LYAPUNOV STABILITY FOR REGULAR EQUATIONS AND APPLICATIONS TO THE LIEBAU PHENOMENON

. We study the existence and stability of periodic solutions of two kinds of regular equations by means of classical topological techniques like the Kolmogorov-Arnold-Moser (KAM) theory, the Moser twist theorem, the averaging method and the method of upper and lower solutions in the reversed order. As an application, we present some results on the existence and stability of T -periodic solutions of a Liebau-type equation.


1.
Introduction. Let us consider the differential equation where f : R × R → R is a continuous function, T -periodic in the first variable and smooth enough, for example f ∈ C 0,4 (R/T Z × R). It is known that when a T -periodic solution x of (1) is of the twist type then it is Lyapunov stable (see Section 2 for more details). This means that the so-called first twist coefficient of the Birkhoff normal form does not vanish and an explicit expression for that coefficient in terms of the third order approximation y + a(t)y + b(t)y 2 + c(t)y 3 + o(y 3 ) = 0, where a(t) = f x (t, x(t)), b(t) = 1 2 f xx (t, x(t)), c(t) = 1 6 f xxx (t, x(t)).
was firstly obtained by Ortega [15] (see also [12,21]) . Some applications of Ortega's works can be found in [1,2,4]. In a recent work [3], based on the above ideas, the existence of Lyapunov stable periodic solutions for the combined attractive-repulsive singularity has been studied by the first author, Chu and Torres. For further results on mathematical models with singularities, we refer to the recent book [19], and the references therein. However, up to now, there are few results about the existence of periodic solutions of the regular systems [10,16]. The main aim of this paper is to develop some new existence and stability criteria so that the regular case can be studied following the ideas in [3] and an application to the Liebau phenomenon could be obtained. More precisely, we will study the existence and stability of T -periodic solutions of the regular equation and the equation with a small parameter ε where 0 < α < β < 1, and r, s are continuous and T -periodic functions. The Liebau phenomenon or "valveless pumping effect" is referred to a preferential direction of the flow obtained in mechanical systems without valves as consequence of asymmetric periodic oscillations, [14]. A simple model which shows this effect, the so called "one pipe-one tank" configuration, was presented in [17] and a detailed description of it is also available in [5] and [19,Chapter 8]. That model leads to the searching of positive T -periodic solutions for the following singular second-order differential equationü where a = r 0 and the meaning of the involved functions and parameters are the following r 0 ≥ 0 friction coefficient on the tube ρ > 0 density of the fluid ζ ≥ 1 junction coefficient g acceleration of gravity A P > 0 cross section of the pipe A T > 0 cross section of the tank V 0 > 0 total volume of the fluid p(t) T -periodic forcing So, from the physical point of view, it is natural to assume in equation (5) that a ≥ 0, b > 1, c > 0 and e is continuous and T − periodic.
Notice that we could even assume that b ≥ 3/2. Recently, some results on the existence and stability of periodic positive solutions for (5) with friction were presented in [5,6,7,19]. Furthermore, Liao obtained in [13] the existence of T -periodic solutions of a generalized Liebau-type differential equation by using the fixed point theorem in cones. However, up to now, there are few works on the conservative case a = 0, which is an idealization but it is interesting from both the mathematical and the physical point of view. Our results in the present paper shall fill partially this gap.
So, let us consider equation (5) without friction, that is, By means of the change of variables u = x κ , where κ = 1/(b + 1) (see [5]), we rewrite the singular problem (6) as which is a regular equation in the form (3). Notice that it has physical sense to consider c as a small parameter, meaning that the section of the pipe is much less than the section of the tank, and then equation (7) fits also in the form (4). The paper is organized as follows: after this Introduction, in Section 2 for the convenience of the reader we collected some general results, well known by the specialists, used in order to prove our main theorems. Section 3 is devoted to our main existence and stability criteria. The importance of such result relies in that, for the first time in this topic, we have stability theorems for the regular equations (3) and (4). In Section 4 the previous results are applied to the Liebau model (6) and some illustrative examples are given.
Throughout this paper the following notations will be used. For a given Tperiodic continuous function h we denotē and h = h M hm . This quantity h ≥ 1 can be regarded as a measure of the ratio of h M to h m and will play a key role in our main results. Furthermore, we define γ = 1 β−α . Since 0 < α < β < 1, we have γ ∈ (1, ∞).

2.
Preliminaries. The linearized equation of (2) is the Hill's equation We say that (8) is elliptic, or linearly stable, if its Floquet multipliers λ 1 , λ 2 satisfy λ 1 = λ 2 , |λ 1 | = 1, λ 1 = ±1. In this case the rotation number ρ is defined by the relation λ = exp(±iρT ), and for convenience we write θ = ρT . It is well known that the linear stability of (8) is not enough to ensure the Liapunov stability of (2). The T -periodic solution x of (1) is called 4-elementary if the multipliers λ of (8) satisfy λ q = 1 for 1 ≤ q ≤ 4. If x is 4-elementary then we say that x is of the twist type if the first twist coefficient µ of the Birkhoff normal form of the Poincaré map is non-zero. In that case Moser's invariant curve theorem can be applied to show that a solution of twist type is Lyapunov stable (see §32- §34 in [18]) and, moreover, around it the complex dynamics prescribed by KAM theory arises. Clearly, a major difficulty in this approach is the computation of the corresponding first twist coefficient µ. This task was firstly accomplished by Ortega in [15] where he found an explicit formula for µ, later reformulated in [21] (see also [12]) as where R and ϕ denote the polar coordinates, Ψ(t) = R(t)exp(iϕ(t)) is the complex solution of (8) with initial conditions Ψ(0) = 1, Ψ (0) = i and the kernel χ θ is given by 2.1. Existence lemmas.
Analogously, an upper solution σ 2 is defined by reversing the respective inequalities in the previous definition. A lower solution (resp. upper solution) is called strict if the inequality in (i) is strict. It is well known that the classical method of lower and upper solutions is a quite effective and flexible tool for studying existence of T -periodic solutions of (1). Actually, a couple of upper and lower solutions such that σ 1 (t) ≤ σ 2 (t) for all t typically leads to unstable solutions lying between σ 1 and σ 2 (see [9]). In order to obtain a stable solution, we now assume that σ 1 and σ 2 are ordered in the reversed way, provided the partial derivative of f with respect to x is not too large. The following result is a consequence of [8, Theorem 3.7].
Lemma 2.2. Assume that there exist upper and lower solutions of (1) such that σ 2 (t) ≤ σ 1 (t) for all t. Under the assumption for every t. The following lemma summarises the averaging method and provides the existence of periodic solutions on differential equations like (4) containing a small parameter (for more details see [11, Chapter V, Theorem 3.2]). Lemma 2.3. We consider the following differential systeṁ where f : R × D × [0, +∞) → R n is a continuous function, T -periodic in the first variable, of class C 1 in x, ε and D is an open subset of R n . We define and assume that for x 0 ∈ D with f 0 (x 0 ) = 0 we have the determinant of the Jacobian matrix J f0 (x 0 ) = 0. Then there exist ε 0 > 0 and a function x(t, ε), continuous for Assume that there exists a T -periodic solution x of (1) such that: Then the solution x is of the twist type and the Moser twist theorem [18] implies that such a solution is stable.
The following asymptotic behavior of R and the rotation number ρ ≡ ρ(a) are very useful in order to get the sign of the first twist coefficient. (8) is nonnegative and has a positive meanā > 0. Then R(t) =: R(t, a) and θ =: θ(a) in formula (9) satisfy the asymptotic behavior

Main results.
3.1. Existence results. Let us define are constant upper and lower solutions of (3), respectively. (ii) For all t ∈ R and x > 0, Proof. It is easy to check that (i) holds and that for all t ∈ R and x > 0 we have , and g is increasing on (0, x 0 ) and decreasing on (x 0 , +∞). Therefore, either Thus claims (iii), (iv) and (v) also hold.
As consequence of Lemmas 3.1 and 2.2 we have the following existence result.
Then equation (3) has at least one T -periodic solution x such that Corollary 1. Assume that r, s are positive T -periodic continuous functions and the following inequality holds Then equation (3) has at least one T -periodic solution x such that (11) holds.
Proof. Taking into account that from Lemma 3.1 and condition (12) it follows that Theorem 3.2 applies.
We complete this section with the application of Lemma 2.3 to equation (4). Moreover, the following asymptotic behavior holds where ω =r s .
Remark 2. The asymptotic behavior (13) will be the key in order to prove stability results in the next section. Moreover, (13) implies that the solution x(t, ε) is positive for all t and ε > 0 small enough, which is essential for the application to the Liebau model.

3.2.
Stability results. Let x be a T -periodic solution of equation (3). A computation of the coefficients in (2) for that equation gives: Whenever a localization for the solution x is available we can obtain some estimates for the previous coefficients. We will derive carefully those estimates in the following three lemmas.
is satisfied, then we have: Proof. Take into account that since α < β and ∆ ≥ 1, assumption (S) implies Claim (i).-Note that a(t) = f x (t, x(t)). Hence, from (11), (17) and Lemma 3.1 we obtain On the other hand, using (11), (17) and reasoning as in the proof of Lemma 3.1 we get Claim (ii).-From (S) and (11) it follows that Claim (iii).-Now, from (11) and (17) we deduce Finally, taking into account that b(t) < 0, (11), (17), (S) and reasoning as in the proof of Lemma 3.1 we havẽ The following lemma, which is easy to check in a similar way to Lemma 3.1, provides us with computable bounds for c M .
If (S) is satisfied, then the following claims hold: (i) For all t ∈ R and x > 0, and the only positive solution of g 1 (x) = 0 is Moreover, g 1 is increasing on (0, x 1 ) and decreasing on (x 1 , +∞).
The computable bounds for c m are given in the following lemma.
Lemma 3.6. Assume that r, s are positive T -periodic continuous functions and that x is a solution of equation (3) satisfying (11). If (S) holds, then: (i) For all t ∈ R and x > 0, and the only positive solution of g 2 (x) = 0 is Moreover, g 2 is increasing on (0, x 2 ) and decreasing on (x 2 , +∞).
Now, we present our first stability result for equation (3).
Theorem 3.7. Let us assume that r, s are positive T -periodic continuous functions and and Then there exists a stable T -periodic solution of (3) satisfying (11).
Proof. Since (18) and (19) imply condition (C1) of Theorem 3.2, the existence of a solution of (3) satisfying (11) follows. The stability of such solution is a consequence of Lemma 2.4, (i), (ii) and (iii), taking into account the estimates obtained in Lemmas 3.4 and 3.5.
Remark 3. Condition (20) allows us to use part (iv) of Lemma 3.5 in order to get condition (21). Assuming the hypotheses of parts (ii) or (iii) of Lemma 3.5 instead of (20) will lead to alternate versions of (21).
Our next stability result, analogous to Theorem 3.7, follows from (i), (ii) and (iii') of Lemma 2.4 and the estimates from Lemmas 3.4 and 3.6. and then there exists a stable T -periodic solution of (3) satisfying (11).
Remark 4. Condition (22) allows us to use part (ii) of Lemma 3.6 in order to get condition (23). Assuming the hypotheses of parts (iii) or (iv) of Lemma 3.6 instead of (22) will lead to alternate versions of (23).
In the remainder of this section we provide a stability criterium for (4).
Corollary 2. Suppose that r and s are T -periodic continuous functions, r and s are positive, α, β satisfy (24)-(25). Then the equation has a stable T -periodic solution x(t, λ) if λ is large enough.
Proof. If we introduce the variable then (36) is changed to the equation Note that ε → 0 if and only if λ → +∞. So Corollary 2 holds from Theorem 3.9.

4.
Application to the Liebau model. In our next result we give explicit conditions in terms of the data in order to obtain the existence, localization, and stability of positive solutions of (6).
Remark 5. Note that in Theorem 4.1 the stability conditions in parts (II) and (III) also imply existence.
On the other hand, since ∆ e ≥ 1, a necessary condition in order to apply either part (II) or part (III) of Theorem 4.1 is   Finally, by considering c as a small parameter in (6), we obtain the following result about the existence and stability of periodic solutions.  Proof. Since r(t) = (b + 1)e(t) and s(t) = b + 1, condition e > 0 implies that r > 0. Now the existence follows by using Theorem 3.3. Concerning the stability, we apply Theorem 3.9. Remark 6. We point out thatē > 0 assumed in Theorem 4.2 is a necessary condition for the existence of a periodic positive solution of (6), see [5]. It is an open problem explicitly stated in [19,Chapter 8] whetherē > 0 is also a sufficient condition for existence. So, Theorem 4.2 can be viewed as a partial answer to this open problem which in addition provides stability information.
Using Theorem 4.2, we can prove the following result with a sign changing nonlinearity.
Example 2. The equationü has at least one stable 2π-periodic positive solution provided that 1 < b ≤ 3 2 and c is small enough. Remark 7. Note that Theorem 4.1 cannot be applied to (40) in order to guarantee the existence and stability of 2π-periodic solution since e(t) = cos t+ 1 2 takes negative values.