Abstract
We prove existence and multiplicity results for periodic solutions of Hamiltonian systems, by the use of a higher dimensional version of the Poincaré–Birkhoff fixed point theorem. The first part of the paper deals with periodic perturbations of a completely integrable system, while in the second part we focus on some suitable global conditions, so to deal with weakly coupled systems.
1 Introduction
This paper provides some results on the existence of periodic solutions for Hamiltonian systems which may be considered as time-periodic perturbations of an autonomous system of the type
Here,
There is a large bibliography on this problem, mainly motivated by models from classical mechanics. Remarkably, we observe that the literature on this issue can be split into two quite disjoint streams.
One of the two currents has a more topological spirit, and aims at minimal regularity assumptions. This kind of results finds a fertile ground mainly in the planar case, where, among others, a powerful tool, the Poincaré–Birkhoff fixed point theorem, can be used to prove the existence of periodic solutions of the perturbed system. Indeed, the case
On the other hand, when
The aim of this paper is to provide a common framework for the two kinds of approach depicted above. Using a recent result by the first author and Ureña [31], where an extension of the Poincaré–Birkhoff theorem to higher dimensional Hamiltonian systems has been proposed, we will be able, on one hand, to relax the usual structural assumptions on the Hamiltonian function and, on the other hand, to extend to higher dimensions some existence results already established in the planar case.
Before entering into details, we will now spend a few words on the framework where our results are to be settled.
The general framework
A classical approach to the study of the Hamiltonian system (HS) is the search for constants of motion, since they can be used for suitably transforming the system into a simpler one. The most remarkable case occurs when (HS) has N constants of motion which are independent and in involution: In this case, the system is said to be completely integrable, and one has a foliation of the space in N-dimensional surfaces, which are invariant for the flow.
The Liouville–Arnold theorem then assures that, when one of these surfaces is bounded and connected, it has to be an N-dimensional torus.
Moreover, for any such invariant torus Γ, there exists an open neighborhood
Each value
Since for every general Hamiltonian system (HS) a constant of motion is always given by the Hamiltonian function
As a matter of fact, completely integrable Hamiltonian systems are rare, and most often the Hamiltonian function is their unique constant of motion [8, 50]. Yet, generic Hamiltonian systems may be considered as perturbations of completely integrable systems [42, 47], usually called nearly integrable systems. A glance of this scenario was already grasped by Henri Poincaré [46], who referred to Hamiltonian perturbation theory as the Problème général de la Dynamique. The efforts made by Poincaré and, among many others, by Birkhoff, led to a broad development of the theory. We suggest [5, 7] for a detailed introduction to Hamiltonian perturbation theory and [26] for a friendly overview.
As we have seen, complete integrability reveals strong properties of the dynamics. A natural question is: How much of this structure is preserved under a small perturbation? In particular, one could wonder whether, near an invariant torus of the unperturbed system, it is possible to find periodic or quasiperiodic solutions for the perturbed system with the same frequency.
A series of positive results are known for a large family of nonresonant tori, those with a Diophantine frequency. These results are usually collected under the name of KAM theory, recalling its main contributors Kolmogorov, Arnold and Moser. We remark that, beyond a nondegeneracy assumption on the torus, strong smoothness of the perturbation is always needed, cf. [2, 35, 48]. While these strongly nonresonant tori survive under small perturbations, the same is not true for the other tori [11, 41, 51] and, in particular, for those made of periodic solutions. Still, some traces of these tori can be found.
For instance, in the planar case, after the pioneering papers [39, 40], the survival of two periodic solutions was obtained as a consequence of the Poincaré–Birkhoff theorem (see, e.g., [17] and [29], where an overview on the use of the Poincaré–Birkhoff theorem for this kind of problems can be found). The required twist condition is satisfied, in this case, under some rather weak nondegeneracy assumptions. A fainter kind of traces of an invariant torus is provided by the so called Aubry–Mather theory (cf. [43] and the references therein), showing the existence of a Cantor set, called cantorus, that preserves, in a generalized sense, the rotational properties of the original torus.
For higher dimensional Hamiltonian systems, a local approach to the problem has been proposed by Bernstein and Katok [9], who showed the survival under small perturbations of
A rather different type of problem arises when one looks for the existence and multiplicity of periodic solutions when only the global behavior of the nonlinearity is assumed to be known. In this case, the approach is no longer perturbative, and it usually combines topological and variational methods.
In this respect, there is a large literature in the planar case, mainly motivated by some models involving scalar second order differential equations, where the Poincaré–Birkhoff theorem has been successfully applied (see, e.g., [15, 25, 30, 34, 36], or again the review in [29]). The twist condition is generated by assuming a difference between the growth of the nonlinearity near a given periodic solution and at infinity, producing a gap in the rotation numbers of the corresponding solutions in the phase plane. A sharp use of the Poincaré–Birkhoff theorem then ensures that the larger this gap, the larger the number of solutions found. Furthermore, the same strategy applies also to the search for subharmonic solutions (see, for instance, [14, 25]).
Incidentally, the twist geometry has sometimes been recovered by detecting, in the unperturbed system, an annulus of periodic orbits displaying a gap between the periods of the boundary orbits. This picture displays the same features already discussed when considering completely integrable systems. A quite common way of producing this geometry is to require the strict monotonicity of the period function associated with system (HS), a feature which has been studied by many authors (see, e.g., [22, 33, 45]) and which ensures its nondegeneracy.
The first multiplicity results extending the Poincaré–Birkhoff philosophy to higher dimensions are due to Amann and
Zehnder [3], who introduced a twist condition between zero and infinity. A different perspective was followed by Conley and Zehnder [24], where the existence of
The main tool and an overview of our results
Let us now recall the result in [31], which will be our main tool in the search for periodic solutions. Consider the Hamiltonian system
where the continuous function
Moreover, let
Theorem 1.1 ([31])
If every solution
then system (1.1)
has at least
such that
We recall that two solutions of system (1.1) are geometrically distinct if one of them cannot be obtained just by adding suitable integer multiples of
We now briefly describe the main results of this paper, obtained by the use of Theorem 1.1.
The first part deals with small time-dependent perturbations of completely integrable systems. In Section 2, taking an invariant torus made of periodic solutions of the unperturbed system, and assuming a rather weak nondegeneracy condition, we prove the survival of
In Section 3, still dealing with completely integrable systems, we gradually abandon the local point of view and move to a large scale perspective. Assuming a twist-type condition on the product of N planar annuli, which is shown to persist for small perturbations, we thus obtain the survival of
In Section 4, we deal with
weakly coupled systems with a T-periodic forcing term, depending on some parameters. We impose suitable
conditions at zero and infinity for each of the N equations, producing a gap in the rotation numbers of the uncoupled systems.
Using Theorem 1.1, we then prove the existence of
In all the paper,
2 Periodic perturbations of completely integrable systems
Let us consider a completely integrable Hamiltonian system on
For every
We are interested in the case when the dynamics on the torus
and
A standard approach to study such a system, defined on
where
Let us now consider a general nearly integrable Hamiltonian system on
The Hamiltonian function
We now fix an
Here, we only ask
where
Clearly, this function
We will show that the nondegeneracy condition (2.2) extends by continuity to a neighborhood
then the perturbed system (CIper) has at least
Here is our main result.
Suppose that there exist
then for every
with
for every
Proof.
We can assume, without loss of generality, the function
Let us fix any
By continuity, there is an open neighborhood
For any arbitrary
and consider the Hamiltonian system
For any fixed positive real numbers
Proof of the claim.
Arguing by contradiction, assume that there is a sequence
one has that
and, accordingly, a sequence
Since the Hamiltonians
given by
On the other hand, passing to the limit in (2.9) yields
which is a contradiction, since the left-hand side is equal to zero. The claim is thus proved. ∎
We can now conclude the proof of Theorem 2.1. Let
We now focus our attention on those
Let
We can therefore apply Theorem 1.1, so to get
such that
Notice that, taking
A simple case is given by the choice
Suppose that there exists
then system (CIper) has at least
with the same rotational properties of the torus
for every
3 Twist conditions for weakly coupled period annuli
In the previous section, we have described the local phenomenon of the survival of some periodic solutions of system (CI) for the perturbed system (CIper). We now turn our attention to finding some conditions at a larger scale which guarantee the existence of multiple periodic solutions.
We still consider system (CIper) as a perturbation of system (CI), but we now look for periodic solutions
contained in
Suppose that there exist N couples of real numbers
Let
and there exist two positive integers
preserving the same rotational properties of
Proof.
By the Poincaré–Miranda theorem (cf. [28, 37]), there exists an
with
Let us pick any
By the same argument used in the claim within the proof of Theorem 2.1, there exists
for every
If
The same is true if
If
and the same is true if
Let us define the
Let us now describe a particular situation when Theorem 3.1 can be applied, generalizing the planar setting studied in [29]. We start by considering the autonomous Hamiltonian system
where
with
Hence, for every
for each of which we assume the following:
The planar system (${\mathrm{HS}_{i}}$) has a periodic solution
Each of such solutions has a corresponding planar open tubular neighborhood
There exist two positive real numbers
Let us define the set
and consider the Hamiltonian system
where
For every
Denoting by
we set
and define the frequency vector
Moreover, we choose the two least positive integers
In the above setting, there exists
has at least
whose orbits lie in
Proof.
By standard arguments (cf. [29]), each of the systems (${\mathrm{HS}_{i}}$) admits a canonical transformation in action-angle coordinates
For each
Theorem 3.1 then applies, and the proof is readily completed. ∎
4 Weakly coupled pendulum-like systems
In this section, we consider a weakly coupled system of the type
where J is the
and
We assume that system (P) can be reduced to a Hamiltonian system by a linear change of variables. More precisely, there exist N invertible
transforms system (P) into a Hamiltonian system. With such an assumption, we will say that (P) is a positive transformation of a Hamiltonian system.
Let us introduce the following notation for a closed cone in
We are now ready to state the main theorem of this section.
Let (P) be a positive transformation of a Hamiltonian system. For every
There exists
There exist
For every
(4.2)
Then, for every fixed positive integers
for every
such that for every
Some comments on the hypotheses of Theorem 4.1 are in order.
Assumption (A1) is needed to ensure the global existence of the solutions to the Cauchy problems associated with (P). Concerning (A2), it will guarantee that the small amplitude planar components of the solutions do rotate around the origin, clockwise, with a least positive angular speed.
Our hypothesis (A3), on the contrary, will ensure a small rotation number for large amplitude components. It could be compared with assumption
We now start the proof of Theorem 4.1. For a solution
Our first lemma concerns solutions
By (A2), if
For any
then any solution
Proof.
Let
We now claim that if (4.4) holds with a suitable choice of
Indeed, set
and assume by contradiction that
for every
a contradiction.
Consequently, if
so that the rotation number of
using (A2) and (4.4), we thus have
Choosing finally
we easily conclude the proof. ∎
Now we need a control on the rotation number of the large planar components of the solutions.
For any
Proof.
Fix
In view of assumption (A1), there exists
as long as
Consequently, the time needed to clockwise cross the sector
Proof of Theorem 4.1.
For any
Take
We now use the fact that (P) is a positive transformation of a Hamiltonian system, and consider the linear transformation
is thus of the type
such that for every
we obtain the solutions of (P) we are looking for, and the proof is thus completed. ∎
Theorem 4.1 exploits a gap between the rotation numbers of the solutions at zero and at infinity.
With reference to the assumption at infinity, another possibility could be to replace (A3) with the requirement that for some
As a particular case, we can deal with a system of scalar second order equations like
where the continuous function
which is in the form (P), with
where
Notice that (4.6) is a positive transformation of a Hamiltonian system, with the linear function
As a consequence, we have the following statement, where, for simplicity, we only consider the case
Assume that the continuous functions
Moreover, for every
Then, there exists
with minimal period T.
Moreover, for every
Proof.
First, we notice that (A1) is fulfilled, in view of the growth assumption on the nonlinearities.
Let us now check (A2). We know that there exist
Then, if
as desired.
We now verify (A3). Fix
we have that if
Taking
The proof is thus completed, noticing that it suffices to choose
As an example, Corollary 4.5 directly applies to the following system of N coupled pendulums,
where
Funding statement: The authors have been partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA – INdAM).
References
[1] Abbondandolo A., Morse Theory for Hamiltonian Systems, CRC Res. Notes Math. 425, Chapman & Hall/CRC, Boca Raton, 2001. 10.1201/9781482285741Search in Google Scholar
[2] Albrecht J., On the existence of invariant tori in nearly-integrable Hamiltonian systems with finitely differentiable perturbations, Regul. Chaotic Dyn. 12 (2007), 281–320. 10.1134/S1560354707030033Search in Google Scholar
[3] Amann H. and Zehnder E., Nontrivial solutions for a class of non-resonance problems and applications to nonlinear differential equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 7 (1980), 539–603. Search in Google Scholar
[4] Ambrosetti A., Coti Zelati V. and Ekeland I., Symmetry breaking in Hamiltonian systems, J. Differential Equations 67 (1987), 165–184. 10.1016/0022-0396(87)90144-6Search in Google Scholar
[5] Arnold V. I., Kozlov V. V. and Neishtadt A. I., Mathematical Aspects of Classical and Celestial Mechanics, Encyclopaedia Math. Sci. 3, Springer, Berlin, 2006. 10.1007/978-3-540-48926-9Search in Google Scholar
[6] Bambusi D. and Berti M., A Birkhoff-Lewis type theorem for some Hamiltonian PDEs, SIAM J. Math. Anal. 37 (2005), 83–102. 10.1137/S0036141003436107Search in Google Scholar
[7] Benettin G., The elements of Hamiltonian perturbation theory, Hamiltonian Systems and Fourier Analysis, Adv. Astronom. Astrophys. 9, Cambridge Scientific Publishers, Cambridge (2005), 1–98. Search in Google Scholar
[8] Benettin G., Galgani L. and Giorgilli A., Poincaré’s nonexistence theorem and classical perturbation theory for nearly integrable Hamiltonian systems, Advances in Nonlinear Dynamics and Stochastic Processes (Florence 1985), World Scientific, Singapore (1985), 1–22. Search in Google Scholar
[9] Bernstein D. and Katok A., Birkhoff periodic orbits for small perturbations of completely integrable Hamiltonian systems with convex Hamiltonians, Invent. Math. 88 (1987), 225–241. 10.1007/BF01388907Search in Google Scholar
[10] Berti M., Biasco L. and Valdinoci E., Periodic orbits close to elliptic tori and applications to the three-body problem, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2004), 87–138. 10.2422/2036-2145.2004.1.06Search in Google Scholar
[11] Bessi U., An analytic counterexample to the KAM theorem, Ergodic Theory Dynam. Systems 20 (2000), 317–333. 10.1017/S0143385700000146Search in Google Scholar
[12] Biasco L. and Di Gregorio L., A Birkhoff–Lewis type theorem for the nonlinear wave equation, Arch. Ration. Mech. Anal. 196 (2010), 303–362. 10.1007/s00205-009-0240-ySearch in Google Scholar
[13] Birkhoff G. D. and Lewis D. C., On the periodic motions near a given periodic motion of a dynamical system, Ann. Mat. Pura Appl. (4) 12 (1934), 117–133. 10.1007/BF02413852Search in Google Scholar
[14] Boscaggin A., Subharmonic solutions of planar Hamiltonian systems: A rotation number approach, Adv. Nonlinear Stud. 11 (2011), 77–103. 10.1515/ans-2011-0104Search in Google Scholar
[15] Boscaggin A. and Garrione M., Resonance and rotation numbers for planar Hamiltonian systems: Multiplicity results via the Poincaré–Birkhoff theorem, Nonlinear Anal. 74 (2011), 4166–4185. 10.1016/j.na.2011.03.051Search in Google Scholar
[16] Boscaggin A. and Ortega R., Monotone twist maps and periodic solutions of systems of Duffing type, Math. Proc. Cambridge Philos. Soc. 157 (2014), 279–296. 10.1017/S0305004114000310Search in Google Scholar
[17] Buttazzoni P. and Fonda A., Periodic perturbations of scalar second order differential equations, Discrete Contin. Dyn. Syst. 3 (1997), 451–455. 10.3934/dcds.1997.3.451Search in Google Scholar
[18] Celletti A., Stability and Chaos in Celestial Mechanics, Springer, Berlin, 2010. 10.1007/978-3-540-85146-2Search in Google Scholar
[19] Celletti A. and Chierchia L., KAM stability and celestial mechanics, Mem. Amer. Math. Soc. 187 (2007). 10.1090/memo/0878Search in Google Scholar
[20] Chang K. C., Methods in Nonlinear Analysis, Springer, Berlin, 2005.Search in Google Scholar
[21] Chen W. F., Birkhoff periodic orbits for small perturbations of completely integrable Hamiltonian systems with nondegenerate Hessian, Twist Mappings and Their Applications, IMA Vol. Math. Appl. 44, Springer, Berlin (1992), 87–94. 10.1007/978-1-4613-9257-6_5Search in Google Scholar
[22] Chicone C., The monotonicity of the period function for planar Hamiltonian vector fields, J. Differential Equations 69 (1987), 310–321. 10.1016/0022-0396(87)90122-7Search in Google Scholar
[23] Conley C. C. and Zehnder E., An index theory for periodic solutions of a Hamiltonian system, Geometric Dynamics, Lecture Notes in Math. 1007, Springer, Berlin (1983), 132–145. 10.1007/BFb0061415Search in Google Scholar
[24] Conley C. C. and Zehnder E., The Birkhoff–Lewis fixed point theorem and a conjecture of V. I. Arnold, Invent. Math. 73 (1983), 33–49. 10.1007/BF01393824Search in Google Scholar
[25] Ding T. and Zanolin F., Periodic solutions of Duffing’s equation with superquadratic potential, J. Differential Equations 97 (1992), 328–378. 10.1016/0022-0396(92)90076-YSearch in Google Scholar
[26] Dumas H. S., The KAM Story: A Friendly Introduction to the Content, History, and Significance of classical Kolmogorov–Arnold–Moser Theory, World Scientific, Singapore, 2014. 10.1142/8955Search in Google Scholar
[27] Ekeland I., A perturbation theory near convex Hamiltonian systems, J. Differential Equations 50 (1983), 407–440. 10.1016/0022-0396(83)90069-4Search in Google Scholar
[28] Fonda A. and Gidoni P., Generalizing the Poincaré–Miranda theorem: The avoiding cones condition, Ann. Mat. Pura Appl. (4), 10.1007/s10231-015-0519-6. 10.1007/s10231-015-0519-6Search in Google Scholar
[29] Fonda A., Sabatini M. and Zanolin F., Periodic solutions of perturbed Hamiltonian systems in the plane by the use of the Poincaré–Birkhoff theorem, Topol. Methods Nonlinear Anal. 40 (2012), 29–52. Search in Google Scholar
[30] Fonda A. and Sfecci A., Periodic solutions of weakly coupled superlinear systems, J. Differential Equations 260 (2016), 2150–2162. 10.1016/j.jde.2015.09.056Search in Google Scholar
[31] Fonda A. and Ureña A. J., On the higher dimensional Poincaré–Birkhoff theorem for Hamiltonian flows. 1. The indefinite twist, preprint 2013, www.dmi.units.it/~fonda/2013\_Fonda-Urena.pdf. Search in Google Scholar
[32] Fonda A. and Zanolin F., Periodic oscillations of forced pendulums with very small length, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), 67–76. 10.1017/S0308210500023519Search in Google Scholar
[33] Garijo A., Gasull A. and Jarque X., A note on the period function for certain planar vector fields, J. Difference Equ. Appl. 16 (2010), 631–645. 10.1080/10236190903241689Search in Google Scholar
[34] Hartman P., On boundary value problems for superlinear second order differential equations, J. Differential Equations 26 (1977), 37–53. 10.1016/0022-0396(77)90097-3Search in Google Scholar
[35] Herman M. R., Sur les courbes invariantes par les difféomorphismes de l’anneau, Astérisque 103, Société Mathématique de France, Paris, 1983. Search in Google Scholar
[36]
Jacobowitz H.,
Periodic solutions of
[37] Kulpa W., The Poincaré–Miranda theorem, Amer. Math. Monthly 104 (1997), 545–550. 10.1080/00029890.1997.11990676Search in Google Scholar
[38]
Lacomba E. A. and Llibre J.,
Integrals, invariant manifolds, and degeneracy for central force problems in
[39] Lazer A. C., Small periodic perturbations of a class of conservative systems, J. Differential Equations 13 (1973), 438–456. 10.1016/0022-0396(73)90003-XSearch in Google Scholar
[40]
Loud W. S.,
Periodic solutions of
[41] MacKay R. S. and Percival I. C., Converse KAM: Theory and practice, Comm. Math. Phys. 98 (1985), 469–512. 10.1007/BF01209326Search in Google Scholar
[42] Markus L. and Meyer K. R., Generic Hamiltonian dynamical systems are neither integrable nor ergodic, Mem. Amer. Math. Soc. 144 (1974). 10.1090/memo/0144Search in Google Scholar
[43] Moser J., Recent developments in the theory of Hamiltonian systems, SIAM Rev. 28 (1986), 459–485. 10.1137/1028153Search in Google Scholar
[44] Moser J. and Zehnder E., Notes on Dynamical Systems, Courant Lect. Notes Math. 12, American Mathematical Society, Providence, 2005. 10.1090/cln/012Search in Google Scholar
[45] Pinsky M. A. and Zevin A. A., Monotonicity criteria for an energy-period function in planar Hamiltonian systems, Nonlinearity 14 (2001), 1425–1432. 10.1088/0951-7715/14/6/301Search in Google Scholar
[46] Poincaré H., Les Méthodes Nouvelles de la Méchanique Céleste, Gauthier–Villars, Paris, 1892. Search in Google Scholar
[47] Robinson R. C., Generic properties of conservative systems, Amer. J. Math. 92 (1970), 562–603. 10.1007/BFb0070136Search in Google Scholar
[48] Salamon D. A., The Kolmogorov–Arnold–Moser theorem, Math. Phys. Electron. J. 10 (2004), Article ID 3. Search in Google Scholar
[49] Sell G. R., Differential equations without uniqueness and classical topological dynamics, J. Differential Equations 14 (1973), 42–56. 10.1016/0022-0396(73)90075-2Search in Google Scholar
[50] Siegel C. L., Über die Existenz einer Normalform analytischer Hamiltonscher Differentialgleichungen in der Nähe einer Gleichgewichtslösung, Math. Ann. 128 (1954), 144–170. 10.1007/BF01360131Search in Google Scholar
[51] Treshchëv D. V., A mechanism for the destruction of resonance tori in Hamiltonian systems, Mat. Sb. 180 (1989), 1325–1346. 10.1070/SM1991v068n01ABEH001371Search in Google Scholar
[52] Willem M., Perturbations of nondegenerate periodic orbits of Hamiltonian systems, Periodic Solutions of Hamiltonian Systems and Related Topics, NATO ASI Series C 209, Springer, Dordrecht (1987), 261–265. 10.1007/978-94-009-3933-2_24Search in Google Scholar
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