This is a preprint of: “Periodic solutions and their stability for some perturbed Hamiltonian systems”, Juan Luis Garcia Guirao, Jaume Llibre, Juan A. Vera, Bruce A. Wade, Int. J. Geom. Methods Mod. Phys., vol. 18(1), 2021. DOI: [10.1142/S0219887821500134] PERIODIC SOLUTIONS AND THEIR STABILITY FOR SOME PERTURBED HAMILTONIAN SYSTEMS JUAN L.G. GUIRAO1 , JAUME LLIBRE2 JUAN A. VERA3 AND BRUCE A. WADE4 Abstract. We deal with non–autonomous Hamiltonian systems of one degree of freedom. For such differential systems we compute analytically some of their periodic solutions, together with their type of stability. The tool for proving these results is the averaging theory of dynamical systems. We present some applications of these results. 1. Introduction and main results We consider the following perturbed first order differential systems dp ∂H dq ∂H (1) =− + εP1 (q, p, t) , = + εP2 (q, p, t) , dt ∂q dt ∂p where H = H(q, p) is a Hamiltonian function defined in an open set U of R2 , q is the position, p its associated momentum, the functions Pi defined in U × R are smooth 2π− periodic in t, and ε is a small parameter. All the lemmas, theorems and corollaries stated in this section are proved in the next two sections. The theorems are proved using the averaging theory, see in the appendix a summary of the results on this theory that we need for proving our theorems. For computing analytically periodic solutions of the differential equations we shall use the averaging theory, see for instance [2, 3, 4], but in those papers we studied periodic solutions of autonomous Hamiltonian systems, and in the present one we are working with non- autonomous differential systems. We assume that the Hamiltonian H is expressed in action–angle variables (I, θ) as H(p, q) = H0 (I). If the change of variable (p, q) 7→ (I, θ) is given by I = I(p, q) and θ = θ(p, q), then we consider the functions ∂I ∂I F1 (I, θ, t) = P1 (q, p, t) + P2 (q, p, t) , ∂p ∂q ∂θ ∂θ F2 (I, θ, t) = P1 (q, p, t) + P2 (q, p, t) . ∂p ∂q The following result holds taking into account that the change of variables to action–angle variables is canonical. 1991 Mathematics Subject Classification. [2010]Primary 34C10, 34C25. Key words and phrases. Hamiltonian system, periodic solutions, averaging theory. 1 2 J.L.G. GUIRAO, J. LLIBRE, J.A. VERA, B. WADE Lemma 1. System (1) in action–angle variables (I, θ) becomes dI dθ (2) = εF1 (I, θ, t), = ω(I) + εF2 (I, θ, t), dt dt where ω(I) = dH0 (I)/dI. The next result provides sufficient conditions for the existence and the stability of 2π–periodic solutions of system (2) when the functions Pi ’s are independent of t, and consequently the functions Fi ’s are also independent of t. Theorem 2. If the functions Fi ’s do not depend on the time t, are 2π– periodic in the variable θ, and if I0 is a simple zero of the function Z 2π 1 F1 (I, θ) F(I) = dθ, 2π 0 ω then there exists a 2π–periodic solution Iε (θ) of system (2) such that Iε (0) → I0 when ε → 0. Moreover, if F 0 (I0 ) > 0 then the periodic solution Iε (θ) is unstable, and if F 0 (I0 ) < 0 it is stable. Assume that the functions Pi are periodic in the variable t with period T = 2π/ω0 , being ω0 a real positive number. We shall study the existence of T periodic solutions of system (2). We shall consider the Hamiltonian H linearizable with ω(I) = ω0 constant. In this case by means of the change of variables (I, θ) ←→ (I, ϕ) given by I = I and θ = ϕ + ω0 t, system (2) is transformed into the system dI dϕ (3) = εF1 (I, ϕ + ω0 t, t), = εF2 (I, ϕ + ω0 t, t). dt dt The following result provides sufficient conditions for the existence and the stability of T –periodic solutions of system (3). Theorem 3. If the perturbed functions Fi are T -periodic in the variable t and (I0 , ϕ0 ) is a zero of the function Z T Z T  1 F(I, ϕ) = F1 (I, ϕ + ω0 t, t) dt, F2 (I, ϕ + ω0 t, t) dt , T 0 0 and the Jacobian det(DF(I0 , ϕ0 )) 6= 0, then there exists a T –periodic solu- tion γε (t) of system (3) such that γε (0) → (I0 , ϕ0 ) when ε → 0. Moreover, if one of the two roots of the characteristic polynomial of the Jacobian matrix DF(I0 , ϕ0 ) have positive real part the periodic solution γε (t) is unstable. If the two roots of that characteristic polynomial have negative real part, then the periodic solution γε (t) is stable. Now we are going to present some applications of the previous theorems. PERIODIC SOLUTIONS OF PERTURBED HAMILTONIAN SYSTEMS 3 Using Theorem 2 we will study the periodic solutions of the following second–order differential equation   d2 x k dx (4) + = εP x, , dt2 x2 dt where k ∈ R+ and ε is a small parameter. Note that when ε = 0 we have the 1–dimensional Kepler problem, see for instance [6] and [7]. As we shall see in subsection 3.1 doing the change of time t 7→ E given by t = (E − sin E)/ω, the elliptic collision solutions of the 1–dimensional Kepler problem become periodic solutions. We recall that the variable E is named as the eccentric anomaly. We want to study which of these periodic solutions persist under the perturbation given in (4). Theorem 4. If I0 is a simple zero of the function Z 2π  2  1 k I k sin E F(I) = sin E P (1 − cos E), dE, 2π 0 2ω 2 I k 2I(1 − cos E) then it exists a 2π–periodic solution xε (E) of the differential equation (4) such that xε (0) → I0 when ε → 0. Moreover, if F 0 (I0 ) > 0 then the periodic solution xε (E) is unstable, and if F 0 (I0 ) < 0 it is stable. See a numerical example of the periodic solution of Theorem 4 after the proof of this theorem. Corollary 5. For ε 6= 0 sufficiently small the differential equation d2 x k dx + 2 = ε(a + bxn ) , dt2 x dt where n ∈ N and a, b ∈ R, has a periodic solution if ab < 0. Corollary 6. For ε 6= 0 sufficiently small and for any positive integer N the differential equation   d2 x k dx + 2 = ε sin x , dt2 x dt has at least N periodic solutions. Using Theorem 3 we will study the periodic solutions of the second–order differential equation d2 x (5) + ω02 x = εx2 (a + bx2 ) sin(ω0 t), dt2 with ω0 > 0 and ab > 0. Corollary 7. The differential equation (5) has at least four periodic solu- tions r 2Ii xi (t, ε) = cos(ϕi + ω0 t) + O(ε), ω0 4 J.L.G. GUIRAO, J. LLIBRE, J.A. VERA, B. WADE for i = 1, 2, 3, 4, with (I1 , ϕ1 ) = (aω0 /b, 0), (I2 , ϕ2 ) = (aω0 /b, π), (I3 , ϕ3 ) = (2aω0 /(3b), π/2), (I4 , ϕ4 ) = (2aω0 /(3b), π). The periodic solution x1 is sta- ble and the periodic solutions xi for i = 2, 3, 4 are unstable. A numerical application of Corollary 7 and consequently of Theorem 3 appears after the proof of Corollary 7. 2. Proofs of our main results Proof Lemma 1. The unperturbed system of system (1) dp ∂H dq ∂H =− , = , dt ∂q dt ∂p has a very simple formulation in action–angle variables, namely dI dθ = 0, = ω. dt dt Since the Poisson Bracket of two smooth functions f and g is defined as ∂f ∂g ∂f ∂g {f, g} = − , ∂q ∂p ∂p ∂q we have that dI ∂I ∂H ∂I ∂H = − = {I, H} = 0 dt ∂q ∂p ∂p ∂q and dθ ∂θ ∂H ∂θ ∂H = − = {θ, H} = ω. dt ∂q ∂p ∂p ∂q The variation with the time of the action I of the perturbed system (1) satisfies dI ∂I dp ∂I dq = + dt ∂p dt ∂q dt     ∂I ∂H ∂I ∂H = − + εP1 (q, p) + + εP2 (q, p) ∂p ∂q ∂q ∂p   ∂I ∂I = ε P1 (q, p) + P2 (q, p) . ∂p ∂q The variation with the time of the angle θ satisfies dθ ∂θ dp ∂θ dq = + dt ∂p dt ∂q dt     ∂θ ∂H ∂θ ∂H = − + εP1 (q, p) + + εP2 (q, p) ∂p ∂q ∂q ∂p   ∂θ ∂θ = ω+ε P1 (q, p) + P2 (q, p) . ∂p ∂q PERIODIC SOLUTIONS OF PERTURBED HAMILTONIAN SYSTEMS 5 Then the differential system (1) is transformed into the differential system (2).  Proof Theorem 2. If the functions Fi are independent of the time t, then system (2) is autonomous. Taking as new independent variable the variable θ system (2) becomes the differential equation dI εF1 (I, θ) F1 (I, θ) (6) = =ε + O(ε2 ). dθ ω + εF2 (I, θ) ω Now using the first order averaging theory (see Theorem 8 of the appendix) we obtain the averaged function Z 2π 1 F1 (I, θ) dθ, 2π 0 ω and the statements of the theorem follows.  Proof Theorem 3. Applying again the first order averaging theory but now to the differential system (3) we obtain the averaged function F(ρ, ϕ). Then the statement of Theorem 3 follows directly from the statement of Theorem 8 of the appendix.  3. Applications of our main results As an application of Theorems 2 and 3 we will study the following prob- lems. 3.1. Periodic solutions of the perturbed 1–dimensional Kepler Prob- lem. dx Proof of Theorem 4. Doing the change of variables x = q and = p, the dt second order differential equation (4) becomes the first order differential system dp k dq = − 2 + εP (q, p) , = p. dt q dt I H : R×(0, +∞)→ R We compute the action–angle variables for the Hamiltonian p2 k 1 given by H(p, q) = − . The action is given by I = p dq computed 2 q 2π in a closed curve of the phase space at the energy level H = h < 0. Then Z −k/h s 1 2k k I = lim + 2h dq = √ . π s→0 s q 2 −h 6 J.L.G. GUIRAO, J. LLIBRE, J.A. VERA, B. WADE k k2 Solving the equation I = √ we obtain h = − 2 . Then H(p, q) = 2 −h 4I k2 − = H0 (I). We consider the generating function 4I 2 Z 4I 2 /k r 2k k2 W(q, I) = ± − 2 ds. q s 2I Then we obtain s r ! Z 4I 2 /k ∂W 2k k2 ∂W ∂ 2k k2 p= =∓ − 2, θ= = ± − 2 ds . ∂q q 2I ∂I ∂I q s 2I dθ dH0 k2 k2 Since =ω= = 3 , we get θ = 3 t = ωt. dt dI 2I 2I p2 k dq Fixed the energy level h, from − = h and using p = , we obtain r 2 q dt dq 2k dq k2 = ± + 2h and dt = ± v u . From the relation h = − , dt q u 2k t 4I 2 +2h q we obtain Z 4I 2 /k 1 (7) t=± r ds. q 2k k2 − 2 s 2I Note that the t in this last equality is the time that needs the particle for going from the position q to the maximal value of its position obtained solving the equation s dq 2k = + h = 0. dt q I2 Doing the change of variable q = (1 − cos E) in (7) we obtain k 2I 3 1 t= 2 (E − sin E) = (E − sin E), k ω and consequently θ = E − sin E. k2 p2 k Since − = − , we have 4I 2 2 q     ∂ k2 k 2 ∂I ∂ p2 k − 2 = 3 = − = p, ∂p 4I 2I ∂p ∂p 2 q k 2 ∂I therefore p = . 2I 3 ∂p PERIODIC SOLUTIONS OF PERTURBED HAMILTONIAN SYSTEMS 7 k2 ∂I p Using ω = we obtain = . By the chain rule and the equation 2I 3 ∂p ω of motion we get   dθ dθ dp dθ k ω= = = − 2 . dt dp dt dp q Then dθ ω I = − q 2 = − (1 − cos E)2 . dp k 2k Finally, from I2 k sin E ∂I p q= (1 − cos E), p= and = , k 2I 1 − cos E ∂p ω we obtain ∂I F1 (I, θ, t) = P (q, p) ∂p   k sin E I2 k sin E = P (1 − cos E), 2ωI 1 − cos E k 2I(1 − cos E) = G1 (I, E), and using ∂θ I = (1 − cos E)2 , ∂p k we have ∂θ F2 (I, θ) = P (q, p) ∂p   I I2 k sin E = (1 − cos E)2 P (1 − cos E), k k 2I(1 − cos E) = G2 (I, E). Then from Theorem 2 and since dθ/dE = 1 − cos E, we obtain  2  dI k I k sin E = ε 2 sin E P (1 − cos E), + O(ε2 ). dE 2ω I k 2I(1 − cos E) Again from Theorem 2 it follows the statement of Theorem 4.  1.194 7.665 1.192 7.664 1.190 1.188 7.663 1.186 7.662 1.184 1 2 3 4 5 6 1 2 3 4 5 6 Figure 1 Figure 2 8 J.L.G. GUIRAO, J. LLIBRE, J.A. VERA, B. WADE The two periodic solutions of the 1-D Kepler differential equation with P(x, ẋ) = (x2 − 1)ẋ and P(x, ẋ) = sin(xẋ). In the first figure k = 1, ε = 1/100 and I(0) = 21/4 . In the second figure k = 1, ε = 10−6 and I(0) = 7.663411940415026 the first zero of the Bessel function of first kind. The horizontal axis there is the eccentric anomaly E varying from 0 to 2π, and the vertical axis the action I(E). d2 x k 3.2. Periodic solutions of the differential equation 2 + 2 = ε(a + dt x dx bxn ) . dt Proof of Corollary 5. From the statement of Theorem 4 and since P(q, p) = (a + bq n )p we have Z 2π  2  1 k I k sin E F(I) = sin E P (1 − cos E), dE 2π 0 2ω 2 I k 2I(1 − cos E) Z 2π   1 k 2 sin2 E bI 2n (1 − cos E)n = a + dE 2π 0 4ω 2 I 2 (1 − cos E) kn Z 2π   1 I 4 sin2 E bI 2n (1 − cos E)n = a + dE, 2π 0 k 2 (1 − cos E) kn because ω = k 2 /(2I 3 ). Therefore ! I4 2n b I 2n Γ n + 21 F(I) = 2 a+ √ n , k π k Γ(n + 2) where Γ(z) is the Euler gamma function, see for more details [1]. The function F(I) has the unique positive zero !1 1 a k n Γ(n + 2) 2n I0 = π 4n − n  . 2 b Γ n + 12 if ab < 0. So, from Theorem 4 the corollary follows.    d2 x k dx 3.3. Periodic solutions of the differential equation 2 + 2 = ε sin x . dt x dt Proof of Corollary 6. From the statement of Theorem 4 and since P(q, p) = sin(qp) we have Z 2π  2  1 k I k sin E F(I) = sin E P (1 − cos E), dE 2π 0 2ω 2 I k 2I(1 − cos E) Z 2π 5   1 2I I sin E = sin E sin dE, 2π 0 k 3 2 PERIODIC SOLUTIONS OF PERTURBED HAMILTONIAN SYSTEMS 9 because ω = k 2 /(2I 3 ). Therefore 2I 5 J1 (I/2) F(I) = , k3 where J1 (z) is the Bessel function of first kind. The function J1 (z) has infinitely many positive simple zeros accumulating at infinity, see for instance [1]. Hence, from Theorem 4 the corollary follows.  d2 x 3.4. Periodic solutions of the differential equation + ω02 x = dt2 εx2 (a + bx2 ) sin(ω0 t). Proof of Corollary 7. The corresponding first order differential system of the second–order differential equation of the statement of this corollary is dp dq = −ω02 q + εq 2 (a − bq 2 ) sin(ω0 t), = p. dt dt p2 + ω02 q 2 The Hamiltonian H(p, q) = has the action–angle variables (I, θ) 2 r √ 2I given by p = 2ω0 I sin θ and q = cos θ, see [5] for details on the ω0 computation of these action–angle variables. The Hamiltonian H in the action–angle variables is H(I) = ω0 I. Com- puting F1 and F2 we obtain √  2I 2ω0 I aω0 − 2bI cos2 θ cos2 θ sin(ω0 t) F1 (I, θ, t) = ω03 and √  2ω0 I aω0 − 2bI cos2 θ cos θ sin (ω0 t) F2 (I, θ, t) = − . ω03 The map F is given by  √ √  I 2ω0 I (aω0 − bI) cos ϕ 2ω0 I (2aω0 − 3bI) sin ϕ F(I, ϕ) = , . 4ω03 4ω03 The system F(I, ϕ) = (0, 0) has the four solutions (I1 , ϕ1 ) = (aω0 /b, 0), (I2 , ϕ2 ) = (aω0 /b, π), (I3 , ϕ3 ) = (2aω0 /(3b), π/2), (I4 , ϕ4 ) = (2aω0 /(3b), π) and   λ 0 DF|(I,ϕ)=(I1 ,ϕ1 ) = − DF|(I,ϕ)=(I2 ,ϕ2 ) = , 0 λ   0 λ1 DF|(I,ϕ)=(I3 ,ϕ3 ) = − DF|(I,ϕ)=(I4 ,ϕ4 ) = , λ2 0 with √ √ 2√ √ a a 3a a 3ab λ=− √ , λ1 = − √ , λ2 = − 2 √ . 2 2bω0 27b bω0 2ω0 b 10 J.L.G. GUIRAO, J. LLIBRE, J.A. VERA, B. WADE The two eigenvalues corresponding to the√linearized Poincaré√map asso- a a a a ciated to the periodic orbit γ1 are λ = − √ and λ = − √ . 2 2bω0 2 2bω0 The two eigenvalues corresponding to the√ linearized Poincaré √ map asso- a a a a ciated to the periodic orbit γ2 are λ = √ and λ = √ . 2 2bω0 2 2bω0 The two eigenvalues corresponding to the√linearized Poincaré map asso- ciated to the periodic orbit γ3 and γ4 are ± λ1 λ2 . Applying Theorem 8 and the canonical change of variables we obtain immediately the results stated in the corollary.  0.00015 3.14155 0.00010 3.14150 0.00005 0.99970 0.99975 0.99980 0.99985 0.99990 0.99995 1.00000 0.99970 0.99975 0.99980 0.99985 0.99990 0.99995 1.00000 Figure 3 Figure 4 0.0005 3.1416 0.0004 3.1415 0.0003 3.1414 0.0002 3.1413 0.0001 3.1412 1.4980 1.4985 1.4990 1.4995 1.5000 1.4980 1.4985 1.4990 1.4995 1.5000 Figure 5 Figure 6 The four periodic solutions of Corollary 7 corresponding to the differential equation (5) with a = b = ω = 1 and ε = 1/1000 are shown in Figures 3, 4, 5 and 6. In the horizontal axis there is the action I(t) and in the vertical one the angle ϕ(t). 4. Conclusion section In this paper we consider non–autonomous Hamiltonian systems of one degree of freedom, and we show how to compute analytically some of their periodic solutions, together with their type of stability, using the averag- ing theory. We illustrate this tool studying two kinds of non–autonomous Hamiltonian systems of one degree of freedom, see Theorems 2 and 3 and its applications Corollaries 5, 6 and 7. Appendix: Averaging theory of first order We deal with the two initial value problems (8) ẋ = εF1 (t, x) + ε2 F2 (t, x, ε), x(0) = x0 , PERIODIC SOLUTIONS OF PERTURBED HAMILTONIAN SYSTEMS 11 and (9) ẏ = εf (y), y(0) = x0 , where the variables x, y and x0 are in the open set Ω ⊂ Rn , t ∈ [0, ∞) and ε ∈ (0, ε0 ]. The functions F1 and F2 which appear in (8) are T –periodic in the variable t. The averaged function of system (8) is defined by Z 1 T f (y) = F1 (t, y)dt. T 0 Theorem 8. Suppose that the functions F1 , Dx F1 , Dxx F1 , F2 and Dx F2 are continuous, bounded by a constant independent of ε in [0, ∞)× Ω×(0, ε0 ], and that y(t) ∈ Ω for t ∈ [0, 1/ε]. Then the following statements hold. (a) x(t) − y(t) = O(ε) as ε → 0 for t ∈ [0, 1/ε]. 6 0 satisfies f (p) = 0 and the Jacobian det(Dy f (p)) 6= 0, then (b) If p = there is a T –periodic solution x(t, ε) of the differential system (8) such that φ(0, ε) = p + O(ε). (c) If a real part of some eigenvalue of the Jacobian matrix Dy f (p) is positive, then the periodic solution x(t, ε) is unstable. The periodic solution x(t, ε) is stable if all the real parts of the eigenvalues of the Jacobian matrix Dy f (p) are negative. For a proof of Theorem 8 see for instance Chapter 11 of [8]. Acknowledgements The first author is partially supported by Ministerio de Ciencia, Inno- vación y Universidades grant number PGC2018-097198-B-I00 and Fundación Séneca de la Región de Murcia grant number 20783/PI/18. The second author is partially supported by the Ministerio de Cien- cia, Innovación y Universidades, Agencia Estatal de Investigación grant MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. References [1] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, National Bureau of Standards Applied Mathematics Series Vol. 55, Tenth Printing, Washington, 1972. [2] M. Kulesza and J. Llibre, The 3–dimensional cored and logarithm potentials: peri- odic orbits, J. Math. Physics 55 (2014), 112702–pp 10. [3] F.E. Lembarki and J. Llibre, Periodic orbits for the generalized Yang-Mills Hamil- tonian system in dimension 6, Nonlinear Dynamics 76 (2014), 1807–1819. [4] J. Llibre and C. Vidal, New 1 : 1 : 1 periodic solutions in 3–dimensional galactic– type Hamiltonian systems, Nonlinear Dynamics 78 (2014), 969–980, 12 J.L.G. GUIRAO, J. LLIBRE, J.A. VERA, B. WADE [5] K.R. Meyer, G.R. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the N -Body Problem, Applied Mathematical Sciences 90, Second Edition, Springer-Verlag, 2009. [6] A.E. Roy, Orbital motion, Halsted Press,John Wiley & Sons, New York, 1978. [7] E. L. Stiefel and G. Scheifele. Linear and Regular Celestial Mechanics. Springer- Verlag Berlin Heidelberg. 1971. [8] F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Universitext, Springer, New York, 1996. 1 Departamento de Matemáca Aplicada y Estadı́stica. Universidad Politécnica de Cartagena, 30202-Cartagena, Región de Murcia, Spain, Spain Email address: juan.garcia@upct.es 1 Departament de Matemàtiques. Universitat Autònoma de Barcelona, Bel- laterra, 08193-Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat 3 Centro Universitario de la Defensa. Academia General del Aire. Uni- versidad Politécnica de Cartagena, 30720-Santiago de la Ribera, Región de Murcia, Spain Email address: juanantonio.vera@upct.es 4 Mathematics Department, P.O. Box 43568, University of Louisiana at Lafayette, Lafayette, LA 70504-3568, USA Email address: bwade@louisiana.edu