AN APPLICATION OF MOSER’S TWIST THEOREM TO SUPERLINEAR IMPULSIVE DIFFERENTIAL EQUATIONS

. In this paper, we consider a simple superlinear Duﬃng equation with impulses, where p ( t + 1) = p ( t ) is an integrable function in R . In order to apply Moser’s twist theorem, we need to ensure that the corresponding Poincar´e map of (0.1) is quite close to a standard twist map but it is not usually achieved due to the existence of impulses. Two types of impulsive functions which overcome this problem with diﬀerent eﬀects in the Poincar´e map are provided here. In both cases, there are large invariant curves diﬀeomorphism to circles surrounding the origin and going to the inﬁnity, which conﬁne the solutions in its interior and therefore lead to the boundedness of all solutions. Furthermore, it turns out that the solutions starting at t = 0 on the invariant curves are quasiperiodic.

In addition, assume that the impulsive time is 1-period, that is, t j+1 = t j + 1 for j ∈ Z, and p(t + 1) = p(t) being integrable, is bounded. It is well known that every solution of the Duffing equation x + x 3 = p(t), p(t + 1) = p(t) being continuous, is bounded. This result, prompted by questions of [2,6,16,19,21,24] and the applications in theory of control [9,13,20] and so on. Among these, the existence of periodic solutions of impulsive differential equations has been discussed as a hot topic via fixed point theory in [3,10,11], topological degree theory in [14,18] and variational method in [1,12,22,23,25]. Generally speaking, the behaviors of solutions with impulsive effects may have great changes compared with solutions without. Here we take a simple linear autonomous Duffing equation for example. Let consider x + (2π) 2 x = 0 (1.2) with the impulsive conditions where t j = j 2 for j ∈ Z. It is obvious to see that without impulses all solutions of (1.2) are 1-periodic and they satisfy (2πx(t)) 2 + (x (t)) 2 = C, (1.4) where C is a constant related to the initial values. However, under the influence of impulses in (1.3), all solutions except for the trivial one are unbounded. In fact, with the initial point (x(0), x (0)) = (x 0 , 0) (x 0 = 0), the solution at each t j ± is located on the x-axis and the radius of the trajectory (1.4) at t j + becomes two times larger than the previous one at t j −, that is, (x( 1 2 +), x ( 1 2 +)) = (−2x 0 , 0), (x(1+), x (1+)) = (4x 0 , 0), · · · , which implies that the solution tends to infinity as t → +∞.
From the example above, we find some behaviors, such as boundedness and periodic or quasiperiodic solutions, worth exploring for impulsive Duffing equations. Besides the commonly used variational method and some functional methods, KAM theory which includes Moser's twist theorem and related fixed point theorems for instance Poincaré-Birkhoff twist theorem is well applied to solving the existence of periodic solutions (see [5,15,17]).
In [17], D. Qian etc considered the superlinear impulsive differential equation with general impulsive functions The authors proved via Poincaré-Birkhoff twist theorem the existence of infinitely many periodic solutions of (1.5) with p = p(t), and also the existence of periodic solutions for non-conservative case with degenerate impulsive terms by developing a new twist fixed point theorem. Subsequently, Y. Niu and X. Li in [15] discussed the periodic solutions of (1.5) with sublinear g. The impulsive functions are special ones ∆x| t=tj = ax (t j −) , ∆x | t=tj = ax (t j −) , j ∈ Z. Basing on the Poincaré-Bohl fixed point theorem and the fixed point theorem established by D. Qian etc in [17], they obtained the existence of harmonic solutions and subharmonic solutions respectively. However, the study about the boundedness of solutions and the existence of quasiperiodic solutions for impulsive equations by the twist theorem is few up to now.
As we all know, one sufficient condition for applying the twist theorem on a twist map is that f, g, together with its derivatives, should be small enough with the form O(r − ) ( > 0) as r → +∞. Due to the existence of impulses, some items will be added to the right hands of Eq. (1.6). If these items in r 1 and θ 1 are small enough or if the added item in θ 1 is a constant while the added item in r 1 is 0, we can treat them equally with f and g or α(r). In both cases, the existence of impulses does not influence the application of the twist theorem. This provides us with two ideas for finding the impulsive functions. One type of impulsive functions is that I, J satisfy assumption (i): where r, s are any non-positive integers with r + s ≤ 5, h 0 (x, y) = 1 2 (x 4 + y 2 ) and ε > 0 is a given number.
We suppose the item brought by impulses in the expression of r 1 has the form O(r −ε1 ) (ε 1 > 0). This hypothesis is reasonable. In fact, if the added item in the expression of r 1 is O(r ε2 ) (ε 2 ≥ 0), r 1 dose not satisfy the twist theorem, let alone the bondedness. According to Eq. (1.1) and its Poincaré map, we can calculate the added item in the expression of θ 1 is O(r −ε3 ), where ε 3 > 0 is uniquely determined by ε 1 . Assumption (i) exactly satisfies the relation between ε 1 and ε 3 when we unify the variables, and it is the weakest condition which guarantees the application of the twist theorem from the discussions above.
The another type of impulsive functions has specific form At impulsive times, the point (x(t j −), x (t j −)) jumps to the origin-symmetric point (x(t j +), x (t j +)) = (−x(t j −), −x (t j −)) on the (x, x ) plane. Under these impulses, the item caused by impulses in the expression of r 1 is 0 and the change in the expression of θ 1 is only a constant. We regard this constant as a part of α(r), which implies the twist theorem is still valid. Consequently, we obtain the boundedness of solutions and the existence of quasiperiodic solutions of (1.1) with two types of impulsive functions mentioned above. The main results are as follows.
Then the results in Theorems 1.1 and 1.2 are also valid.
Remark 1. Our discussions start from the relatively simple Eq. (1.1) with only one impulsive time in (0, 1). Actually, for the general Duffing type equations with polynomial potentials like and with finite number of impulsive times in (0, 1), the discussion is similar and one can also obtain the results in all above Theorems.
Remark 2. Except the quasiperiodic solutions, one also obtain infinitely many periodic solutions of (1.1) with minimal period m (m ≥ 1, m ∈ N) by the Poincaré-Birkhoff fixed point theorem. The proof is similar with the proof in [4] then we omit here.
The structure of the paper is as follows. In section 2 we transform equation (1.1) into Hamiltonian system (3.1) under action and angle variables. Then some estimations of the impulsive functions and the Poincaré mapping of (3.1) under a scalar transformation are given in section 3. Finally, we prove the main results by the twist theorem in the last section.
2. Action and angle-variables. Dropping the time dependent term, we first consider equation (1.1) without impulse effects which can be written as an equivalent system of the form

This is a time independent Hamiltonian system on
Clearly h 0 > 0 on R 2 except at the only equilibrium point (x, y) = (0, 0) where h 0 = 0. All solutions of (2.1) are periodic, the periods tending to zero as h 0 = E tends to infinity. We take (C(t), S(t)) for a solution of system (2.1) with the initial conditions (C(0), S(0)) = (1, 0). Let T * > 0 be its minimal period. Then functions C and S satisfy (3) S 2 (t) + C 4 (t) = 1.
The action and angle variables are now defined by the map ϕ : R × S 1 → R 2 \ 0, where S 1 = R/Z and (x, y) = ϕ(λ, θ) with λ > 0 and with θ (mod 1) is given by the formulae: ϕ : We claim that ϕ is a symplectic diffeomorphism from R + ×S 1 onto R 2 \0. Indeed, from the Jacobian ∆ of ϕ one finds by (2) and (3) |∆| = 1, so that ϕ is measure preserving. Moreover since (C, S) is a solution of a differential equation having T * as minimal period, one concludes that ϕ is one to one and onto, which proves the claim.
As for (1.1) without the time dependent term, it can be written in the form of Hamiltonian system X h0 : where h 0 is defined in (2.2). Here and hereafter we denote (x(t j −), y(t j −)) by (x(t j ), y(t j )) for simplicity. By the transformation (2.3) and property (3) of C(t) and S(t), the Hamiltonian function of (2.4) becomes The trajectories h 0 = c (c is a parameter) of (2.1) in (x, y) plane are closed curves which are symmetric with respect to the x-axis and y-axis. In the new coordinates (λ, θ), action variable λ denotes the area encircled by the solution orbit h 0 = c, while angle variable θ is the angle with the x-axis along the orientation of solutions.
which together with (1.8) implies that and According to the initial values and the motions of solutions of Eq. (2.4), we can define θ(t j −) well. One difficulty we encountered is the definition and estimation of θ(t j +), since the arguments which are up to any integer multiple of 1 are generally regarded as the same ones. Therefore we need an exact expression of θ(t j +) to make sure the jumping map Λ to be a homeomorphism.
By letting z = (x, y) and then we define where k is chosen to satisfy and arg z denotes the argument of z with arg z ∈ [0, 1). Particularly, if for some This implies that in the new coordinates the jumping map is a homeomorphism.
Before the estimation of P , we first give some estimations about I * and J * .  Under the assumption (i) mentioned in (1.7), it holds that for any non-negative integers k 1 , k 2 (k 1 + k 2 ≤ 5),

YANMIN NIU AND XIONG LI
Proof. By (2.6) and the assumption (i), we have For k 1 = k 2 = 0, it is obvious from (3.5) and (3.6) that ∂J(x, y) ∂y · ∂y ∂λ and again by (2.3), (3.5) and (3.6), we conclude For k 1 = 0, k 2 = 1, The other cases where k 1 , k 2 are any non-negative integers with k 1 + k 2 ≤ 5 can be discussed by the similar way and we obtain The proof then is finished.

Lemma 3.2. Under the assumption (i) mentioned in
Proof. By (2.6) and (2.10), we have the last line of which is due to Taylor's formula. The estimation above and Lemma 3.1 imply that for any non-negative integers k 1 , k 2 (k 1 + k 2 ≤ 5), With the property (2) of C and S, we take the partial derivative of θ on both sides of Eq. (3.9) and therefore we figure out By (3.7), (3.10) and Lemma 3.1, Similarly, taking the partial derivative of λ on both sides of the Eq. (3.9), we have which along with (3.7), (3.10) and Lemma 3.1 implies When ( 3 4 ) 1 4 < |C(θT * )| ≤ 1, |S(θT * )| ≤ 1 2 , we discuss the second equation of (3.8) and obtain (3.11) in the similar way with the discussions above. Thus we finish the proof.
It follows Lemma 3.2 that for any non-negative integers k 1 , k 2 (k 1 + k 2 ≤ 5), Now we give the Poincaré map of (3.13) in the following lemma.
4. Proof of Theorems 1.1-1.3. In this section, we will conclude that the time 1 map P is close to a twist map. To apply the twist theorem, we need another essential condition-the intersection property. Although there exist impulses for system X h1 , the Poincaré map P defined in (3.2) is always area-preserving or symplectic. Thus the following lemma appeared in [4] is valid for P and we omit the proof. From Lemma 3.3, if ρ is sufficiently large, the map P is, with its derivatives, close to a standard twist map. Moreover, it has the intersection property by the transformation (3.12) and Lemma 4.1, so that the assumptions of Moser's twist theorem are met.
It follows that for ω ≥ ω * ( ω * sufficiently large) with for two constants β > 0 and c > 0 and for all integers p and q = 0, there is an embedding ψ : S 1 → A ρ * of a circle, which is invariant under the map P . Here ρ * is a large number defined in Lemma 3.3. More specifically, on this invariant curve the map P is conjugated to a rotation with number ω P • ψ(s) = ψ(s + ω) with s(mod 1).
Under the scalar transformation (3.12), (4.2) also holds for P on A λ0 in (λ, θ) coordinates, where λ 0 = (ρ * /a) 3 . May as well denote the invariant curve by ψ with rotation number ω. The solution of the Hamiltonian system (3.1) starting at time t = 0 on this invariant curve determines a 1-periodic cylinder in the space (λ, θ, t) ∈ A λ0 × R. Since the Hamiltonian vector field X h1 is time-periodic, the phase space is A λ0 × S 1 .
Here we give some explanations about the quasiperiodic solutions. The discontinuous of Φ t results that X(t) is piecewise continuous. On each interval without impulses, X(t) has the form of quasiperiodic solutions while at impulsive times X(t) satisfies impulsive conditions. Strictly speaking, Ψ(ωt, t) is not a continuous shell function for all t ∈ R. But according to the form of X(t) at t = t j , j ∈ Z, we also regard X(t) is quasiperiodic with piecewise continuous shell function Ψ.
Through the symplectic transformation (2.3), solutions of impulsive system with Hamiltonian (2.11) are also quasiperiodic in (x, y) plane. This prove the statement of Theorem 1.2.
In order to verify the statement of Theorem 1.1, we transform the invariant curves obtained for the Poincaré map P of X h1 into the (x, y) coordinates. Since the transformation ϕ (2.3) is symplectic, there are invariant curves of the time 1 map of (1.1) in (x, y) plane. Meanwhile, the solutions starting at t = 0 on the invariant curves are quasiperiodic. These families of quasiperiodic solutions can be visualized as invariant cylinders in the space (t, x, y). By uniqueness, the solution starting inside a cylinder will never escape. The solution is therefore confined in the interior of the time periodic cylinder above the invariant curve and hence is bounded. This ends the proof of Theorem 1.1.