This is a preprint of: “Existence of periodic solutions for a class of second order ordinary differential equations”, Antonio Garcı́a, Jaume Llibre, Acta Appl. Math., 2019. DOI: [10.1007/s10440-019-00295-9] EXISTENCE OF PERIODIC SOLUTIONS FOR A CLASS OF SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS ANTONIO GARCIA1 AND JAUME LLIBRE2 Abstract. We provide sufficient conditions for the existence of a periodic solution for a class of second order differential equations of the form ẍ + g(x) = εf (t, x, ẋ, ε), where ε is a small parameter. 1. Introduction and statement of the results The second order differential equations of the form ẍ + g(x) = εf (t, x, ẋ, ε), have been studied by many authors because they have many applica- tions, see for instance [2, 3, 7, 8, 9, 12, 15]. Two of the main families studied are the Duffing equations see [5, 6], ... or the forced pendulum see the nice survey [11] and the references quoted therein. The aim of this work is to study periodic solutions of the second order differential equation (1) ẍ + g(x) = µ2n+1 p(t) + µ4n+1 q(t, x, y, µ), where n is a positive integer, µ is a small parameter, and the functions g(x) = x + x2n+1 (b + xh(x)) , and h(x) are smooth, b ̸= 0, p(t) and q(t, x, y, µ) are smooth and peri- odic with period 2π in the variable t. Let Γ(x) the Gamma function, see for more details [1], and let α and β the first Fourier coefficients of the periodic function p(t), i.e. ∫ ∫ 1 2π 1 2π α= p(t) cos t dt, β= p(t) sin t dt. π 0 π 0 Then our main result is the following. 2010 Mathematics Subject Classification. 37G15, 37C80, 37C30. Key words and phrases. periodic orbit, second–order differential equation, averag- ing theory. 1 http://www.gsd.uab.cat 2 A. GARCIA AND J. LLIBRE Theorem 1. If αβ ̸= 0 then for µ ̸= 0 sufficiently small the differential equation (1) has a 2π–periodic solution x(t, µ) such that ( ) 2n+1 1 ( 2 )−n 1 Γ(n + 2) β x(0, µ) = π 4n+2 ( ) α +1 + O(µ2n ). 2bΓ n + 32 α2 Theorem 1 is proved in section 3. Its proof uses the averaging theory for computing periodic solutions, see section 2 for a summary of the results on this theory that we shall need. 2. The averaging theory We want to study the T –periodic solutions of the periodic differential systems of the form (2) x′ = F0 (t, x) + εF1 (t, x) + ε2 F2 (t, x, ε), with ε > 0 sufficiently small, where F0 , F1 : R × Ω → Rn and F2 : R × Ω × (−ε0 , ε0 ) → Rn are C 2 functions, T –periodic in the variable t, and Ω is an open subset of Rn . Let x(t, z, ε) be the solution of the differential system (2) such that x(0, z, ε) = z. Suppose that the unperturbed system (3) x′ = F0 (t, x), has an open set V with V ⊂ Ω such that for each z ∈ V , x(t, z, 0) is T –periodic. Let y be an n × n matrix, and consider the first order variational equation (4) y′ = Dx F0 (t, x(t, z, 0))y, of the unperturbed system (3) on the periodic solution x(t, z, 0). Let Mz (t) be the fundamental matrix of the linear differential system (4) with periodic coefficients such that Mz (0) is the n × n identity matrix. Theorem 2. Consider the function F : V → Rn ∫ T (5) f (z) = Mz−1 (t)F1 (t, x(t, z, 0))dt. 0 If there exists α ∈ V with f (α) = 0 and (6) det ((d f /dz) (α)) ̸= 0, then there exists a T –periodic solution x(t, ε) of system (2) such that x(0, ε) = α + O(ε). PERIODIC SOLUTIONS 3 The existence of the periodic solution of Theorem 2 is due to Malkin [10] and Roseau [13], for a shorter and easier proof see [4]. The proof for the stability follows in a similar way to the proof of Theorem 11.6 of [14]. 3. Proof of Theorem 1 The differential equation of second order (1) can be written as the first order differential system ẋ = y, (7) ẏ = −x − x2n+1 (b + xh(x)) + µ2n+1 p(t) + µ4n+1 q(t, x, y, µ). In order to apply the averaging theory described in section 2 to this differential system we do the scaling x → µx and y → µy. Hence the differential system (7) becomes ẋ = y, (8) ẏ = −x + µ2n (−bx2n+1 + p(t)) + µ4n q ∗ (t, x, y, µ). This system is written into the normal form (2) for applying the aver- aging theory described in section 2, where x = (x, y), ε = µ2n , (9) F0 (x) = (y, −x), F1 (x, t) = (0, −bx2n+1 + p(t)), F2 (x, t, ε) = (0, q̄(t, x, y, ε)) . From section 2 the solution x(t, z, 0) = (x(t, z, 0), y(t, z, 0)) of system (8) with ε = 0 satisfies x(0, z, 0) = z = (x0 , y0 ), and consequently x(t, z, 0) = x0 cos t + y0 sin t, y(t, z, 0) = −x0 sin t + y0 cos t. The fundamental matrix Mz (t) = M (t) of the the first order varia- tional equation (4) satisfying (9) is ( ) cos t sin t M (t) = . − sin t cos t 4 A. GARCIA AND J. LLIBRE According with Theorem 2 in order to compute the 2π–periodic so- lutions of the differential system (8) we must compute the integral ( ) f1 (x0 , y0 ) f (z) = f2 (x0 , y0 ) ∫ 2π = M −1 (t)F1 (t, x(t, z, 0))dt 0  ∫ 2π ∫ 2π  2n+1  b sin t(x0 cos t + y0 sin t) dt − p(t) sin tdt   0 0  =   ∫ 2π ∫ 2π .   2n+1  −b cos t(x0 cos t + y0 sin t) dt + p(t) cos tdt 0 0 Doing induction with respect to n it is not difficult to show that ∫ 2π √ ( ) 2n+1 2 π Γ 32 + n ( )n sin t (x0 cos t + y0 sin t) dt = y0 x20 + y02 , 0 Γ(2 + n) ∫ 2π √ ( ) 2n+1 2 π Γ 32 + n ( )n cos t (x0 cos t + y0 sin t) dt = x0 x20 + y02 . 0 Γ(2 + n) Therefore we must solve the system  √ (3 )  ( ) 2 π b Γ + n y0 (x20 + y02 ) − πβ1  ( ) 2 n f1 (x0 , y0 )  Γ(2 + n) 0   = √ (3 ) = . f2 (x0 , y0 )  −2 π b Γ 2 + n  0 2 2 n x0 (x0 + y0 ) + πα1 Γ(2 + n) This system has a unique solution  ( 2 )−n  ( ) ( ) 2n+1 1 x∗0 Γ(n + 2) α αβ 2 + 1 1 ( )   = π 4n+2  ( ) . y0∗ 2bΓ n + 32 α2 −n β β2 + 1 The determinant (6) of the Jacobian matrix Df (x∗0 , y0∗ ) is ( )− 2n+1 2 1 2n Γ(n + 2) det(Df (x∗0 , y0∗ )) = 4 2n+1 (2n + 1)π 2n+1 +1 ( ) bΓ n + 32 (( )2n ( 2 )−n ) 2n+1 2 ( ( )−n ) 2n+1 2 α β2 β β2 + 1 + α α2 + 1 , and by assumptions it is positive because αβb ̸= 0. 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