existence of periodic solutions of differential systems in the plane, preprint, available online at: Matematici/ pdf [8] E. Gaines and J. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Lect. Notes Math. 568, Springer, Berlin, 977. [9] E. A. Jackson, Nonlinear coupled oscillators I. Perturbation theory; ergodic problem, J. Math. Phys. 4 (963), [0] J. Mawhin and J. R. Ward, Periodic solutions of some forced Liénard differential equations at resonance, Arch. Math. 4 (983), [] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems. Applied Mathematical Sciences, 74, Springer, Berlin, 989. [2] R. E. Mickens, A regular perturbation technique for nonlinearly coupled oscillators in resonance, J. Sound Vibration 8 (982), [3] M. Struwe, Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems, fourth edition, Springer, Berlin,

17 Authors addresses: Alessandro Fonda Dipartimento di Matematica e Informatica Università di Trieste P.le Europa I-3427 Trieste Italy a.fonda@units.it Andrea Sfecci SISSA - International School for Advanced Studies Via Bonomea 265 I-3436 Trieste Italy sfecci@sissa.it Mathematics Subject Classification: 34C25, 34C5 Keywords: periodic solutions, coupled oscillators, nonlinear dynamics. 7