This is a preprint of: “On the C 1 non-integrability of the autonomous differential systems”, Jaume Llibre, Nonlinear Anal. Real World Appl., vol. 74, 103943 (9 pages), 2023. DOI: [10.1016/j.nonrwa.2023.103943] ON THE C 1 NON-INTEGRABILITY OF THE AUTONOMOUS DIFFERENTIAL SYSTEMS JAUME LLIBRE Abstract. In the study of the dynamics of the autonomous differential sys- tems to know the existence or non-existence of first integrals is a relevant fact. These last decades the meromorphic non-integrability of the autonomous differential systems have been studied intensively using the Ziglin’s and the Morales-Ramis’ theories. Here we study the C 1 non-integrability of the au- tonomous differential systems, these studies goes back to Poincaré. It is known that the semiclassical Jayne-Cummings differential system of dimension five has only two independent meromorphic first integrals, namely H and F , and of course any meromorphic function in the variables H and F . Here we illustrate how to study the C 1 non-integrability of the autonomous differential systems showing that the semiclassical Jayne-Cummings differen- tial system of dimension five has only two independent C 1 first integrals H and F , and of course any C 1 function in the variables H and F . 1. Introduction Ziglin’s theory on the meromorphic non-integrability was inspired in the studies of the integrability of the rigid body done by Kovalevskaya. This theory study the non-integrability of an autonomous differential system using the monodromy group of the variational equation associated to some non-equilibrium solution of the analysed differential system. Ziglin’s theory was improved by the Morales-Ramis theory that considers the Galois differential group instead of the monodromy group of the variational equation, and in general the Galois group is easier to study, see [12]. But both theories only allow to study the non-existence of meromorphic first integrals. As Arnold said in [1] the mentioned both theories of non-integrability, in a be- ginning inspired in Kovalevskaya’s ideas, go back to Poincaré, because Poincaré for studying the non-integrability of the autonomous differential systems already used the multipliers of the monodromy group of the variational equations associated to periodic orbits. Apparently the mathematical community forgot the results of Poincaré until that some Russian mathematicians published on them, see [1, 5]. In what follows we shall summarize the results of Poincaré. 2020 Mathematics Subject Classification. Primary 34C05, 34A34, 34C14. Key words and phrases. C 1 integrability, differential systems, periodic orbits, Jayne-Cummings differential system. 1 2 J. LLIBRE Let U be an open subset of Rn and let f : U → Rn be C 2 function. We shall work with an autonomous differential system of the form dx (1) = ẋ = f (x). dt As usual we denote by ϕ(t, x0 ) the solution of system (1) such that ϕ(0, x0 ) = x0 ∈ U , and where t varies in the maximal interval of definition of this solution. When ϕ(T, x0 ) = x0 and ϕ(t, x0 ) ̸= x0 for t ∈ (0, T ) we say that ϕ(t, x0 ) is a periodic solution of period T > 0, or simply a T -periodic solution. The set γ = {ϕ(t, x0 ) : t ∈ [0, T ]} ⊂ U is the T -periodic orbit associated to the T -periodic solution ϕ(t, x0 ). As usual the differential equation   ∂f (x) (2) Ṁ = M, ∂x x=ϕ(t,x0 ) where M is an n × n matrix, is the variational equation associated to the T -periodic solution ϕ(t, x0 ). Here ∂f (x)/∂x denotes the Jacobian matrix of f . Let M (t, x0 ) be the solution of the differential equation (2) such that M (0, x0 ) is the identity matrix. Then the matrix M (T, x0 ) is the monodromy matrix associated to the T - periodic solution ϕ(t, x0 ). And the multipliers of the periodic solution ϕ(t, x0 ) are the eigenvalues of the monodromy matrix M (T, x0 ). A non-constant function F : U → R of class C 1 satisfying that ∇F (x) · f (x) = 0, where ∇F (x) is the gradient vector of the function F and the · indicates the usual inner product of Rn , is a first integral of the differential system (1), because F is constant on the solutions of system (1). Let F : U → R and G : U → R be two first integrals. They are independent if their gradients are independent in all the points of the open set U except perhaps in a set of zero Lebesgue measure. The following result goes back to Poincaré, see [13]. For a detailed proof see Corollary 3 of [9]. Theorem 1. Consider the C 2 differential system (1). If there is a periodic orbit γ having only s + 1 multipliers equal to 1, then system (1) has at most F1 , . . . , Fs C 1 linearly independent first integrals defined in a neighborhood of γ if the s + 1 vectors ∇F1 (x), . . . , ∇Fs (x) and f (x) are linearly independent on the points x ∈ γ. Theorem 1 provides a tool for studying the C 1 non-integrability of an au- tonomous differential system (1) in a neighborhood of the periodic orbit γ, and consequently in the domain of definition U of the differential system (1). Probably the main reason that Theorem 1 has almost not used for studying the C 1 non-integrability of an autonomous differential system (1) is that to find a peri- odic orbit γ of that differential system for which we can compute its multipliers and verify that vectors ∇F1 (x), . . . , ∇Fs (x) of the known first integrals of the differen- tial system and f (x) are linearly independent on the points x ∈ γ is not in general an easy task. Now the recent developments of the averaging theory for computing periodic solutions facilitates this task. ON THE C 1 NON-INTEGRABILITY OF THE DIFFERENTIAL SYSTEMS 3 Maciejewski and Szumiński in [10] proved that the semiclassical Jaynes-Cummings system (see [2, 4, 6, 7, 11]) defined by the autonomous differential system (3) ẋ = −y, ẏ = x + zu, ż = −yu, u̇ = v, v̇ = αx − µ2 u, only has the two meromorphic first integrals, namely µ2 2 1 2 (4) H = α(z − x)u + u + v , and F = x2 + y 2 + z 2 , 2 2 or meromorphic functions in the variables H and F . Our main result is the following one. Theorem 2. The only C 1 first integrals of the semiclassical Jaynes-Cummings system (3) are the first integrals H, F and the C 1 functions in the variables H and F. Theorem 2 is proved in section 3. The tools that we shall use for proving Theorem 2 can be used for studying the C 1 non-integrability in arbitrary autonomous differential systems. The key steps in the proof of Theorem 2 are: 1. To write the differential system (3) in the normal form for applying the averaging theory for computing periodic orbits, see section 2 for a summary of this averaging theory. 2. Compute a good analytical approximation of a periodic solution γ of the differential system (3). 3. Compute the multipliers of the periodic solution γ and verify the assump- tions of Theorem 1. 2. The averaging theory for computing periodic orbits Since the semiclassical Jaynes-Cummings differential system (3) is of class C 2 we can apply the classical averaging theory for computing periodic solutions, see [14]. If the differential system is only continuous, or a piecewise discontinuous differential system we can use the results on the averaging theory of [3] or [8] for computing periodic solutions, respectively. In Theorem 11.5 of Verhulst [14] is it proved the following result which gives a first order approximation for the periodic solutions of a periodic differential system. Consider first the differential system (5) ẋ = εF (t, x) + ε2 R(t, x, ε), x(0) = x0 , with x ∈ U where U is an open subset of R , and t ≥ 0. We suppose that the n function F (t, x) is T periodic in t. Consider now in U the averaged differential system (6) ẏ = εf (y), y(0) = x0 , where Z T 1 (7) f (y) = F (t, y)dt. T 0 4 J. LLIBRE In the next theorem we shall see, under convenient assumptions, that the equi- librium solutions of the averaged system (6) provide T –periodic solutions of the differential system (5). Theorem 3. Consider the differential systems (5) and (6), and assume the fol- lowing three conditions: (i) The function F , its Jacobian matrix ∂F/∂x, its Hessian matrix ∂ 2 F/∂x2 are continuous and bounded by an independent constant ε in [0, ∞) × U and ε ∈ (0, ε0 ]. (ii) The funciton F (t, x) is T –periodic in t, and its period T does not depend on ε. (iii) The curve y(t) is in U for all t ∈ [0, 1/ε] Then for every equilibrium point p of the averaged equation (6) satisfying   ∂f (8) det ̸= 0, ∂y y=p there is a T –periodic solution x(t, ε) of the differential system (5) such that x(0, ε) → p as ε → 0. 3. Proof of Theorem 2 It is easy to verify that the differential system (3) has two straight lines filled up with equilibrium points, namely (0, 0, z, 0, 0) for all z ∈ R and (µ2 u/α, 0, −µ2 /α, u, 0) for all u ∈ R. 3.1. Writting the differential system (3) into the normal form (5) of the averaging theory. First we translate the equilibrium point (0, 0, −µ2 /α, 0, 0) to the origin of coordinates doing the change of variables z → σ doing z = σ − µ2 /α, in the new variables (x, y, σ, u, v) system (3) writes   µ2 (9) ẋ = −y, ẏ = x + u σ − , σ̇ = −uy, u̇ = v, v̇ = αx − µ2 u, α and the two first integrals H and F given in (4) now become    2 µ2 µ2 2 1 2 µ2 H = α σ − xu − + u + v , and F = x2 + y 2 + σ − . α 2 2 α The linear part of the differential system (9) at the origin of coordinates is   0 −1 0 0 0  1 0 0 −µ2 /α 0    (10)  0 0 0 0 0   .  0 0 0 0 1  α 0 0 −µ2 0 So the eigenvalues of thepequilibrium point localized at the origin of coordinates of system (9) are 0, 0, 0, ± µ2 + 1 i. ON THE C 1 NON-INTEGRABILITY OF THE DIFFERENTIAL SYSTEMS 5 Doing the change of variables (x, y, σ, u, v =) → (X, Y, Z, U, V ) given by    α   X 0 0 0 1 x    p µ2 p    Y   −α µ2 + 1/µ2 µ2 + 1 0  y  = 0 0   , (11)  Z   0  σ    0 0 1 0    U  αε 0 0 ε 0  u V 0 −α 0 0 1 v we write the matrix (10) in its real Jordan normal form  p  p 0 − µ2 + 1 0 0 0  µ2 + 1 0 0 0 0     0 0 0   0 0 ,  0 0 0 0 ε  0 0 0 0 0 where ε > 0 will be the small parameter which appears in the normal form of the differential system (5) for applying the averaging theory. And the differential system (9) in the new variables (X, Y, Z, U, V ) becomes p  2  α µ2 + 1U Z − εµ2 Y µ2 + 1 − αZ Ẋ = 3/2 , p εµ2 (µ2 + 1) Ẏ = µ2 + 1X,   p µ2 (V − X) µ2 + 1U + εµ2 Y (12) Ż = , 5/2 εα (µ2 + 1) U̇ = εV,   p αZ µ2 + 1U + εµ2 Y V̇ =− 3/2 . ε (µ2 + 1) Now the first integrals H and F for the system (12) are µ2 µ2 1  H̃ = ε2 H = − U 2 + ε U Y + ε 2 (µ2 + 1)V 2 2(µ2 + 1)2 (µ2 + 1)5/2 2(µ2 + 1)3 +2µ2 (µ2 + 1)V X + µ2 − 2µ6 + µ4 (X 2 +Y 2 − 6)  +µ2 (X 2 + 2Y 2 − 6) − 2 + 2α(µ2 + 1)3 Z , µ4 2µ4 F̃ = ε2 F = 2 2 2 U2 − ε 5/2 UY α (µ + 1) α2 (µ2 + 1) 4 2 4 2   ! 2 2 µ (V − X) µ Y µ +ε2 2 + 2 3 + Z − α . α2 (µ2 + 1) α (µ2 + 1) Since the normal form of the differential system (5) for applying the averaging theory needs the small parameter ε in front of all the equations of the differential system, first we do the change of variables (X, Y ) = (r cos θ, r sin θ) and after the rescaling (r, Z, U, V ) = (ε2 R, ε2 S, ε2 Q, ε2 W ), so in the new variables (R, θ, S, Q, W ) 6 J. LLIBRE the differential system (12) becomes αQS cos θ αRS sin θ cos θ Ṙ = ε 4 2 + ε2 3/2 , µ +µ (µ2 + 1) p αQS sin θ 2 2 αS sin θ θ̇ = µ2 + 1 − ε 4 − ε , (µ + µ2 ) R (µ2 + 1) 3/2 (13) µ2 Q(W − R cos θ) µ4 R sin θ(R cos θ − W ) Ṡ = ε 2 − ε2 5/2 , α (µ2 + 1) α (µ2 + 1) Q̇ = εW, αQS αµ2 RS sin θ Ẇ = −ε 2 − 3/2 , µ +1 (µ2 + 1) and the two first integrals for the system (13) become ! 2 2 4 µ2 Q2 µ2 QR sin θ 1 H = −ε µ + ε αS − 2 + ε5 5/2 + ε6 3 2 2 (µ + 1) 2 2 (µ + 1)  (µ2 + 1)  1 4 2 6 2 4 2 3µ R 2  2 2 2 2 − µ R cos(2θ) + µ R + + 2 µ + 1 µ RW cos θ + µ W + W , 2 2! µ4 µ2 µ2 Q2 2µ4 QR sin θ 1 F = ε2 2 + ε4 2 2 − 2αS − ε5 5/2 + ε6 3 α α 2 (µ + 1) 2 2 α (µ + 1) 2 2 α (µ + 1) µ4 R2 sin2 θ + (µ2 + 1)µ4 R2 cos2 θ −2(µ2 + 1)µ4 RW cos θ  2  + µ2 + 1 α2 µ2 + 1 S 2 + µ4 W 2 . We shall restrict the differential system (13) to the invariant levels µ4 4 2hµ 2 f H = −ε2 µ2 + ε4 h, and F = ε2 2 − ε 2 − ε5 2 2 , α α α (µ + 1)5/2 where h and f are constants. Then from this first equation we obtain 2 2h µ2 + 1 + µ2 Q2 (14) S= 2 + O(ε), 2α (µ2 + 1) and from the second one we get f (15) R= + O(ε). 2µ4 Q sin θ Additionally taking the variable θ as the new independent variable the differential system (13) becomes a 2π-periodic differential system in the normal form (5) for applying the averaging theory, i.e. dQ W = εp + O(ε2 ) dθ µ2+ 1  (16) 2 2 2 2 dW Q 2h µ + 1 + µ Q = −ε 7/2 + O(ε2 ). dθ 2 (µ2 + 1) 3.2. Computing an explicit periodic solution of the differential system (16). Now the averaged function (7) for the differential system (16) is f (Q, W ) = ON THE C 1 NON-INTEGRABILITY OF THE DIFFERENTIAL SYSTEMS 7 (f1 (Q, W ), f2 (Q, W )) where W f1 (Q, W ) = p , µ2+ 1  2 Q 2h µ2 + 1 + µ2 Q2 f2 (Q, W ) = − 7/2 , 2 (µ2 + 1) Hence the averaged differential system (6) in our case have the equilibrium points   µ2 + 1 √ p = (0, 0), p± = ± −2h, 0 . µ Of course the equilibria p± only exist if h < 0. Note that the Jacobian (8) at the points p and p± takes the nonzero values h/(µ2 + 1)2 and −2h/(µ2 + 1)2 if h ̸= 0. Then from Theorem 3 we know that the differential system (16) has a 2π-periodic solution of the form  2  µ + 1√ (Q(θ), W (θ)) = −2h, 0 + O(ε), µ associated to the equilibrium point p+ . We have such a periodic solution for every value of the constants h < 0 and f . Then we fix one of these periodic solutions for a value of f ̸= 0, and denote this periodic orbit by γ. From (14) and (15) the periodic orbit γ of the differential system (16) in the differential system (13) becomes (R(t), θ(t), S(t), Q(t), W (t)) equal to  p  f 2 + 1, 0, µ2 + 1 √ √ , t µ −2h, 0 + O(ε), 2 −2hµ3 (µ2 + 1) sin θ µ p and has period 2π/ µ2 + 1. This periodic orbit in the coordinates of the differential system (12) is (X(t), Y (t), Z(t), U (t), V (t)) equal to  p   f cot t µ2 + 1 f µ2 + 1 √ ε2 √ , ε2 √ , 0, ε2 −2h, 0 + O(ε3 ). 2 −2hµ3 (µ2 + 1) 2 −2hµ3 (µ2 + 1) µ Using the change of variables (11) this periodic orbit for the differetnial system (9) becomes (x(t), y(t), σ(t), u(t), v(t)) equal to  p  √ f cot t µ2+1 ε −2hµ − ε2 √ f α 5/2 , ε2 √ 2 , 0, 2α −2hµ (µ2 + 1) 2α −2hµ (µ2 + 1) p  √ f cot t µ2 + 1 −2h f  + O(ε3 ). ε + ε2 √ 5/2 , ε2 √ 2 µ 2 −2hµ (µ2 + 1) 2 −2hµ (µ2 + 1) p Finally the periodic orbit γ of period 2π/( µ2 + 1) in the coordinates of the dif- ferential system (3) is (x(t), y(t), z(t), u(t), v(t)) equal to  p  f cot t µ2+1 √ ε −2hµ − ε2 √ f 2 µ2 α 5/2 , ε √ 2 , − , 2α −2hµ (µ2 + 1) 2α −2hµ (µ2 + 1) α (17) p  √ f cot t µ2 + 1 −2h f  + O(ε3 ). ε + ε2 √ 5/2 , ε2 √ 2 µ 2 −2hµ (µ2 + 1) 2 −2hµ (µ2 + 1) 8 J. LLIBRE 3.3. Computing the multipliers of the periodic solution γ. The variational equation (2) on the periodic orbit (17) of the differential system (3) restricted to the dominant terms of order ε is (18) Ẏ = M Y, where   y1′ (t) y2′ (t) y3′ (t) y4′ (t) y5′ (t)  y6′ (t) y7′ (t) y8′ (t) y9′ (t) y10 ′ (t)    Ẏ (t) =   ′ ′ ′ ′ ′ y11 (t) y12 (t) y13 (t) y14 (t) y15 (t) ,   ′ y16 ′ (t) y17 ′ (t) y18 ′ (t) y19 ′ (t) y20 (t)  ′ ′ ′ ′ ′ y21 (t) y22 (t) y23 (t) y24 (t) y25 (t)   0 −1 √0 0 0  −2h µ2   1 0 ε − 0   µ α   √  M =  0 −2h ,  −ε 0 0 0    µ   0 0 0 0 1  α 0 0 −µ2 0 and   y1 (t) y2 (t) y3 (t) y4 (t) y5 (t)  y6 (t) y7 (t) y8 (t) y9 (t) y10 (t)    Y (t) =   y11 (t) y12 (t) y13 (t) y14 (t) y15 (t) .   y16 (t) y17 (t) y18 (t) y19 (t) y20 (t)  y21 (t) y22 (t) y23 (t) y24 (t) y25 (t) The solution Y (t) at order O(ε) of the variational equation (18) such that Y (0) is the 5 × 5 identity matrix is  √ p   2 2 √ 2 2 2µ2 t µ2 + 1 + S  µ +C tµ (C − 1) −2h µ − Cµ   − 2   µ2 + 1 2µ2 + 2  p + 1)µ (µ2  αµ2 + α α (µ2 + 1) 3/2   √   −h t µ2 + 1µ2 + 2S    p S µ2 + C Sµ2 (C − 1)µ 2    √ − p   µ2 + 1 µ2 + 1 2µ (µ2 + 1) 3/2 α µ2 + 1 2 α (µ + 1)   √  p    −2hµ t µ2 + 1 − S   ,  0 0 1 0   2 3/2    α (µ +p 1)    √ 2 S Sµ + t µ + 1 2   α − Cα tα (C − 1) −hα Cµ2 + 1   − 2 −   µ2 + 1 2µ + 2 2 + 1) 2 µ2 + 1 (1 + µ ) 2 3/2   µ (µ    √ p   −hα t µ2 + 1 − 2S   Sα (C − 1)α Sµ2 2 Cµ + 1  p − √ −p µ2 + 1 µ2 + 1 2µ (µ2 + 1) 3/2 µ2 + 1 µ2 + 1 ON THE C 1 NON-INTEGRABILITY OF THE DIFFERENTIAL SYSTEMS 9 p p C = cos( µ2 + 1 t) and S = sin( µ2 + 1 t). Then the monodromy matrix where p Y (2π/ µ2 + 1) is  √  πµ2 2 2πµ2  1 − 3/2 0 0 3/2   (µ2 + 1) √ √ α (µ2 + 1)   i 2 hπµ     0 1 0 0   2 (µ + 1) 3/2   √ √   2i 2 hπµ   0 0 1 0 ,  3/2   2 α (µ + 1)   πα 2πS   0 − 0 1   3/2 3/2   2 (µ + 1) 2 (µ + 1)   √ √   i 2 hπα  0 0 − 3/2 0 1 2 µ (µ + 1) and their eigenvalues √ √ 2π h 2π h 1, 1, 1, 1− 3/2 , 1+ 3/2 , (µ2 + 1) (µ2 + 1) are the multipliers of the periodic orbit γ. That is, three multipliers equal to 1 and two complex multipliers. In order to apply Theorem 1 and to obtain that the unique C 1 first integrals of the differential system (3) are H, F and any C 1 function in the variables H and F we need to verify that the three vectors ∇H(x, y, z, u, v), ∇F (x, y, z, u, v) and (ẋ, ẏ, ż, u̇, v̇) of system (3) are linearly independent on the points (x, y, z, u, v) of the periodic orbit (17). Indeed, an easy computation shows that the matrix 3 × 5 having the three files formed by the components of the vectors ∇H(x, y, z, u, v), ∇F (x, y, z, u, v) and (ẋ, ẏ, ż, u̇, v̇) has rank three. This completes the proof of The- orem 2. Acknowledgements The author is partially supported by the Agencia Estatal de Investigación grant PID2019-104658GB-I00 and the H2020 European Research Council grant MSCA- RISE-2017-777911. References [1] V.I. Arnold, Forgotten and neglected theories of Poincaré, Russian Math. Surveys 61 No. 1 (2006), 1–18. [2] P.I. Belobrov, G.M. Zaslavsky and G.Kh. Tartakovsky, Stochastic breaking of bound states is a system of atoms interacting with a radiation field, Sov. Phys. JETP 44 (1977) 945. Translation of Zh. Eksp. Teor. Fiz. 71 (1976), 1799–1812. [3] A. Buica and J. Llibre, Averaging methods for finding periodic orbits via Brouwer degree, Bull. Sci. Math. 128 (2004), 7–22. [4] M. Jeleńska-Kuklińska, M. Kuś, Exact solution in the semiclassical Jaynes-Cummings model without the rotating-wave approximation, Phys. Rev. A 41 (1990), 2889–2891. [5] V.V. Kozlov, Integrability and non-integrability in Hamiltonian mechanics, Russian Math. Surveys 38 No. 1 (1983), 1–76. [6] A. Kujawski, Exact periodic solution in the semiclassical Jaynes-Cummings model without the rotating-wave approximation, Phys. Rev. A 37 (1988), 1386–1387. 10 J. LLIBRE [7] A. Kujawski and M. Muntz, Exact periodic solution and chaos in the semiclassical Jaynes- Cummings model, Z. Phys. B, Condensed Matter 37 (1989), 273–276. [8] J. Llibre, A.C. Mereu, D.D. Novaes, Averaging theory for discontinuous piecewise differ- ential systems, J. Differential Equations 258 (2015), 4007–4032. [9] J. Llibre and C. Valls, On the C 1 non-integrability of differential systems via periodic orbits, Euro. J. of Applied Mathematics 22 (2011), 381–391. [10] A. Maciejewskia and W. Szumiński, Non-integrability of the semiclassical Jaynes- Cummings models without the rotating-wave approximation, Appl, Math. Letters 82 (2018), 132–139. [11] P.W. Milonni, J.R. Ackerhalt and H.W. Galbraith, Chaos in the semiclassical N-atom Jaynes-Cummings model: Failure of the rotating-wave approximation, Phys. Rev. Lett. 50 (1983), 966–969. [12] J.J. Morales-Ruiz, Differential Galois Theory and non-integrability of Hamiltonian sys- tems, Progress in Math. Vol. 178, Birkhauser, Verlag, Basel, 1999. [13] H. Poincaré, Sur l’intégration des équations différentielles du premier ordre et du premier degré I and II, Rendiconti del circolo matematico di Palermo 5 (1891), 161–191; 11 (1897), 193–239. [14] F. Verhulst, Nonlinear differential equations and dynamical systems, Universitext, Springer-Verlag, Berlin, 1996. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat