Perturbed Euler top and bifurcation of limit cycles on invariant Casimir surfaces
aa r X i v : . [ m a t h - ph ] O c t Perturbed Euler top and bifurcation of limit cycleson invariant Casimir surfaces
Isaac A. Garc´ıa , ∗ and Benito Hern´andez-Bermejo Departament de Matem`atica. Universitat de Lleida.Avda. Jaume II, 69. 25001 Lleida, Spain.E–mail: [email protected] (2)
Departamento de F´ısica. Universidad Rey Juan Carlos.Calle Tulip´an S/N. 28933–M´ostoles–Madrid, Spain.E-mail: [email protected]
Abstract
Analytical perturbations of the Euler top are considered. The perturba-tions are based on the Poisson structure for such a dynamical system, in sucha way that the Casimir invariants of the system remain invariant for the per-turbed flow. By means of the Poincar´e-Pontryagin theory, the existence oflimit cycles on the invariant Casimir surfaces for the perturbed system is in-vestigated up to first order of perturbation, providing sharp bounds for theirnumber. Examples are given.
Keywords:
Poisson systems; Casimir invariants; Hamiltonian systems; perturba-tion theory; limit cycles; Poincar´e-Pontryagin theory.
PACS codes: ∗ Corresponding author. Telephone: (+34) 973702728. Fax: (+34) 973702702.1
The Euler top
The basis of this work is the following system of ODEs known as Euler equations,which describe the rotation of a rigid body, or Euler top: ˙ x = µ − µ µ µ x x , ˙ x = µ − µ µ µ x x , ˙ x = µ − µ µ µ x x . (1)Here x = ( x , x , x ) ∈ R where x i denotes the i th component of angular momen-tum, and constants µ i are the moments of inertia about the coordinate axes, bothfor i = 1 , ,
3. Energy is conserved for this system, but of course the flow is odddimensional and a classical Hamiltonian formulation is excluded. However this is aPoisson system (see [17, 20, 25] for general references on Poisson systems) in termsof the following structure matrix: J ( x ) = − x x x − x − x x . Notice that the rank of the structure matrix is 2 everywhere in R except at theorigin, in which the rank vanishes. The Hamiltonian is the total energy (kineticenergy, in this case): H ( x ) = 12 (cid:18) x µ + x µ + x µ (cid:19) . This SO (3)-based Lie-Poisson structure of the Euler top equations was first rec-ognized by Arnol’d [1] (see [24] for a modern classification of Lie-Poisson struc-tures). In addition, other Poisson formulations are also possible for system (1),for instance see [11, 13, 14], although we shall not be concerned with them in thiswork. Since its characterization, the SO (3) Lie-Poisson structure for the Eulertop has been repeatedly investigated from a variety of perspectives and has pro-vided the basis for a number of mathematical and physical developments, e.g. see[2, 8, 10, 11, 12, 15, 21, 22, 23].Since the rank of J is 2 (excluding the origin from the analysis) there must beone independent Casimir invariant, which can be chosen to be: D ( x ) = x + x + x = k x k . (2)Namely, the distinguished invariant (2) is the square of the Euclidean norm ofthe angular momentum, which is a conserved quantity during the system rotation.2herefore, the symplectic foliation is given by: x + x + x = constant (3)which are concentric spheres in R . We thus see that the symplectic leaves (3) areeven dimensional (two dimensional, in this case) and therefore Darboux’s theoremensures that on the symplectic leaves the dynamics is Hamiltonian in the classicalsense, at least locally (in the neighborhood of each point). The actual trajectories ofthe system in phase space are obtained by the intersection of the symplectic leaveswith the energy level sets H ( x ) = 12 (cid:18) x µ + x µ + x µ (cid:19) = constantwhich in geometric terms are ellipsoids in phase space. Oscillations play a prominent role in many physical systems where an importantproblem is to determine if spontaneous oscillatory activity persists when subjectedto a small external stimulus. In this sense, it is worth recalling here the relationshipbetween the number of zeros of the so-called Poincar´e–Pontryagin function givenby a generalization of an Abelian integral and the number of limit cycles (isolatedperiodic solutions) of the corresponding planar analytic differential systems.We consider an analytic Hamiltonian function H ( x, y ) defined on some opensubset U ⊆ R . We shall assume that, the corresponding Hamiltonian vector field X H has a family of periodic orbits filling up an annulus and given by the ovals γ h ⊂ H − ( h ), continuously depending on a parameter h ∈ ( a, b ) ⊂ R . Now, weperturb the system as follows˙ x = ∂H∂y + ǫP ( x, y ) , ˙ y = − ∂H∂x + ǫQ ( x, y ) , (4)where P and Q are analytic functions on U and ǫ is a real small parameter. Weemphasize here to the reader that the considered perturbations (4) are not necessaryHamiltonian ones.Then, we may define the Poincar´e–Pontryagin function I ( h ) as the following lineintegral I ( h ) = I γ h P ( x, y ) dy − Q ( x, y ) dx . (5)Notice that, in case that H , P and Q be polynomials, then I ( h ) is just an Abelianintegral. For small values of ǫ , the following question arises: How many orbits γ h ǫ of the perturbed system (4) bifurcates from γ h if Γ ǫ tends to γ h in the sense ofHausdorff distance as ǫ → I ( h )
0, the answer of the aforementioned question is givenby the next theorem, see for instance [4] for its proof and the state–of–the–art ofthis and other related topics.
Theorem 1.
Let I ( h ) be given by (5). Assuming I ( h ) for h ∈ ( a, b ) , thefollowing statements hold: (i) If system (4) has a limit cycle bifurcating from γ h ∗ , then I ( h ∗ ) = 0 . (ii) If I ( h ) has a simple zero at h ∗ ∈ ( a, b ) (that is, I ( h ∗ ) = 0 and I ′ ( h ∗ ) = 0 ), thensystem (4) has a unique limit cycle bifurcating from γ h ∗ and, moreover, thislimit cycle is hyperbolic. In this work, analytical perturbations of the Euler top (1) shall be investi-gated, in such a way that either some invariant surface S c = { ( x , x , x ) ∈ R : D ( x , x , x ) = c } for an arbitrary fixed c > S c for any c > S c for the perturbedsystem by means of Theorem 1. To conclude, some examples are given.Let us recall here that, when the phase space has dimension greater than 2, aperiodic orbit γ is called limit cycle if it is α or ω –limit set of another orbit. Thus,in this case γ needs not to be isolated inside the set of periodic orbits. Severalauthors have studied (by using different techniques) bifurcations of limit cycles inperturbations of a vector field with an invariant two–dimensional manifold, assumingthat the restriction of the field on this manifold is Hamiltonian (see for instance [5,16]). To the authors’ knowledge, this kind of analysis is carried out for the Casimirinvariants of a Poisson system for the first time in the present work. Moreover, thepreservation of the Casimir invariants is physically relevant, as far as it amounts toa conservation of the angular momentum for the perturbed system. By suppressingthe explicit restriction about the existence of invariant two–dimensional manifolds,in [3] and using Poincar´e maps, a study of the existence of T –periodic solutions for T –periodic perturbations of the symmetric ( µ = µ ) Euler top is performed. Let us rewrite the Euler top (1) as˙ x = αx x , ˙ x = βx x , ˙ x = γx x , (6)4ith parameters α := µ − µ µ µ , β := µ − µ µ µ , γ := µ − µ µ µ . Notice that α + β + γ = 0. The Euler top is an integrable system having the firstintegrals H ( x , x , x ) = 12 (cid:18) x µ + x µ + x µ (cid:19) , D ( x , x , x ) = x + x + x . Without loss of generality, we can assume the condition αβ <
0. Therefore, givenany real value c = 0, the invariant spheres S c := { ( x , x , x ) ∈ R : D ( x , x , x ) = c } , (7)are foliated by periodic orbits of (6). We will also define the semispheres S + c := { ( x , x , x ) ∈ S c : x > } , S − c := { ( x , x , x ) ∈ S c : x < } . (8)Now, we consider the following analytic perturbation in R \{ x = 0 } of the Eulertop (6) leaving invariant the semispheres S + c and S − c .˙ x = αx x + ǫA ( x , x , x ) , ˙ x = βx x + ǫB ( x , x , x ) , (9)˙ x = γx x + ǫC ( x , x , x ) , where A ( x , x , x ) = x P ( x , x , D ( x , x , x )) ,B ( x , x , x ) = x Q ( x , x , D ( x , x , x )) , (10) C ( x , x , x ) = D ( x , x , x ) − c x R ( x , x , D ( x , x , x )) − x P ( x , x , D ( x , x , x )) − x Q ( x , x , D ( x , x , x )) , being P , Q and R analytic functions in all R . The following theorem is one of themain results of this work. Without loss of generality, in statements (ii) and (iii) ofthe theorem we shall focus on S + c . Theorem 2.
Let us consider the Euler top (6) with αβ < , therefore having theinvariant sphere S c foliated with periodic orbits. Then, the following holds: (i) The most general analytic 1–parameter perturbation of (6) in R \{ x = 0 } leaving invariant the semispheres S + c and S − c is given by (9–10). ii) Assume that I ( h ) = I H = h P ( x, y, c ) dy − Q ( x, y, c ) dx , (11) where H ( x, y ) = ( αy − βx ) . Then, the periodic orbits γ h ∗ ⊂ S + c of (6) fromwhich bifurcates a limit cycle of the perturbed system (9) are given by γ h ∗ = { ( x , x , x ) ∈ S + c : H ( x , x , x ) = c / (2 µ ) − h ∗ } , where I ( h ∗ ) = 0 , α h ∗ > and h ∗ < c (cid:18) µ − max {− α , β } (cid:19) , or h ∗ > c (cid:18) µ − min {− α , β } (cid:19) if α > or α < , respectively. (iii) If P and Q are polynomials of maximum degree n in the first two variables x and y , then a sharp upper bound of the maximum number of limit cycles ofsystem (9) on S + c is (cid:26) ( n − / , if n is odd ( n − / , if n is even . (12) Proof.
We perform the change of variables given by the diffeomorphism( x , x , x ) ( x, y, z ) , z = D ( x , x , x ) , (13)defined in { ( x , x , x ) ∈ R : x > } . Observe that under such transformation, thesemisphere S + c is transformed into the open disk Ω = { ( x, y, c ) ∈ R : x + y < c } .The perturbed system (9) restricted to the semispace x > x = p z − ( x + y ) (cid:18) ∂H∂y + ǫP ( x, y, z ) (cid:19) , ˙ y = p z − ( x + y ) (cid:18) − ∂H∂x + ǫQ ( x, y, z ) (cid:19) , (14)˙ z = ǫ ( z − c ) R ( x, y, z ) , with H ( x, y ) = ( αy − βx ). The perturbed system (9) restricted to the semispace x < x and ˙ y . Theexpression of (14) contains the most general perturbation of the Euler top (6) writ-ten in ( x, y, z )–coordinates which leaves invariant the disk Ω. Therefore, undoingthe change of coordinates done, statement (i) is proved.6e emphasize that, condition αβ < ǫ = 0possesses on the invariant disk Ω a center at ( x, y ) = (0 , γ h = H − ( h ) with certain values of h to be specified later.Clearly, h > α > h < α <
0, if γ h is to be an ellipse. Moreover,since H ( x, y, p c − ( x + y )) = c / (2 µ ) − H ( x, y ), it follows that the level curve γ h corresponds to the level curve H ( x, y, p c − ( x + y )) = ¯ h with ¯ h = c / (2 µ ) − h .Finally, since γ h ⊂ Ω, we must impose that both semiaxes p h/β and p − h/α of γ h be smaller than the disk radius c of Ω. This last condition leads to twopossibilities: • If α > β <
0, then it must be:¯ h > c {− α , β }• In the complementary case α < β >
0, we have:¯ h < c {− α , β } . Hence, the restriction of system (14) to the invariant disk Ω is given by theanalytic system ˙ x = p c − ( x + y ) (cid:18) ∂H∂y + ǫP ( x, y, c ) (cid:19) , ˙ y = p c − ( x + y ) (cid:18) − ∂H∂x + ǫQ ( x, y, c ) (cid:19) . Now, using Theorem 1, it is found that the periodic orbits of the center that persistunder the perturbation for small ǫ are given by the zeros of (11). Then, statement(ii) is proved.In order to compute (11), we will assume α > β < t → − t ). The ellipses H = h with h > x = p − h/β cos θ , y = p h/α sin θ with θ ∈ [0 , π ). Then, I ( h ) = √ h Z π ( ¯ P ( θ, h, c ) cos θ + ¯ Q ( θ, h, c ) sin θ ) dθ , (15)where ¯ P ( θ, h, c ) := p /α P ( p − h/β cos θ, p h/α sin θ, c ) , ¯ Q ( θ, h, c ) := p − /β Q ( p − h/β cos θ, p h/α sin θ, c ) . P and Q are polynomials in x and y of degree at most n , that is, P ( x, y, z ) = P ni + j =0 a ij ( z ) x i y j and Q ( x, y, z ) = P ni + j =0 b ij ( z ) x i y j with analytic coefficients a ij ( z ) , b ij ( z ) ∈ C ω ( R ). After some alge-bra, we get that I ( h ) = √ hM n ( h ) with M n ( h ) a polynomial in the variable √ h ofmaximum degree n . More precisely, M n ( h ) = n X i + j =0 h ˜ a ij I ij + ˜ b ij J ij i ( √ h ) i + j where ˜ a ij and ˜ b ij are real constants, and: I ij = Z π cos i +1 θ sin j θ dθ , J ij = Z π cos i θ sin j +1 θ dθ . Taking into account that Z π sin i θ cos j θdθ = ( i +12 )Γ( j +12 )Γ( i + j +22 ) if i and j even , i, j ∈ N ∪ { } i or j odd , (16)where Γ is Euler’s gamma function, it can be seen that the following recurrenceholds: M n ( h ) = (cid:26) M n − ( h ) , if n is even M n − ( h ) + χ n ( √ h ) n , if n is oddwhere M ( h ) = M ( h ) = χ √ h , and χ n (with n ≥
1) is a real constant depending on a ij ( c ), b ij ( c ), I ij and J ij , in all cases for those i, j such that i + j = n . Consequently, M n ( h ) is an odd polynomial of √ h , and therefore h = 0 is always a root. Moreover,the remaining roots are distributed symmetrically around the origin. In particular,the maximum number of positive roots of M n ( h ) is (12). Finally, we shall now provethat there exist suitable P ( x, y, z ) and Q ( x, y, z ) such that this maximum numberis achieved for every n because of the arbitrariness in the constants χ n , which inturn arises from the arbitrariness in the coefficients a ij ( c ) and b ij ( c ). Remark 3.
Note that statement (iii) of Theorem 2 agrees with a classical resultin the theory of limit cycles (see [18]) that an n –degree polynomial perturbation ofthe harmonic oscillator ˙ x = − y + ǫ ( ax + P ( x, y )), ˙ y = x + ǫ ( ay + Q ( x, y )) has atmost ( n − / n is odd and ( n − / n is even. Theorem 4.
Let us consider the Euler top (6) with αβ < , therefore having allthe invariant spheres S c ( c > ) foliated with periodic orbits. Then, the followingholds: i) The most general analytic 1–parameter perturbation of (6) in R leaving invari-ant all the spheres S c ( c > ) is given by (9), with A ( x , x , x ) , B ( x , x , x ) and C ( x , x , x ) any analytic functions in R satisfying x A ( x , x , x ) + x B ( x , x , x ) + x C ( x , x , x ) = 0 . (17) In addition, a family of solutions of (17) is given by A = x M − x N , B = x N − x L , C = x L − x M (18) for arbitrary analytic functions L ( x , x , x ) , M ( x , x , x ) and N ( x , x , x ) .Moreover, if the perturbation ( A, B, C ) is polynomial and homogeneous of de-gree m , then the family (18) provides the general solution of (17) for L , M and N some homogeneous polynomials of degree m − . (ii) Assume that I ( h ) = I H = h A (cid:16) x, y, p c − ( x + y ) (cid:17) dy − B (cid:16) x, y, p c − ( x + y ) (cid:17) dx p c − ( x + y ) , (19) where H ( x, y ) = ( αy − βx ) . Then, for all c > , the periodic orbits γ h ∗ ⊂ S + c of (6) from which bifurcates a limit cycle of the perturbed system (9)satisfying (17) are given by γ h ∗ = { ( x , x , x ) ∈ S + c : H ( x , x , x ) = c / (2 µ ) − h ∗ } , where I ( h ∗ ) = 0 , α h ∗ > and h ∗ < c (cid:18) µ − max {− α , β } (cid:19) , or h ∗ > c (cid:18) µ − min {− α , β } (cid:19) if α > or α < , respectively. (iii) If A ( x , x , x ) = x P ( x , x , x ) and B ( x , x , x ) = x Q ( x , x , x ) verifyingcondition (17) with polynomials P and Q of maximum degree n , then a sharpupper bound of the maximum number of limit cycles of system (9) on S + c forany c > is n − . Proof.
Recalling that α + β + γ = 0 and imposing that the Casimir function D ( x , x , x ) = x + x + x must be a first integral also for the perturbed system(9) for all ǫ , we get that the functions A ( x , x , x ), B ( x , x , x ) and C ( x , x , x )must satisfy (17). In the particular case that the components of the perturbationfield ( A, B, C ) are homogeneous polynomials of degree m , Darboux showed [9] thatcondition (17) is equivalent to the existence of homogeneous polynomials L , M and9 of degree m − A = zM − yN , B = xN − zL and C = yL − xM .Thus, statement (i) is proved.Regarding statement (ii), we perform again the change of variables (13), whichis a diffeomorphism in { ( x , x , x ) ∈ R : x > } . Recall that S + c is mapped intothe disk Ω = { ( x, y, c ) ∈ R : x + y < c } . Following analogous steps to those inthe proof of Theorem 2, the perturbed system (9), once reduced to the semispace x >
0, restricted to the invariant disk Ω for any c >
0, and submitted to the timerescaling dτ = p c − ( x + y ) dt , takes the form ˙ x = ∂H∂y + ǫ A (cid:16) x, y, p c − ( x + y ) (cid:17)p c − ( x + y ) , ˙ y = − ∂H∂x + ǫ B (cid:16) x, y, p c − ( x + y ) (cid:17)p c − ( x + y ) (20)with H ( x, y ) = ( αy − βx ). Note that the previous system is an analytic perturbationin the disk Ω of a Hamiltonian vector field.Taking again into account that condition αβ < ǫ = 0possesses on the invariant disk Ω a center at ( x, y ) = (0 ,
0) with period annulus foliatedby the ellipses γ h = H − ( h ), by imposing the condition γ h ⊂ Ω and taking into accountTheorem 1, the same reasoning employed in the proof of Theorem 2 leads to the proof ofstatement (ii).In order to prove statement (iii), we consider the case A ( x , x , x ) = x P ( x , x , x )and B ( x , x , x ) = x Q ( x , x , x ), as indicated. Assuming α > β < γ h with h > x = p − h/β cos θ and y = p h/α sin θ with θ ∈ [0 , π ), we get I ( h ) = √ h Z π ( ¯ P ( θ, h, c ) cos θ + ¯ Q ( θ, h, c ) sin θ ) dθ , (21)where¯ P ( θ, h, c ) := r α P s − hβ cos θ, r hα sin θ, c + 2 h (cid:20) cos θβ − sin θα (cid:21)! , ¯ Q ( θ, h, c ) := r − β Q s − hβ cos θ, r hα sin θ, c + 2 h (cid:20) cos θβ − sin θα (cid:21)! . Since P and Q are polynomials of degree n , that is, P ( x, y, z ) = P ni + j + k =0 a ijk x i y j z k and Q ( x, y, z ) = P ni + j + k =0 b ijk x i y j z k , taking (16) into account and using similar argumentsto those in the proof of statement (iii) of Theorem 2, it is straightforward to show that I ( h ) = hM n − ( h ) with M n − ( h ) a polynomial in the variable h of degree at most n − M n − ( h ) is n −
1, which can beachieved for suitable P ( x, y, z ) and Q ( x, y, z ). This completes the proof.A corollary of Theorem 4 for a class of homogeneous perturbations is the following: orollary 5. Let us consider the Euler top (6) with αβ < , therefore having all theinvariant spheres S c ( c > ) foliated with periodic orbits. Assume that the perturbationfield ( A, B, C ) in (9) is polynomial and homogeneous of degree m satisfying (17) and ofthe form A ( x , x , x ) = x P ( x , x , x ) and B ( x , x , x ) = x Q ( x , x , x ) . Then, theupper bound stated in Theorem 4 (iii) for the maximum number of limit cycles of system(9) on S + c for any c > is not achieved. More precisely, if I ( h ) is the function (19), thefollowing holds: (i) If m = 3 or m = 5 , then I ( h ) ≡ . (ii) If either m = 4 or m = 6 , then I ( h ) = hM k ( h ) where M k is a polynomial of degree k with k = 1 or k = 2 , respectively. In particular, the maximum number of limitcycles of system (9) on S + c for any c > is k . (iii) If m = 7 and I ( h ) , then I ( h ) has either one unique positive root h ∗ > or noneaccording to whether α = β or not, respectively. In addition, h ∗ = αβc / ( β − α ) does not depend on the perturbation field ( A, B, C ) . Example 6.
In case (iii) of Corollary 5 we have I ( h ) = λ h [ αβc + ( α − β ) h ] where theconstant λ = π ( λ β − λ α ) / ( − αβ ) / being λ and λ the coefficient of P in x x x andthe coefficient of Q in x x x , respectively. Thus, I ( h ) λ β = λ α . Inthis case, I ( h ∗ ) = 0 where h ∗ = αβc / ( β − α ). Therefore when α > β <
0) andaccording to statement (ii) of Theorem 4, in order to have a limit cycle of system (9) on S + c for any c > h ∗ < c (cid:18) µ − max {− α , β } (cid:19) (22)must be satisfied. It is easy to see that always exists a µ > µ such that condition (22) is fulfilled is as follows: if α + β > < µ < ( β − α ) / ( β ( α + β )) and when α + β < < µ < ( β − α ) / ( α ( α + β )). The symmetric case α + β = 0 gives no restriction except µ > Example 7.
In [7], the problem of stabilization of permanent rotations of the free rigidbody with two controls about the intermediate principal axis is considered. In short,system (9) with A ( x , x , x ) = − kx , B ( x , x , x ) = 0 and C ( x , x , x ) = kx is con-sidered, where k ∈ R is the feedback gain parameter. Here we shall consider a modifiedversion of this problem consisting of the perturbed field: A ( x , x , x ) = − x ( k − x + x x ) ,B ( x , x , x ) = x x (1 + x ) ,C ( x , x , x ) = kx + x ( x − − x (1 + x ) . Notice that this perturbation satisfies the conditions of statement (iii) of Theorem 4 with P ( x , x , x ) = − k + x (1 − x ) and Q ( x , x , x ) = x (1 + x ) and therefore at most1 limit cycle of system (9) can appear on each S + c for any c >
0. Direct computations how that the function I ( h ) of (19) is I ( h ) = λ h [ − αβ + ( α + β ) h ] where λ is a non–vanishing constant. Thus, I ( h ∗ ) = 0 where h ∗ = 2 αβ/ ( α + β ). Taking α > β <
0) and using statement (ii) of Theorem 4, the additional condition (22) must besatisfied to have a limit cycle of system (9) on S + c . It is easy to show that (22) is alwayssatisfied for any c > µ > α + β >
0. On the contrary, when α + β < c is greater ornot than c ∗ := 2 p β/ ( α + β ). More precisely, when α + β <
0, either 0 < c < c ∗ and0 < µ < ( α + β ) c / [4 αβ − α ( α + β ) c ] or c ≥ c ∗ and µ > Acknowledgments.
The first author (I.G.) is partially supported by a MCYT/FEDER grant number MTM2008-00694 and by a CIRIT grant number 2009 SGR 381. The second author (B.H.-B.) wouldlike to acknowledge the kind hospitality at Lleida University during which part of thiswork was developed. eferences [1] V. I. Arnol’d, The Hamiltonian nature of the Euler equations in the dynamics of arigid body and an ideal fluid, Usp. Mat. Nauk. 24 (1969) 225–226 (in Russian).[2] A. Ay, M. G¨urses and K. Zheltukhin, Hamiltonian equations in R , J. Math. Phys.44 (2003) 5688–5705.[3] A. Buic˘a and I.A. Garc´ıa, Periodic solutions of the perturbed symmetric Euler top,preprint (2009).[4] C. Christopher and C. Li, Limit cycles of differential equations. Advanced Coursesin Mathematics. CRM Barcelona. Birkh¨auser Verlag, Basel, 2007.[5] A. Cima, J. Llibre and M.A. Teixeira, Limit cycles of some polynomial differentialsystems in dimension 2, 3 and 4, via averaging theory, Appl. Anal. 87 (2008) 149–164.[6] F. Cong, J. Hong and Y. Han, Near-invariant tori on exponentially long time forPoisson systems, J. Math. Anal. Appl. 334 (2007) 59–68.[7] M. Craioveanu and M. Puta, On the rigid body with two linear controls, Differentialgeometry and applications (Brno, 1995), 373–380, Masaryk Univ., Brno, 1996.[8] P. Crehan, Variational Principles and Poisson Structures, Prog. Theor. Phys. Suppl.110 (1992) 321–328.[9] G. Darboux, M´emoire sur les ´equations diff´erentielles du premier ordre et du premierdegr´e, Bull. Sci. Math. 1 (series 2) (1878), 60–96; 2 (1878) 123–200.[10] J. Grabowski, G. Marmo and A. M. Perelomov, Poisson structures: towards a clas-sification, Mod. Phys. Lett. A 8 (1993) 1719–1733.[11] H. G¨umral and Y. Nutku, Poisson structure of dynamical systems with three degreesof freedom, J. Math. Phys. 34 (1993) 5691–5723.[12] B. Hern´andez-Bermejo, New solutions of the Jacobi equations for three-dimensionalPoisson structures, J. Math. Phys. 42 (2001) 4984–4996.[13] B. Hern´andez-Bermejo, Characterization and global analysis of a family of Poissonstructures, Phys. Lett. A 355 (2006) 98–103.[14] B. Hern´andez-Bermejo, An integrable family of Poisson systems: Characterizationand global analysis, Appl. Math. Lett. 22 (2009) 187–191.[15] D. D. Holm and J. E. Marsden, The rotor and the pendulum, in: Symplectic Ge-ometry and Mathematical Physics. Eds. P. Donato, C. Duval, J. Elhadad and G. M.Tuynman. Progress in Mathematics, Vol. 99, Birkhauser (Boston, USA), 1991, pp.189–203.
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