May 16, 2016 Integrable deformations of Ro¨ssler and Lorenz systems from Poisson–Lie groups A´ngel Ballesterosa, Alfonso Blascoa and Fabio Mussoa,b aDepartamento de F´ısica, Universidad de Burgos, 09001 Burgos, Spain 6 bI.C. “Leonardo da Vinci”, Via della Grande Muraglia 37, I-0014 Rome, Italy 1 0 e-mail: [email protected], [email protected], [email protected] 2 y a M Abstract A method to construct integrable deformations of Hamiltonian systems of ODEs endowed with 3 Lie–PoissonsymmetriesisproposedbyconsideringPoisson–LiegroupsasdeformationsofLie–Poisson 1 (co)algebras. Moreover, the underlying Lie–Poisson symmetry of the initial system of ODEs is used toconstructintegrablecoupledsystems,whoseintegrabledeformationscanbeobtainedthroughthe ] I construction of the appropriate Poisson–Lie groups that deform the initial symmetry. The approach S is applied in order to construct integrable deformations of both uncoupled and coupled versions of . certain integrable types of Ro¨ssler and Lorenz systems. It is worth stressing that such deformations n i are of non–polynomial type since they are obtained through an exponentiation process that gives nl rise to the Poisson–Lie group from its infinitesimal Lie bialgebra structure. The full deformation [ procedure is essentially algorithmic and can be computerized to a large extent. 2 v 7 5 3 3 MSC: 37J35; 34A26; 34C14; 17B62; 17B63 0 . 1 KEYWORDS: integrability, deformations, ordinary differential equations, coupled systems, R¨ossler sys- 0 tem, Lorenz system, Poisson–Lie groups, coalgebras, Lie bialgebras. 6 1 : v i X r a 1 Introduction The integrability problem for systems of first order ODEs is indeed a relevant issue in the theory of dynamical systems (see, for instance, [1, 2, 3, 4, 5] and references therein), and the same question for coupled systems of ODEs could be also meaningful from the viewpoint of applications (for instance, regarding synchronization problems [6]). In this context, the aim of this paper is two fold: on one hand, to present a novel approach to the explicit construction of Liouville–integrable coupled systems of ODEs endowed with Lie–Poisson symmetries and, on the other hand, to show how integrable deformations of the previous systems can be systematically obtained. Theconstructionherepresentedestablishesa(tothebestofourknowledge)novelconnectionbetween Poisson–Lie groups and integrable deformations of dynamical systems. Such a link is based on the well- known result by Drinfel’d that establishes the one-to-one correspondence between Poisson–Lie groups and Lie bialgebras [7], as well as on the general construction of integrable Hamiltonian systems from Poissoncoalgebrasthatwasintroducedin[8,9,10],thusgivingrisetotheso-called“coalgebrasymmetry approach”tofinitedimensionalintegrablesystems(see[11,12]forapplicationsofthismethodtoH´enon– HeilesandLotka–Volterrasystems,respectively,and[13]forthecoalgebra–basedconstructionofnonlinear superposition rules arising in nonautonomous Lie–Hamilton systems [14]). In particular, we will apply the formalism here introduced in order to obtain new integrable de- formations of certain Lorenz [15] and R¨ossler [16] systems, as well as of some coupled versions of them. Nevertheless,westressthatthemethodpresentedinthepaperiscompletelygeneralandcouldbeapplied to any other interesting Hamiltonian systems of ODEs endowed with a Lie–Poisson symmetry. The basics of the Poisson coalgebra approach to integrability will be summarized in the next Section. In short, we will consider a finite–dimensional integrable Hamiltonian system of ODEs that is defined throughaLie–Poissonalgebra(F(g∗),{,}),whereF(g∗)isthealgebraofsmoothfunctionsong∗ andthe Lie–Poissonbracketis{,}. AsetoffunctionsininvolutionwillbegivenbyasingleHamiltonianfunction H∈F(g∗)andanumberofCasimirfunctionsC ∈F(g∗),k =1,...,r. Then,sinceLie–Poissonalgebras k are endowed with a (primitive or non–deformed) coalgebra structure, the coalgebra symmetry approach will straightforwardly provide, by following [8], a coupled integrable generalization of the system. This approachwillbeillustratedinsection3throughtheexplicitconstructionofanintegrablecoupledR¨ossler system. A detailed technical presentation of the construction of Poisson–Lie groups as deformations of Lie bialgebras will be given in Section 4. If we denote by g the abstract Lie algebra associated with the Lie– Poissoncoalgebra(F(g∗),{,}),wewillassumethatweareabletofindacocommutatormapδ compatible with g, so that (g,δ) defines a Lie bialgebra structure [17, 18]. Then, it is well–known that the dual of the cocommutator δ will define a second Lie algebra structure (that we will denote by d), and the dual of the structure tensor for g will provide a compatible cocommutator map γ for d, in such a way that (d,γ)isjustthedualLiebialgebrato(g,δ). Onceagivenδ∗ (cid:39)disidentified, the(connectedcomponent containing the identity) of the Lie group D with Lie algebra d can be constructed by exponentiation, and the unique Poisson–Lie structure having g∗ as its linearization at the identity can be computed, thus providing the deformed Poisson structure {,} we were looking for. Simultaneously, the group law η for D will give us the explicit deformed coproduct map ∆ on the algebra of smooth functions F(D). η Moreover, the deformation nature of this construction arises in a more transparent way if we consider the one-parametric cocommutator γ = ηγ, since the parameter η will give us information concerning η the orders in the deformation process that arises from the ‘exponentiation’ of the tangent Lie bialgebra structure. Itisworthtobestressedthattheprocedurethatwehavejustsketchedisalgorithmicandcan be made fully explicit by making use of symbolic computation packages. Therefore, as a Poisson coalgebra, (F(D),{,} ,∆ ) can be interpreted as a smooth η–deformation η η of (F(g∗),{,},∆) on which the initial dynamical system with Hamiltonian function H is defined. As a consequence, if we consider the same Hamiltonian function H on (F(D),{,} ,∆ ), then the associated η η dynamical system is just an integrable η–deformation of the initial system of ODEs. Moreover, the coalgebra symmetry approach [8] guarantees that the set of involutive functions for the deformed system 2 will be given by the deformed coproducts of the Hamiltonian H and of the deformed Casimir functions Cη for the Poisson structure {,} , with k =1,...,r. k η Sections 5 and 6 are devoted to use this Poisson-deformation approach in order to obtain integrable deformations of two coupled R¨ossler and Lorenz systems, respectively. It is worth stressing that, as a result of the exponentiation process that underlies the full construction, such integrable deformations of dynamical systems are provided by non–polynomial vector fields, which is a non–standard finding from the viewpoint of integrability. Also, the underlying Poisson–Lie groups are responsible of the fact that thecouplingbetweenthesystemsisquiteinvolved, sincesuchcouplingisgeneratedbytheirnon–abelian group law. A final Section including some remarks and open problems closes the paper. 2 Lie–Poisson coalgebras Let g be a finite dimensional Lie algebra. As it is well known, the associated Lie–Poisson algebra F(g∗) is the algebra of C∞ functions on g∗, equipped with the Poisson bracket {f,g}(x)=(cid:104)[df,dg],x(cid:105), x∈g∗, f,g ∈F(g∗), (1) where (cid:104),(cid:105) is the non-degenerate pairing between g and g∗. Let us assume that g is generated by the set {X }dim(g), with commutation relations i i=1 [X ,X ]=ck X , (2) i j ij k and let us denote with {x }dim(g) the corresponding dual basis in g∗ i i=1 (cid:104)x ,X (cid:105)=δ , i,j =1,...,dim(g). (3) i j ij Then the Lie–Poisson bracket (1) for the coordinate functions on g∗ is given by the fundamental Poisson brackets {x ,x }=ck x . (4) i j ij k Now,bydefiningthefollowingcoproductmap∆:F(g∗)→F(g∗)⊗F(g∗)forthecoordinatefunctions ∆(x )=x ⊗1+1⊗x , (5) i i i wecansaythat(F(g∗),{,},∆)isendowedwithaPoissoncoalgebrastructure[8],sincethecoassociativity condition (∆⊗id)◦∆=(id⊗∆)◦∆, (6) is fulfilled, and the coproduct map ∆ can be straightforwardly shown to define a Poisson map between F(g∗) and F(g∗)⊗F(g∗), namely ∆({f,g})={∆(f),∆(g)}, ∀f,g ∈F(g∗), (7) with respect to the Poisson structure on F(g∗)⊗F(g∗) induced from (4). ThecoalgebraapproachtointegrabilityisbasedonthefactthatifwestartfromaLiouvilleintegrable system defined on F(g∗), we can use the coproduct map (5) in order to extend it to an integrable system definedon[F(g∗)]N :=F(g∗)⊗...N⊗F(g∗), withN beinganarbitrarynumberofcopiesofF(g∗). The keystone for this result is given by the k-th coproduct maps, which are defined recursively as ∆(k) ≡(∆⊗id)◦∆(k−1), k =3,...,N, ∆(2) ≡∆, (8) and, by construction, satisfy the Poisson homorphism property (9) ∆(k)({f,g})={∆(k)(f),∆(k)(g)}, ∀f,g ∈F(g∗). (9) 3 The simplest possible case in this framework occurs when the integrable system on F(g∗) is defined by a single Hamiltonian function H and an arbitrary number of Casimir functions C , k = 1,...,r. In k this case all the involutive functions defining an integrable system on [F(g∗)]N are obtained as images of the Hamiltonian and Casimir functions under the ∆(k) maps (k = 2,...,N). In this paper we will consider only this case, but we recall that this approach can be generalized for Poisson algebras in which integrability involves either more than a single Hamiltonian or even non-coassociative Poisson maps (see [19, 20, 21]). 3 Integrable Ro¨ssler systems In order to illustrate this integrability approach, let us consider the specific Lie–Poisson algebra that underlies the integrability of a dynamical system of R¨ossler type. We recall that the so–called R¨ossler system x˙ =−y−z, y˙ =x+ay, (10) z˙ =xz+b−cz, was introduced in [16] as a prototype of 3D system of ODEs exhibiting the phenomenon of continuous chaos. Whena=b=c=0,thesystem(10)isknowntobeintegrableintheLiouvillesense[22,23,24,25], and gives rise to x˙ =−y−z, y˙ =x, (11) z˙ =xz. In particular, this system is known to be Hamiltonian [25] with respect to the Poisson bracket {x,y}=−1, {x,z}=−z, {y,z}=0, (12) provided that we consider the Hamiltonian function 1 H= (x2+y2)+z. (13) 2 The bracket (12) can be thought of as a particular case of the Lie–Poisson bracket {x,y} =−w, {x,z} =−z, {y,z} =0, {w,·} =0, (14) 0 0 0 0 where we have introduced a fourth “central” Poisson algebra generator w. It is immediate to show that this is just the Lie–Poisson algebra corresponding to the 4D real solvable Lie algebra g ≡ A (here we 4,3 follow the classification of four–dimensional real Lie algebras given in [26]). The Casimir functions for (14) are given by W =w, C =ze−wy. (15) Indeed,iftheHamiltonianofthesystemisagaingivenby(13),theR¨osslersystem(11)isrecoveredwhen W =w =1. 3.1 Coupled R¨ossler systems from Lie–Poisson coalgebras The Lie–Poisson coalgebra (F(g∗),{,} ,∆) is defined through the primitive coproduct map 0 ∆(x) = x⊗1+1⊗x=x +x , 1 2 ∆(y) = y⊗1+1⊗y =y +y , (16) 1 2 ∆(z) = z⊗1+1⊗z =z +z , 1 2 ∆(w) = w⊗1+1⊗w =w +w , 1 2 4 where in the second equality we have identified the coordinate functions on each copy of F(g∗) through a different subscript. Now, by using the standard Poisson coalgebra structure (14)–(16), we are able to define integrable coupled generalizations of the R¨ossler system (examples of nonintegrable couplings can be found in [27, 28]). Explicitly, if we consider the coproduct of the Hamiltonian (13), we get ∆(H)= 1(cid:0)[∆(x)]2+[∆(y)]2(cid:1)+∆(z)= 1(x2+y2)+z + 1(x2+y2)+z +x x +y y , (17) 2 2 1 1 1 2 2 2 2 1 2 1 2 and the corresponding 6D equations of motion lead to a coupled R¨ossler system x˙ =−w (y +y )−z , 1 1 1 2 1 y˙ =w (x +x ), 1 1 1 2 z˙ =z (x +x ), 1 1 1 2 (18) x˙ =−w (y +y )−z , 2 2 1 2 2 y˙ =w (x +x ), 2 2 1 2 z˙ =z (x +x ), 2 2 1 2 together with w˙ = w˙ = 0. Notice that the coupling between both systems comes from the non–linear 1 2 terms of the Hamiltonian (13), and the restriction of the dynamics to the submanifold x = y = z = 2 2 2 w =0, w =1 (or, alternatively, to x =y =z =w =0, w =1) leads us to the uncoupled R¨ossler 2 1 1 1 1 1 2 system (11). The 6D Hamiltonian system (18) is integrable since, due to the underlying coalgebra symmetry, we havethreeadditional(besidestheHamiltonian)independentintegralsofthemotionthatareininvolution. Two of them are given by the (nontrivial) Casimir functions on each of the copies of the Lie–Poisson algebra, namely C1 =z1e−wy11, C2 =z2e−wy22, (19) and the the third one is just the coproduct of the Casimir C, which reads ∆(C)=∆(z)e−∆∆((wy)) =(z1+z2)e−((wy11++wy22)). (20) TheN–dimensionalgeneralizationofthisresultisstraightforwardbyconsideringtheN-thcoproduct map ∆(N) of the Hamiltonian and of the Casimir functions, that is obtained as the pull-back through the N-th coproduct of the coordinate functions [8] ∆(N)(x)=x⊗1⊗1⊗...N−1)⊗1 +1⊗x⊗1⊗...N−2)⊗1+... +1⊗1⊗...N−1)⊗1⊗x, (21) (and the same for y and z). Explicitly, (cid:32) N (cid:33)2 (cid:32) N (cid:33)2 N 1(cid:16) (cid:17) 1 (cid:88) 1 (cid:88) (cid:88) H(N) :=∆(N)(H)= [∆(N)(x)]2+[∆(N)(y)]2 +∆(N)(z)= x + y + z , (22) 2 2 i 2 i i i=1 i=1 i=1 and the N-coupled R¨ossler system reads (cid:16) (cid:17) x˙ =−w (cid:80)N y −z , i i j=1 j i (cid:16) (cid:17) y˙ =w (cid:80)N x , i=1,...,N (23) i i j=1 j (cid:16) (cid:17) z˙ =z (cid:80)N x . i i j=1 j This system of 3N coupled ODEs has the following (2N −1) integrals of the motion: Ci =zie−wyii, i=1,...,N (24) C(k) :=∆(k)(C)=∆(k)(z)e−∆∆((kk))((wy)) =(cid:32)(cid:88)k zi(cid:33) e−(((cid:80)(cid:80)kiki==11wyii)), k =2,...,N. (25) i=1 By construction, all these functions are in involution and Poisson-commute with the Hamiltonian H(N). 5 4 Lie bialgebras and Poisson–Lie groups The aim of this paper is to construct integrable deformations of coalgebra symmetric systems of the type(23)foranynumberN =1,2,... ofcoupledconstituents. Aswewillshowinthesequel,thiscanbe achieved by considering the Lie–Poisson algebra g∗ as the linearization of a Poisson–Lie group structure on a certain different Lie group D. In his way, the construction of the corresponding full Poisson–Lie groupstructure(F(D),{,} ,∆ )willprovideaPoissoncoalgebraη–deformationof(F(g∗),{,} ,∆),and η η 0 the application of the coalgebra approach to a Hamiltonian defined on the Poisson–Lie group D will automatically provide an integrable η–deformation of the initial system defined on F(g∗). 4.1 Lie bialgebras Given a Lie algebra g with basis {X }dim(g) and Lie bracket i i=1 [X ,X ]=ck X , (26) i j ij k a Lie bialgebra structure (g,δ) is given by a skew-symmetric cocommutator map δ :g →g⊗g δ(X )=fijX ∧X , (27) k k i j such that • i) δ is a 1-cocycle, i.e., δ([X,Y])=[δ(X), Y ⊗1+1⊗Y]+[X⊗1+1⊗X, δ(Y)], ∀X,Y ∈g. (28) • ii) The dual map δ∗ :g⊗g →g is a Lie bracket. Therefore, the dual δ∗ of the cocommutator map defines a second (in general, different) Lie algebra structure, that we will call d, and whose Lie brackets are [xˆi,xˆj]=fijxˆk. (29) k Here the duality relation is established through the canonical pairing (cid:104)xˆj,X (cid:105) = δj. Note that, given k k the initial Lie algebra g, the dual Lie algebra d has to fulfill the 1-cocycle condition, which implies the following compatibility equations between the two Lie algebras g and d: fabck =fakcb +fkbca +fakcb +fkbca . (30) k ij i kj i kj j ik j ik By taking the components of the structure tensor c as initial data, the solutions to the (now linear) equations (30) together with the (quadratic) equations coming from the Jacobi identity for f provide all possible Lie bialgebra structures for g. Finally, if the solutions for the tensor fij are classified into k equivalence classes under the action of all possible automorphisms for g, the classification of all possible Lie bialgebra structures of g is obtained. Two remarks are in order: • It is immediate to prove that if (g,δ) is a Lie bialgebra with structure tensors (c,f), then (d,γ) with γ :d→d⊗d being the cocommutator map γ(xˆk)=ck xˆi∧xˆj, (31) ij defines a Lie bialgebra structure on d with structure tensors (f,c), that is called the dual Lie bialgebra to (g,δ). In other words, d and δ are defined through the same structure tensor f, and we can say that δ∗ ≡ d. In the same manner, g and γ are defined by the same structure tensor c and we can say that γ∗ ≡g. 6 • For our purposes it will be convenient to multiply all the components fij of the cocommutator k map by a real parameter η, that will play the role of a deformation parameter. The resulting cocommutator will be denoted by δ , and the dual Lie algebra structure d will be η η [xˆi,xˆj]=ηfijxˆk. (32) k Obviously, d is just d and these two Lie algebras are isomorphic under the change of basis η=1 xˆi → ηxˆi. Nevertheless, as we will see in the sequel, the explicit appearance of η will allow us to control the full deformation/exponentiation procedure. 4.2 Poisson–Lie groups and Drinfel’d theorem A Poisson–Lie (PL) group is a Lie group D endowed with a Poisson structure on the algebra C∞(D) of functions on the group manifold, such that the Lie group multiplication is a Poisson map. Therefore, by defining the coproduct ∆(f) as the pull back of a function f ∈ C∞(D) by the group multiplication, we obtain a natural structure of Poisson coalgebra on the algebra of smooth functions on the Poisson–Lie group F(D). In algebraic terms, if D is a Lie group, the group multiplication on D induces a coproduct map ∆:C∞(D)→C∞(D)⊗C∞(D), (33) which is coassociative (since the group law is associative). Therefore, PL groups are instances of Poisson coalgebras, since the coproduct map is -by definition- a Poisson algebra homomorphism: {∆(a),∆(b)}=∆({a,b}) ∀a,b ∈C∞(D), (34) where {a⊗b,c⊗d}={a,c}⊗bd+ac⊗{b,d}, (35) is the natural Poisson structure on C∞(D)⊗C∞(D). Now, the relevance of Lie bialgebras arises due to their one-to-one correspondence with Poisson–Lie groups, a result that can be explicitly stated as follows. Theorem [7]. Let D be a Lie group with Lie algebra d: a)If(D,{,})isaPoisson–Liegroup,thendhasanaturalLiebialgebrastructure(d,γ),calledthetangent Lie bialgebra of (D,{,}). b) Conversely, if D is connected and simply connected, every Lie bialgebra structure (d,γ) is the tangent Lie bialgebra of a unique Poisson structure on D which makes D into a Poisson–Lie group. Explicitly, if (D,{,}) is a PL group, the canonical Lie algebra structure on d∗ is given by [ξ1,ξ2] :=(d{f ,f }) =γ∗(ξ1⊗ξ2), ξ1,ξ2 ∈d∗ (36) d∗ 1 2 e where f ,f ∈ C∞(D) are such that (df ) = ξi. In other words, the linearization of the PL bracket 1 2 i e in terms of the local canonical coordinates ξi on the D group manifold is just γ∗, the dual of the cocommutator map that defines the associated tangent Lie bialgebra structure. 4.3 Poisson–Lie groups as deformations of Lie–Poisson coalgebras A novel construction of coalgebra deformations of Lie–Poisson algebras can be envisaged if Drinfel’d theorem is revisited by taking the tangent Lie bialgebra (d,γ) as the initial data for the construction of the associated PL group. Let us assume that we want to construct a Poisson coalgebra deformation of a given Lie–Poisson algebra g∗. Then, we can proceed as follows: 1. Take the Lie algebra g, find a non–trivial Lie bialgebra structure (g,δ) and construct its dual Lie bialgebra (d,γ). Now, recall that Drinfel’d theorem ensures that (d,γ) will be the tangent Lie bialgebra structure of a certain Poisson–Lie group (D,{,}) such that d=Lie(D). 7 2. Introduce the isomorphic Lie algebra d (32) in order to have an explicit deformation parameter in η the tangent Lie bialgebra. 3. Construct the PL group (D,{,}), that will provide the deformation of the Poisson coalgebra (F(g∗),{,} ,∆). In order to obtain it, we have firstly to find a faithful representation ρ of d , 0 η and use it to construct the matrix element of the (connected component) of the Lie group D through the usual exponentiation: dim(d) (cid:89) D = exp(x ρ(xˆi)). (37) i i=1 4. With an abuse of notation, we will make use the same symbol x for the coordinate functions on i g∗ and for the local coordinates of the D group in (37). In this way, the pullback of the group multiplication will provide a coproduct map ∆ :F(D)→F(D)⊗F(D), (38) η where, due to the coordinate identification that we have previously made, the coproduct ∆ is a η η deformation of the primitive one for (F(g∗),{,} ,∆), namely 0 lim∆ (x )=x ⊗1+1⊗x =∆(x ). (39) η i i i i η→0 Note that in the ‘non-deformed case’ η = 0 we have from (32) that the dual Lie algebra d is [xi,xj]=0, and then the group D is abelian. Therefore, its group law is just the addition of group coordinates, which is algebraically expressed through the primitive coproduct (39). 5. Therefore, byDrinfel’dtheorem, thereexistsauniquePoissonbracket{,} onF(D)andsuchthat η (F(D),∆ ,{,} ) is a Poisson–Lie group η η ∆ ({F,G})={∆ (F),∆ (G)}, ∀F,G∈F(G∗). (40) η η η It turns out that the Poisson bracket {,} is also an η deformation of the Lie–Poisson bracket {,} η on g∗, in the sense that lim{x ,x } ={x ,x }. (41) i j η i j η→0 Indeed, the Poisson–Lie bracket {,} has to be explicitly found by solving simultaneously the η equations (40) and (41), but this problem turns out to be computationally feasible. Due to the properties (39) and (41) we can conclude that, as a Poisson coalgebra, the Poisson–Lie group (F(D),∆ ,{,} ) is an η–deformation of the Lie–Poisson algebra (F(g∗),{,},∆). Indeed, there η η will be as many Poisson-coalgebra deformations of g∗ as different PL groups D such that g∗ induces an admissible tangent Lie bialgebra cocommutator γ for d=Lie(D). 5 An integrable deformation of the Ro¨ssler system InthesequelwewillmakeuseofthefactthattheintegrableR¨osslersystemisanoutstandingexampleof Lie–Poisson coalgebra-symmetric system. This will allow the construction of an η–deformed coproduct map that will give rise to integrable coupled systems of ODEs of R¨ossler type. LetusstartbyconsideringtherealLiealgebraA inthebasisgivenin[26](notethatthisisexactly 4,3 the four dimensional algebra (14)) [X,Y]=−W, [X,Z]=−Z, [Y,Z]=0, [W,·]=0. (42) 8 It is straightforward to check that the following cocommutator map δ (X)=ηX∧Z, δ (Y)=ηZ∧W, δ (Z)=δ (W)=0, (43) η η η η is compatible with the Lie–Poisson algebra (14) in the sense of (28), and thus defines a Lie bialgebra structure (A ,δ ). The dual Lie algebra d obtained from the dual cocommutator map is 4,3 η η [xˆ,yˆ]=0, [xˆ,zˆ]=ηxˆ, [xˆ,wˆ]=0, [yˆ,zˆ]=0, [yˆ,wˆ]=0, [zˆ,wˆ]=ηyˆ. (44) It can be checked that the dual Lie algebra (44) is also isomorphic to the A algebra. Therefore, we 4,3 have a self–dual Lie bialgebra structure. Now we have to construct the Lie group D whose Lie algebra is d . If we define the 7×7 matrix ej η i as the one with the only nonvanishing entry 1 in the i–th row and j–th column, a faithful representation ρ of the algebra (44) is given by ρ(xˆ) = ηe3, 1 ρ(yˆ) = ηe7, 5 ρ(zˆ) = −ηe1+ηe4+ηe6, (45) 1 2 5 ρ(wˆ) = −ηe3+ηe7. 2 6 By using ρ, we construct the matrix group element D = exp(xρ(xˆ))exp(yρ(yˆ))exp(zρ(zˆ))exp(wρ(wˆ))=  e−ηz 0 ηx 0 0 0 0   0 1 −ηw ηz 0 0 0     0 0 1 0 0 0 0    =  0 0 0 1 0 0 0 , (46)    0 0 0 0 1 ηz η(y+ηwz)     0 0 0 0 0 1 ηw  0 0 0 0 0 0 1 where {x,y,z,w} are the local coordinate functions on the group D. At this point it is worth stressing that the parameter η appears as as multiplicative factor within all the nonvanishing matrix entries of the fundamental representation of the Lie algebra d (45). Therefore, when we exponentiate in order to obtain the group element D, the powers of η will give us the corresponding contributions coming from different powers of the Lie algebra generators. By solving the functional equations ∆ (D ) = (cid:80)7 D ⊗D through the algorithm presented η ij k=1 ik kj in [29], we get the group law, i.e., the coproduct maps for the coordinate functions of D, which read ∆ (x) = x⊗1+e−ηz⊗x=x +e−ηz1x , η 1 2 ∆ (y) = y⊗1+1⊗y−ηw⊗z =y +y −ηw z , η 1 2 1 2 ∆ (z) = z⊗1+1⊗z =z +z , (47) η 1 2 ∆ (w) = w⊗1+1⊗w =w +w , η 1 2 that are indeed η-deformations of the primitive coproduct (16) such that lim ∆ = ∆. Again, by η→0 η following the approach described in [29], the unique Poisson–Lie bracket {, } for which the coproduct η ∆ is a Poisson map is found to be η e−ηz−1 {x,y} =−we−ηz, {x,z} = , {y,z} =0, {w,·} =0. (48) η η η η η Asexpected,lim {·,·} ={·,·} andwehavethusobtainedaη–deformationoftheLie–Poissonalgebra η→0 η 0 (14). Finally, the Casimir functions for the algebra (48) are found to be W = w, (49) η 1−exp(−ηz) (cid:16) y(cid:17) C = exp − , (50) η η w 9 and, as expected, they are again η–deformations of the Casimir functions (15). Therefore, the Poisson–Lie group (F(D),∆ ,{,} ) has been fully constructed. Now, by taking the η η same Hamiltonian (13) of the R¨ossler system, the PL bracket (48) gives rise to the system of ODEs (cid:18)e−ηz−1(cid:19) (cid:18) z2(cid:19) (cid:18)1 z3(cid:19) x˙ =−ywe−ηz+ =−(wy+z)+ wyz+ η− wyz2+ η2+o[η3], η 2 2 6 (cid:18) (cid:19) 1 y˙ =xwe−ηz =wx−(wxz)η+ wxz2 η2+o[η3], 2 (51) (1−e−ηz) (cid:18)1 (cid:19) (cid:18)1 (cid:19) z˙ =x =xz− xz2 η+ xz3 η2+o[η3], η 2 6 w˙ =0, that, for w = 1, provides an integrable deformation of the R¨ossler system (11), with deformed integrals of the motion given by (49) and (50). The preservation of the closed nature of the trajectories under deformation is clearly appreciated in Figure 1. z z x x y y (a) (b) Figure 1: (a) Closed trajectories of the integrable R¨ossler system (11) for the initial data x(0)=1,y(0)=2,z(0)=3 (black line) and of the deformed integrable R¨ossler system (51) with the same initial data, w = 1 and η = −1.5 (blue line), η = −0.5 (green line), η = 0.5 (red line), η = 1.5 (yellow line). (b) The same figure as (a) but with initial data x(0)=1,y(0)=−1,z(0)=0.5. 5.1 Deformed coupled systems Once the Poisson–Hopf algebra (F(D),∆ ,{,} ) has been fully determined, the construction of an in- η η tegrable deformation for the N–coupled system (23) is straightforward. Namely, in the N = 2 case the deformed coproduct ∆ on the Hamiltonian of the R¨ossler system (13) will be η ∆ (H)= 1(cid:0)[∆ (x)]2+[∆ (y)]2(cid:1)+∆ (z), (52) η 2 η η η and from (47) we get ∆η(H)= 21(cid:0)(x1+e−ηz1x2)2+(y1+y2−ηw1z2)2(cid:1)+(z1+z2) (53) = 21x21+x1x2e−ηz1 + 12(cid:0)e−2ηz1x22+(y1+y2)(y1+y2−2ηw1z2)+η2w12z22(cid:1)+z1+z2. 10