On the periodic orbits and the integrability of the regularized Hill lunar problem Jaume Llibre and Luci Any Roberto Citation: J. Math. Phys. 52, 082701 (2011); doi: 10.1063/1.3618280 View online: http://dx.doi.org/10.1063/1.3618280 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v52/i8 Published by the AIP Publishing LLC. Additional information on J. Math. Phys. Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors Downloaded 18 Jul 2013 to 200.130.19.215. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions http://jmp.aip.org/?ver=pdfcov http://oasc12039.247realmedia.com/RealMedia/ads/click_lx.ads/www.aip.org/pt/adcenter/pdfcover_test/L-37/726139175/x01/AIP-PT/HC_JMPCoverPg_0713/chp_books_banner1640x440.jpg/6c527a6a7131454a5049734141754f37?x http://jmp.aip.org/search?sortby=newestdate&q=&searchzone=2&searchtype=searchin&faceted=faceted&key=AIP_ALL&possible1=Jaume Llibre&possible1zone=author&alias=&displayid=AIP&ver=pdfcov http://jmp.aip.org/search?sortby=newestdate&q=&searchzone=2&searchtype=searchin&faceted=faceted&key=AIP_ALL&possible1=Luci Any Roberto&possible1zone=author&alias=&displayid=AIP&ver=pdfcov http://jmp.aip.org/?ver=pdfcov http://link.aip.org/link/doi/10.1063/1.3618280?ver=pdfcov http://jmp.aip.org/resource/1/JMAPAQ/v52/i8?ver=pdfcov http://www.aip.org/?ver=pdfcov http://jmp.aip.org/?ver=pdfcov http://jmp.aip.org/about/about_the_journal?ver=pdfcov http://jmp.aip.org/features/most_downloaded?ver=pdfcov http://jmp.aip.org/authors?ver=pdfcov JOURNAL OF MATHEMATICAL PHYSICS 52, 082701 (2011) On the periodic orbits and the integrability of the regularized Hill lunar problem Jaume Llibre1,a) and Luci Any Roberto2,b) 1Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain 2Departamento de Matemática, Ibilce – UNESP, 15054-000 São José do Rio Preto, Brazil (Received 16 February 2011; accepted 8 July 2011; published online 10 August 2011) The classical Hill’s problem is a simplified version of the restricted three-body problem where the distance of the two massive bodies (say, primary for the largest one and secondary for the smallest one) is made infinity through the use of Hill’s variables. The Levi-Civita regularization takes the Hamiltonian of the Hill lunar problem into the form of two uncoupled harmonic oscillators perturbed by the Coriolis force and the Sun action, polynomials of degree 4 and 6, respectively. In this paper, we study periodic orbits of the planar Hill problem using the averaging theory. Moreover, we provide information about the C1 integrability or non-integrability of the regularized Hill lunar problem. C© 2011 American Institute of Physics. [doi:10.1063/1.3618280] I. INTRODUCTION AND STATEMENT OF THE MAIN RESULTS In this paper, we study periodic orbits of the Hill lunar problem and its C1 non-integrability. The Hill lunar problem is a limit version of the restricted three-body problem, it is a model originally based on the Moon’s motion under the joint action of the Earth and the Sun.4 In the rotating frame the Hamiltonian of the Hill lunar problem is HHill(x) = 1 2 (x2 3 + x2 4 ) + x2x3 − x1x4 − 1√ x2 1 + x2 2 − x2 1 + 1 2 x2 2 , (1.1) where x = (x1, x2, x3, x4). For more details on this Hamiltonian, see Ref. 13. To avoid the difficulties due to the collision is performed the Levi-Civita regularization as follows. We do the change of variables in the positions given by ( x1 x2 ) = ( x̂1 −x̂2 x̂2 x̂1 )( x̂1 x̂2 ) , (1.2) and the induced change in the conjugate momenta is ( x3 x4 ) = 2 r̂2 ( x̂1 −x̂2 x̂2 x̂1 ) ( x̂3 x̂4 ) , (1.3) where r̂2 = x̂2 1 + x̂2 2 = r = √ x2 1 + x2 2 . To complete the regularization procedure it is necessary to rescale the time doing dτ = 4dt (x̂2 1 + x̂2 2 ) . Applying these changes of variables in (1.1), and considering the Hamiltonian Ĥ (x̂) = r 4 (HHill(x(x̂)) + p0), after omitting the hat of the variables, we get that the a)Electronic mail: jllibre@mat.uab.cat. b)Electronic mail: lroberto@ibilce.unesp.br. 0022-2488/2011/52(8)/082701/8/$30.00 C©2011 American Institute of Physics52, 082701-1 Downloaded 18 Jul 2013 to 200.130.19.215. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions http://dx.doi.org/10.1063/1.3618280 http://dx.doi.org/10.1063/1.3618280 http://dx.doi.org/10.1063/1.3618280 mailto: jllibre@mat.uab.cat mailto: lroberto@ibilce.unesp.br 082701-2 J. Llibre and L. F. Roberto J. Math. Phys. 52, 082701 (2011) new Hamiltonian for the Hill problem becomes H (x) = ( p0 2 ) x2 1 + x2 2 2 + 1 2 (x2 3 + x2 4 ) + 1 2 ( x2 1 + x2 2 ) (x2x3 − x1x4) − 1 4 ( x6 1 − 3x4 1 x2 2 − 3x2 1 x4 2 + x6 2 ) , (1.4) where 1 2 p0 = − 1 2 h = c, with h being the value of the Hamiltonian (1.1). The Hamiltonian (1.4) is still dependent on the parameter c which can be eliminated doing the canonical change of variables x1 = 2c1/4 x̄1, x2 = 2c1/4 x̄2, x3 = 2c3/4 x̄3, x4 = 2c3/4 x̄4, (1.5) and the Hamiltonian of the Hill problem that we shall use is HReg = 1 4c H (2c1/4x1, 2c1/4x2, 2c3/4x3, 2c3/4x4) = 1 2 ( x2 1 + x2 2 + x2 3 + x2 4 ) + 2 ( x2 1 + x2 2 ) (x2x3 − x1x4) − 4 ( x6 1 − 3x4 1 x2 2 − 3x2 1 x4 2 + x6 2 ) , (1.6) where the bar has been suppressed from the variables. The Hamiltonian HReg is called the regularized Hamiltonian of the Hill lunar problem, for more details see Subsection 2.2 in Ref. 16. Notice that the Hamiltonian (1.6) is a polynomial. This is very convenient for numerical com- putations. The terms of degree 2 of the polynomial Hamiltonian for the regularized Hill problem take the form of two uncoupled harmonic oscillators. The fourth degree terms are due to the Cori- olis force because we have changed the inertial reference frame at the center of mass to a rotating frame centered at the secondary. The sixth degree terms are due to the action of the primary. The Hamiltonian equations of motion become ẋ1 = x3 + 2x2 ( x2 1 + x2 2 ) , ẋ2 = x4 − 2x1 ( x2 1 + x2 2 ) , ẋ3 = −x1 + 2x4 ( x2 1 + x2 2 ) + 4 ( 6x5 1 − 12x3 1 x2 2 − 6x1x4 2 ) − 4x1(x2x3 − x1x4), ẋ4 = −x2 − 2x3 ( x2 1 + x2 2 ) + 4 (−6x4 1 x2 − 12x2 1 x3 2 + 6x5 2 ) − 4x2(x2x3 − x1x4). (1.7) As usual the dot denotes the derivative with respect to the independent variable t , the time. In this work, we use the averaging method of the first order to compute periodic orbits, see Appendix. This method allows to find analytically periodic orbits of the Hill lunar problem (1.7) at any positive values of the energy. Roughly speaking, this method reduces the problem of finding periodic solutions of some differential system to the one of finding zeros of some convenient finite- dimensional function. This method was also used by Kozlov in Ref. 8, Llibre and Jiménez-Lara in Refs. 6 and 7, and Llibre and Roberto in Ref. 9. The following theorem is the main result. Theorem 1: At every positive energy level the regularized Hill lunar problem has at least two periodic orbits. The periodic orbits for the Hill lunar problem were also studied by Maciejewski and Rybicki in Ref. 10. They described global bifurcations of non-stationary periodic orbits of the regularized Hill lunar problem which emanate from stationary ones. The symmetric periodic orbits of this problem were studied by Hénon in Ref. 3, and Howison and Meyer in Ref. 5 established the existence of a new family of periodic solutions for the spatial Hill’s lunar problem. Theorem 1 states that at any positive energy level there exist at least two periodic orbits and we can use these particular periodic orbits to prove our second main result about the C1 integrability or non-integrability in the sense of Liouville–Arnold of the regularized Hill lunar problem. Downloaded 18 Jul 2013 to 200.130.19.215. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions 082701-3 Periodic orbits of Hill lunar problem J. Math. Phys. 52, 082701 (2011) Proposition 2: The regularized Hill lunar problem (1.7) is (a) either Liouville–Arnold integrable and the gradients of the two constants of motion are linearly dependent on some points of the periodic orbits found in Theorem 1, or (b) it is not Liouville–Arnold integrable with any second first integral of class C1. The proof of Proposition 2 is based on a result of Poincaré that allows to prove the Liouville– Arnold non-integrability independently of the class of differentiability of the second first integral, see Sec. III. The main difficulty for applying Poincaré’s non-integrability method to a given Hamiltonian system is to find for such system periodic orbits having multipliers different from 1. The non-integrability of Hill lunar problem was studied by some authors. Winterberg and Meletlidou in Refs. 11 and 12 proved the analytic non-integrability of the Hill lunar problem, that is, the problem does not possess a second analytic integral of motion, independent of H . Morales–Ruiz et al. in Ref. 14 presented an algebraic proof of meromorphic non-integrability. But in our case we present some result on the C1 integrability . In Sec. II we prove Theorem 1, and Proposition 2 is proved in Sec. III. II. PROOF OF THEOREM 1 We shall use the averaging theory of first order to analyze the existence of periodic orbits for system (1.7). To apply Theorem 5 we need a small parameter ε, then we rescale system (1.7) doing (x1, x2, x3, x4) = ( √ εX1, √ εX2, √ εX3, √ εX4), and we obtain the Hamiltonian system Ẋ1 = X3 + 2X2ε ( X2 1 + X2 2 ) , Ẋ2 = X4 − 2X1ε ( X2 1 + X2 2 ) , Ẋ3 = −X1 + 2ε ( 3X2 1 X4 − 2X1 X2 X3 + X2 2 X4 ) + 24ε2 X1 ( X4 1 − 2X2 1 X2 2 − X4 2 ) , Ẋ4 = −X2 − 2ε ( X2 1 X3 − 2X1 X2 X4 + 3X2 2 X3 ) − 24ε2 X2 ( X4 1 + 2X2 1 X2 2 − X4 2 ) (2.1) with the Hamiltonian 1 2 ( X2 1 + X2 2 + X2 3 + X2 4 ) − 2ε ( X2 1 + X2 2 ) (X1 X4 − X2 X3) − 4ε2 ( X2 1 + X2 2 ) ( X4 1 − 4X2 1 X2 2 + X4 2 ) . (2.2) We remark that the rescaling used is, in fact, a symplectic rescaling of multiplier ε−1, see for more details Ref. 13. Notice that system (2.1) has the same phase portrait as system (1.7) for all ε �= 0. Another change of variables is done because system (2.1) is not in the normal form (A1) for applying the averaging theory. We do the transformation X1 = r cos θ, X2 = ρ cos(θ + α), X3 = r sin θ, X4 = ρ sin(θ + α). Recall that this is a change of variables when r > 0 and ρ > 0. Moreover, doing this change of variables appear in the system the periodic variables θ and α. Later on, the variable θ will be used for obtaining the periodicity necessary for applying the averaging theory. Downloaded 18 Jul 2013 to 200.130.19.215. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions 082701-4 J. Llibre and L. F. Roberto J. Math. Phys. 52, 082701 (2011) With this change of variables we obtain the following equations of motion: ṙ = ερ ( cos α(cos 2θ (r2 + ρ2 cos 2α) + r2 + ρ2) − sin α sin 2θ (−2r2 + ρ2 cos 2α + ρ2) )+ 24ε2r sin θ cos θ ( r4 cos4 θ − 2r2ρ2 cos2 θ cos2(α + θ ) − ρ4 cos4(α + θ ) ) , θ̇ = −1 + ε ρ r sin α ( 3r2 cos 2θ + 3r2 + ρ2 cos 2(α + θ ) + ρ2 )+ 24ε2 cos2 θ ( r4 cos4 θ − 2r2ρ2 cos2 θ cos2(α + θ ) − ρ4 cos4(α + θ ) ) , ρ̇ = − 1 2εr ( r2 cos(α − 2θ ) + 2(r2 + ρ2) cos α + (r2 − ρ2) cos(α + 2θ ) + 3ρ2 cos(3α + 2θ ) ) + 24ε2ρ sin(α + θ ) cos(α + θ ) (−r4 cos4 θ − 2r2ρ2 cos2 θ cos2(α + θ ) + ρ4 cos4(α + θ ) ) , α̇ = ε 1 rρ sin α ( r4 + ( 3r2ρ2 − ρ4 ) cos 2(α + θ ) + r2 ( r2 − 3ρ2 ) cos 2θ − ρ4 )− 24ε2 ( r4 cos6 θ + r2(r2 − 2ρ2) cos4 θ cos2(α + θ ) − ρ2(ρ2 − 2r2) cos2 θ cos4(α + θ )− ρ4 cos6(α + θ ) ) . (2.3) This change of variables is not canonical, so the system lost the Hamiltonian structure. In these new variables, Hamiltonian (2.2) become the first integral H = 1 2 (r2 + ρ2) − εrρ sin α ( r2 cos 2θ + r2 + ρ2 cos 2(α + θ ) + ρ2 )− 4ε2 ( r2 cos2 θ + ρ2 cos2(α + θ ) ) ( r4 cos4 θ − 4r2ρ2 cos2 θ cos2(α + θ ) + ρ4 cos4(α + θ ) ) . (2.4) However, the derivatives of the left-hand side of Eqs. (2.3) are with respect to the time variable t , which is not periodic. So we change the variable θ as the new independent one, and we denote by a prime the derivative with respect to θ . Moreover, we write the system as a Taylor series in powers of ε and obtain r ′ = ε ( ρ sin α sin 2θ (−2r2 + ρ2 cos 2α + ρ2) − ρ cos α(cos 2θ (r2 + ρ2 cos 2α) + r2 + ρ2) ) + O(ε2), ρ ′ = 1 2εr ( r2 cos(α − 2θ ) + 2(r2 + ρ2) cos α + (r2 − ρ2) cos(α + 2θ ) + 3ρ2 cos(3α + 2θ ) ) + O(ε2), α′ = ε 1 rρ sin α (−r4 + (ρ4 − 3r2ρ2) cos 2(α + θ ) − r2(r2 − 3ρ2) cos 2θ + ρ4 ) + O(ε2). (2.5) Now, system (2.5) is 2π -periodic in the variable θ . We shall apply Theorem 5 in the Hamiltonian level H = h for h > 0, H given by (2.4). Solving H = h for ρ = ρ0 + ερ1 + O(ε2), we have ρ = √ 2h − r2 + ε(2hr sin α cos 2(α + θ ) + 2hr sin α + r3 sin α cos 2θ − r3 sin α cos 2(α + θ )) + O(ε2). (2.6) Substituting ρ in Eq. (2.5) and developing it in power series of ε, we have just two differential equations r ′ = − 1 2ε √ 2h − r2 ( 8h cos α cos2(α + θ ) + 2r2 sin α(sin 2(α + θ ) + 3 sin 2θ ) ) + O(ε2), α′ = 2ε r √ 2h−r2 sin α((2h2 − 5hr2 + 2r4) cos 2(α + θ ) + r2(3h − 2r2) cos 2θ + 2h(h−r2)) + O(ε2). (2.7) Downloaded 18 Jul 2013 to 200.130.19.215. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions 082701-5 Periodic orbits of Hill lunar problem J. Math. Phys. 52, 082701 (2011) Now, system (2.7) satisfies all assumptions of Theorem 5 and it is in the normal form (A1). We define the function f1(r, α) = ( f11, f12) = 1 2π ∫ 2π 0 ( F11, F12 ) dθ = ( −2h √ 2h − r2 cos α, 4h(h − r2) sin α r √ 2h − r2 ) , (2.8) where F11 = −1 2 √ 2h − r2 ( 8h cos α cos2(α + θ ) + 2r2 sin α(sin 2(α + θ ) + 3 sin 2θ ) ) and F12 = 2 r √ 2h − r2 [ sin α ( (2h2 − 5hr2 + 2r4) cos 2(α + θ ) + r2(3h − 2r2) cos 2θ + 2h(h − r2) )] . To find the zeros of f1, first we solve f11 = 0 with respect to r . These zeros are r = ±√ 2h that take f2 indefinite. Then, we solve f11 = 0 for α and obtain α = ±π/2. Substituting in f12 = 0 these two values of α we have the following zeros for f1: s1 = ( √ h, π/2) and s2 = ( √ h,−π/2). Of course r = −√ h is a zero for f2 where α = ±π/2. Now, we shall verify at what zeros of f1 we have a non-zero Jacobian. The Jacobian of f1 is J f1 = ∣∣∣∣∣∣∣∣∣ 2hr cos α√ 2h − r2 2h √ 2h − r2 sin α − 8h3 sin α r2(2h − r2)3/2 4h(h − r2) cos α r √ 2h − r2 ∣∣∣∣∣∣∣∣∣ , (2.9) and the Jacobian restricted at each zero s1 and s2 takes the value 16h2 > 0. In short, by Theorem 5 the solutions s1 and s2 of f1(r, α) = 0 provide two periodic solutions of system (2.7), and consequently of the Hamiltonian system (1.7) on the level h > 0. This completes the proof of Theorem 1. III. PROOF OF PROPOSITION 2 We shall summarize some facts on the Liouville–Arnold integrability of the Hamiltonian systems and on the theory of the periodic orbits of the differential equations, for more details see Refs. 1 and 2 and Subsection 7.1.2 of Ref. 2, respectively. We present these results for Hamiltonian systems of two degrees of freedom, because we are studying a Hamiltonian system with two degrees of freedom associated with the regularized Hill lunar problem, but these results work for an arbitrary number of degrees of freedom. We recall that a Hamiltonian system with Hamiltonian H of two degrees of freedom is integrable in the sense of Liouville–Arnold if it has a first integral G independent of H (i.e., the gradient vectors of H and G are independent in all the points of the phase space except perhaps in a set of zero Lebesgue measure), and in involution with H (i.e., the parenthesis of Poisson of H and G is zero). For Hamiltonian systems with two degrees of freedom the involution condition is redundant, because the fact that G is a first integral of the Hamiltonian system implies that the mentioned Poisson parenthesis is always zero. A flow defined on a subspace of the phase space is complete if its solutions are defined for all time. Now, we shall state the Liouville–Arnold theorem restricted to Hamiltonian systems of two degrees of freedom. Theorem 3: Suppose that a Hamiltonian system with two degrees of freedom defined on the phase space M has its Hamiltonian H and the function G as two independent first integrals in involution. If Ihc = {p ∈ M : H (p) = h and C(p) = c} �= ∅ and (h, c) is a regular value of the map (H, G), then the following statements hold: Downloaded 18 Jul 2013 to 200.130.19.215. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions 082701-6 J. Llibre and L. F. Roberto J. Math. Phys. 52, 082701 (2011) (a) Ihc is a two-dimensional submanifold of M invariant under the flow of the Hamiltonian system. (b) If the flow on a connected component I ∗ hc of Ihc is complete, then I ∗ hc is diffeomorphic either to the torus S1 × S1, or to the cylinder S1 × R, or to the plane R2. If I ∗ hc is compact, then the flow on it is always complete and I ∗ hc ≈ S1 × S1. (c) Under the hypothesis (b) the flow on I ∗ hc is conjugated to a linear flow on S1 × S1, on S1 × R, or on R2. The main result of this theorem is that the connected components of the invariant sets associated with the two independent first integrals in involution are generically submanifolds of the phase space, and if the flow on them is complete then they are diffeomorphic to a torus, a cylinder or a plane, where the flow is conjugated to a linear one. Using the notation of Theorem 3 when a connected component I ∗ hc is diffeomorphic to a torus, either all orbits on this torus are periodic if the rotation number associated with this torus is rational, or they are quasi-periodic (i.e., every orbit is dense in the torus) if the rotation number associated with this torus is not rational. We consider the autonomous differential system ẋ = f (x), (3.1) where f : U → Rn is C2, U is an open subset of Rn , and the dot denotes the derivative respect to the time t . We write its general solution as φ(t, x0) with φ(0, x0) = x0 ∈ U and t belonging to its maximal interval of definition. We say that φ(t, x0) is T -periodic with T > 0 if and only if φ(T, x0) = x0 and φ(t, x0) �= x0 for t ∈ (0, T ). The periodic orbit associated with the periodic solution φ(t, x0) is γ = {φ(t, x0), t ∈ [0, T ]}. The variational equation associated with the T -periodic solution φ(t, x0) is Ṁ = ( ∂ f (x) ∂x ∣∣∣ x=φ(t,x0) ) M, (3.2) where M is an n × n matrix. The monodromy matrix associated with the T -periodic solution φ(t, x0) is the solution M(T, x0) of (3.2) satisfying that M(0, x0) is the identity matrix. The eigenvalues λ of the monodromy matrix associated with the periodic solution φ(t, x0) are called the multipliers of the periodic orbit. For an autonomous differential system, one of the multipliers is always 1, and its corresponding eigenvector is a tangent to the periodic orbit. A periodic orbit of an autonomous Hamiltonian system always has two multipliers equal to 1. One multiplier is 1 because the Hamiltonian system is autonomous, and another is 1 due to the existence of the first integral given by the Hamiltonian. Theorem 4: If a Hamiltonian system with two degrees of freedom and Hamiltonian H is Liouville–Arnold integrable, and G is a second first integral such that the gradients of H and G are linearly independent at each point of a periodic orbit of the system, then all the multipliers of this periodic orbit are equal to 1. Theorem 4 is due to Poincaré.15 It gives us a tool to study the non-Liouville–Arnold integrability, independently of the class of differentiability of the second first integral. The main problem for applying this theorem is to find periodic orbits having multipliers different from 1. Proof of Proposition 2: We know by Theorem 1 that the regularized Hill lunar system at every positive Hamiltonian level has at least 2 periodic solutions corresponding to solutions s1 and s2, and that their associated Jacobians are non-zero, that is, J f1(si ) = 16h2 for i = 1, 2. So the corresponding multipliers are not all equal to 1 (see for more details the last part of the Appendix). Hence, by Theorem 4, either the regularized Hill lunar problem cannot be Liouville–Arnold integrable with any second C1 first integral G, or this system is Liouville–Arnold integrable and the vector gradient of H and G are linearly dependent on some points of these periodic orbits. Therefore, Proposition 2 is proved. � Downloaded 18 Jul 2013 to 200.130.19.215. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions 082701-7 Periodic orbits of Hill lunar problem J. Math. Phys. 52, 082701 (2011) ACKNOWLEDGMENTS The first author is partially supported by Grant Nos. MEC/FEDER MTM 2008-03437, CIRIT 2009SGR 410, and ICREA Academia. The second author is partially supported by CAPES/MECD- DGU 015/2010 Brazil and Spain, Process No. BEX 4251/10-5. APPENDIX: PERIODIC ORBITS VIA AVERAGING THEORY OF FIRST ORDER Now, we shall present the basic results from averaging theory that we need for proving the results of this paper. The next theorem provides a first order approximation for the periodic solutions of a periodic differential system, for the proof see Theorems 11.5 and 11.6 of Verhulst.17 Consider the differential equation ẋ = εF1(t, x) + ε2 F2(t, x, ε), x(0) = x0 (A1) with x ∈ D ⊂ Rn, t ≥ 0. Moreover, we assume that both F1(t, x) and F2(t, x, ε) are T -periodic in t . Separately, we consider in D the averaged differential equation ẏ = ε f1(y), y(0) = x0, (A2) where f1(y) = 1 T ∫ T 0 F1(t, y)dt. Under certain conditions, equilibrium solutions of the averaged equation turn out to correspond with T -periodic solutions of Eq. (A1). Theorem 5: Consider the two initial value problems (A1) and (A2). Suppose: (i) F1, its Jacobian ∂ F1/∂x, and its Hessian ∂2 F1/∂x2, F2 and its Jacobian ∂ F2/∂x are defined, continuous and bounded by a constant independent of ε in [0,∞) × D and ε ∈ (0, ε0]. (ii) F1 and F2 are T −periodic in t (T independent of ε). (iii) y(t) belongs to � on the interval of time [0, 1/ε]. Then the following statements hold: (a) For t ∈ [1, ε], we have that x(t) − y(t) = O(ε), as ε → 0. (b) If p is a singular point of the averaged equation (A2) and det ( ∂ f1 ∂y )∣∣∣∣ y=p �= 0, then there exists a T −periodic solution ϕ(t, ε) of Eq. (A1) which is close to p, such that ϕ(0, ε) → p as ε → 0. (c) The stability or instability of the limit cycle ϕ(t, ε) is given by the stability or instability of the singular point p of the averaged system (A2). In fact, the singular point p has the stability behavior of the Poincaré map associated with the limit cycle ϕ(t, ε). In the following we use the idea of the proof of Theorem 5(c). For more details, see Secs. 6.3 and 11.8 of Ref. 17. Suppose that ϕ(t, ε) is a periodic solution of (A1) corresponding to y = p, a singular point of the averaged equation (A2). We linearize Eq. (A1) in a neighbourhood of the periodic solution ϕ(t, ε) and obtain a linear equation with T -periodic coefficients, ẋ = εA(T, ε)x (A3) with A(t, ε) = ∂ ∂x [F1(t, x) + εF2(t, x, ε)]x=ϕ(t,ε) . Downloaded 18 Jul 2013 to 200.130.19.215. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions 082701-8 J. Llibre and L. F. Roberto J. Math. Phys. 52, 082701 (2011) We introduce the T -periodic matrix B(t) = ∂ F1 ∂x (t, p). From Theorem 5, we have limε→0 A(t, ε) = B(t). We shall use the matrices B1 = 1 T ∫ T 0 B(t)dt and C(t) = ∫ t 0 [B(s) − B1] ds. Note that B1 is the matrix of the linearized averaged equation. The matrix C(t) is T -periodic and it has average zero. The near-identity transformation x → y = (I − εC(t))x (A4) writes Eq. (A3) as ẏ = εB1 y + ε(A(t, ε) − B(t))y + O(ε2). (A5) We note that A(t, ε) − B(t) → 0 as ε → 0, and also that the characteristic exponents of Eq. (A5) depend continuously on the small parameter ε. It follows that for ε sufficiently small, if the determi- nant of B1 is not zero, then 0 is not an eigenvalue of the matrix B1, and then it is not a characteristic exponent of (A5). By the near-identity transformation we obtain that system (A3) has not multipliers equal to 1. 1 Abraham, R. and Marsden, J. E., Foundations of Mechanics (Benjamin, Reading, MA, 1978). 2 Arnold, V. I., Kozlov, V., and Neishtadt, A., Dynamical Systems III. Mathematical Aspects of Classical and Celestial Mechanics, Encyclopedia of Mathematical Science, 4th ed. (Springer-Verlag, Berlin, 2006). 3 Hénon, M., “New families of periodic orbits in Hill’s problem of three bodies,” Celest. Mech. Dyn. Astron. 85(3), 223–246 (2003). 4 Hill, G., “Researches in the Lunar theory,” Am. J. Math. 1, 5–26, 129–147, 245–260 (1878). 5 Howison, R. C. and Meyer, K. R., “Doubly symmetric periodic solutions of Hill’s lunar problem,” in Hamiltonian Systems and Celestial Mechanics, World Scientific Monograph Series in Mathematics Vol. 6 (World Scientific, River Edge, NJ, 2000), pp 186–196. 6 Jiménez–Lara, L. and Llibre, J., “Periodic orbits and non-integrability of Henon–Heiles system,” J. Phys. A: Math. Theor. 44, 205103 (2011). 7 Jiménez–Lara, L. and Llibre, J., “Periodic orbits and non-integrability of generalized classical Yang-Mills Hamiltonian system,” J. Math. Phys. 52, 032901 (2011). 8 Kozlov, V. V., “Integrability and non-integrability in Hamiltonian mechanics,” Russ. Math. Surveys 38(1), 1–76 (1983). 9 Llibre, J. and Roberto, L. A., “Periodic orbits and non-integrability of Armbruster-Guckenheimer-Kim potential,” preprint (2011). 10 Maciejewski, A. J. and Rybicki, S. M., “Global bifurcations of periodic solutions of the Hill lunar problem,” Celest. Mech. Dyn. Astron. 81(4), 279–297 (2001). 11 Meletlidou, E. and Winterberg, F. J., “Non-continuation of integrals of the rotating two-body problem in Hill’s lunar problem,” Celest. Mech. Dyn. Astron. 88(1), 37–49 (2004). 12 Meletlidou, E., Ichtiaroglou, S., and Winterberg, F. J., “Non-integrability of Hill’s lunar problem,” Celest. Mech. Dyn. Astron. 80(2), 145–156 (2001). 13 Meyer, K. R., Hall, G. R., and Offin, D., Introduction to Hamiltonian Dynamical Systems and the N–Body Problem, Applied Mathematical Science Vol. 90, 2nd ed. (Springer, New York, 2009). 14 Morales-Ruiz, J. J., Simó, C., and Simon, S., “Algebraic proof of the non-integrability of Hill’s problem,” Ergod. Theory Dyn. Syst. 25(4), 1237–1256 (2005). 15 Poincaré, H., Les méthodes nouvelles de la mécanique céleste (Gauthier-Villars, Paris, 1899), Vol. 1. 16 Simó, C. and Stuchi, T. J., “Central stable/unstable manifolds and the destruction of KAM tori in the planar Hill problem,” Phys. D 140(1–2), 1–32 (2000). 17 F. Verhulst, Nonlinear Differential Equations and Dynamical Systems (Springer, New York, 1991). Downloaded 18 Jul 2013 to 200.130.19.215. This article is copyrighted as indicated in the abstract. 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