Group actions as stroboscopic maps of ordinary differential equations 9 0 0 Andrzej Okninski 2 Politechnika Swietokrzyska, Physics Division, n a Al. 1000-lecia PP 7, 25-314 Kielce, Poland J 4 January 24, 2009 2 ] h Abstract p - Discrete-time dynamical systems can be derived from group actions. h In the present work possibility of application of this method to systems t a of ordinary differential equations is studied. Invertible group actions are m consideredaspossiblecandidatesforstroboscopicmapsofordinarydiffer- ential equations. It is shown that flow of the Bloch equation is a unique [ suspension of an invertiblemap on the SU(2) group. 3 v 8 1 Introduction 2 1 Discrete-timedynamicalsystemscanbeformulatedintermsofgroupactionsto 2 exploit the group structure and get a better understanding of the correspond- . 4 ing dynamics. This approach was used to study non-invertible discrete-time 0 dynamics on E(2) [1] and SU(2) [1, 2, 3, 4]. The Shimizu-Leutbecher map 8 [5, 6], a tool to study group structure, was solved for an arbitrary Lie group 0 : [1]. Since the Shimizu-Leutbecher sequence generates the logistic map for v G =SU(2) [1, 3] a general solution of the logistic map was thus obtained. For i G X = E(2) this approach led to a better understanding of the Harter-Heighway G r fractal curve [1]. A two-dimensional generalizationof the logistic map was also a introduced as a map on SU(2) and investigated [4]. On the other hand, structure of Kleinian groups is naturally studied in the setting of discrete-time dynamical systems, revealing in this way connections with fractals [7, 8, 9]. There were several attempts to study dynamics on Lie groupswithincontinuous-timeratherthandiscrete-timeapproach. Continuous- time dynamics on SU(2) and SU(2) SU(2) was defined in the setting of × ordinary differential equations (ODEs) [10, 11] while continuous iteration of maps was defined to find a correspondence between mappings and continuous- time evolution [12, 13]. Thequestionnowariseswhetherageneralconnectionbetweendiscrete-time group actions and continuous-time dynamical systems can be established. 1 Weinvestigateapossibilityofrelatinggroupactionswithstroboscopicmaps ofODEs. Letus considera continuous-timedynamicalsystemgivenby asetof ODEs dx =f(x(t),t), (1) dt where x = [x , x , ... , x ], f = [f , f , ... , f ]. Let x(t) be a solution to 1 2 n 1 2 n Eq.(1). Then the invertible map S : x(t) x(t+T), (2) T 7−→ is a stroboscopic map of strobe time T. Stroboscopic maps with several strobe times are standard tools to solve differential equations to mention only the Runge-Kutta methods. On the other hand, a discrete map can be also used to generate a continuous flow. Such a flow, non unique in general, is called a suspension of the map [14, 15]. In the present work we shall consider a class of invertible maps on a group andthentry tofinda flow,preferablyunique,suchthatitstrajectoriesconnect continuously the iterates of the map. The paper is organized as follows. In the next Section maps on a Lie group aredefinedandmethodtodeduceevolutionofgroupparametersisdescribed. G Principal results are described in Sections 3 and 4. In Section 3 a class of invertible maps is considered. These maps are solved in Subsection 3.1. In Subsection 3.2 a simplified map is defined, solved and parameterized on the SU(2) group. In Subsection3.3 symmetry andrestrictions ofthe dynamics are determined. Results of the Subsection 3.2 are used in Section 4 to derive ODE for which the simplified map is a stroboscopic map. The map samples the flow of the ODE exactly and arbitrarily densely and it follows that the construction is unique. This ODE is the Bloch equation [16], also known as the Landau- Lifshitz equation. These results generalize an earlier findings [17]. In Section 5 computations for the simplified map on the SU(2) group are presented to elucidate dynamics of the Bloch equation. In the last Section the obtained results are summarized and perspectives of further researchare outlined. 2 Group dynamical systems Let us consider a dynamical system defined by the following map G =ϕ(G , ...), (3) N+1 N whereG . Let beasimpleLiegroupandgitsLiealgebra. ThenG N N ∈G G ∈G can be written in exponential form G =exp(X ), (4) N N where X g [18]. Any infinitesimal operator X of a n-parameter Lie group N N ∈ is a linear combination of n generators Ik G n X = Ikck , Ik g, (5) N X N ∈ k=1 2 where real parameters c1 ,...,cn are local coordinates of the Lie group element N N X . N Substituting Eqs.(4), (5) into Eq.(3) and using completeness of the basis consisting of the generators Ik and the unit matrix 1 we get a discrete-time dynamical system in parameter space [1, 19] cj =Fj c1 , ..., cn , j =1, ..., n, (6) N+1 (cid:0) N N(cid:1) where Fj are continuous functions [20]. 3 Discrete-time dynamics on a group Let us consider an invertible discrete-time dynamical system on a Lie group G RN+1 =QNRNQN−1RN−1Q−N1−1RN−1Q−N1, N =1,2,..., (7) i.e. ϕ = QNRNQN−1RN−1Q−N1−1RN−1Q−N1 in Eq.(3), where we assume knowl- edge of all group elements Q needed for the computations. Let us note M here that apparently simpler non-invertible Shimizu-Leutbecher map R = N+1 R QR−1 hasacomplicatedsolution[1]. Itturnsout,however,thatarelatively N N simple solution to (7) can be constructed upon introducing new variables S N defined below. 3.1 Exact solution We note that Eq.(7) can be reduced to two simpler equations. Indeed, intro- ducing new quantity df SN =RNQN−1RN−1QN−2, (8) we can rewrite Eq.(7) as RN+1 =SN+1RN−1SN−1+1, N =1,2,... , (9a) S =Q S Q−1 , N =1,2,... . (9b) N+1 N N N−2 To run dynamics defined by (7) or, alternatively, by (9), we have to impose initial conditions for Eq.(7), R , R , from which initial condition for Eq.(9b), 0 1 S =R Q R Q , can be also computed. Equations (9) are easily solved 2 2 1 1 0 R2K =S2KS2K−1...S2R0S2−1...S2−K1−1S2−K1, (10a) SN =QN−1...Q1S1Q−−11...Q−N1−3, (10b) where K = 1,2,... , N = 2,3,... and similar equations can be written for R . 2K+1 3 3.2 Discrete-time dynamics on the sphere In the case = SU(2) the Hamilton’s parameterization can be used. Let unit G vectorsr = r1 , r2 , r3 ,q = q1 , q2 , q3 andanglesχ ,α correspond N (cid:0) N N N(cid:1) N (cid:0) N N N(cid:1) N N to rotation axes and rotation angles, respectively. Then the rotation matrices R , Q are defined as N N R = exp iχNσ r , r =1, (11) N (cid:0) 2 · N(cid:1) | N| Q = exp iαNσ q , q =1, (12) N (cid:0) 2 · N(cid:1) | N| where i2 = 1, and σ = σ1, σ2, σ3 is the pseudo vector with the Pauli − (cid:2) (cid:3) matrices as components [21]. With parameterization(11), (12 ) Eq.(7) induces dynamicsofvectorsr onunitsphere. WeshallconsideraspecialcaseQ Q N N ≡ RN+1 =QRNQRN−1Q−1RN−1Q−1. (13) The solution of (13) is immediately obtained from (10): R =Q2KPKR P−KQ−2K, (14a) 2K 0 R˜ =Q2KPKR˜ PKQ−2K, (14b) 2K+1 1 whereP =df Q−1S Q−1 =Q−1R QR , R˜ =df Q−1R Q, K =0, 1, 2, .... 1 1 0 2K+1 2K+1 It follows that equations (14a), (14b) generate analogous dynamics. Matrices Q, P are parameterized as Q=exp iασ q , q =1, (15a) (cid:0) 2 · (cid:1) | | P =exp iβσ p , p =1, (15b) (cid:16) 2 · (cid:17) | | where α, β are the corresponding angles of rotations while p= p1,p2,p3 and (cid:2) (cid:3) q = q1,q2,q3 are unit vectors. We still have to impose initial condition R 0 while(cid:2)R is com(cid:3)puted as R =QPR−1Q−1. 1 1 0 A sufficient condition that points generated according to (14a) form a peri- odic trajectory (a finite set of points) is that for some integer K the following conditions hold: Kβ =2mπ, 2Kα=2nπ, (16) for some integer m, n; in this case the parameter β/(2α)=m/n is rational. Let us note that due to properties of the Pauli matrices the vector x(γ) defined by σ x(γ)=S σ x S−1 =exp iγσ s σ x exp iγσ s , (17) · · (cid:0) 2 · (cid:1) · (cid:0)− 2 · (cid:1) wherex= x1,x2,x3 ,s= s1,s2,s3 are unit vectorsandγ is a corresponding (cid:2) (cid:3) (cid:2) (cid:3) angle of rotation, is easily computed as x(γ)=cos(γ)x+sin(γ)x s+(1 cos(γ))(s x)s. (18) × − · 4 Using this result it is possible to write the solution (14a) in a closed form. Indeed, equation (14a) can be explicitly written as σ r =exp(iKασ q)exp iKβσ p σ r exp iKβσ p exp( iKασ q). · 2K · (cid:16) 2 · (cid:17) · 0 (cid:16)− 2 · (cid:17) − · (19) Applying Eq.(18) twice we find the closed form solution of Eq.(13): r(Kα)=cos(Kα)t(Kα)+sin(Kα)t(Kα) q+(1 cos(Kα))(q t(Kα))q, × − · (20a) t(Kα)=cos(λKα)r +sin(λKα)r p+(1 cos(λKα))(p r )p, (20b) 0 0 0 × − · df df where r(Kα)=r , λ=β/(2α) and K =0,1,2,... . 2K Thesequenceofvectorsr , r , r ,...,generatedaccordingtoEq.(20),sam- 0 2 4 ples a continuous curve . Indeed, it follows from Eq.( 20) that for very small C α and fixed λ vectors r(Kα) and r((K+1)α) are very close on a unit sphere. Thereforefor decreasingα, β andfixed λ=β/(2α) the sequence r , r , r ,... 0 2 4 approximates the curve more and more exactly. It is useful to introduce new C df variable θ = Kα which can treated as continuous since α is arbitrary. Using this we can rewrite Eq.(19) as: σ r(θ)=exp(iθσ q)exp(iλθ σ p) σ r exp( iλθσ p)exp( iθσ q), (21) 0 · · · · − · − · while Eq.(20) leads to explicit formula for the curve : C r(θ)=cos(θ)t(θ)+sin(θ)t(θ) q+(1 cos(θ))(q t(θ))q, (22a) × − · t(θ)=cos(λθ)r +sin(λθ)r p+(1 cos(λθ))(p r )p. (22b) 0 0 0 × − · 3.3 Symmetry and restrictions of dynamics Dynamical system (13) has continuous symmetry: R QκR Q−κ, κ R. (23) N N → ∀ ∈ This symmetry is equivalent to rotation of r around q about an arbitrary N angle κα. It can be thus expected that dynamics of the quantity r q should N · decouple from other degrees of freedom in (13) [2, 4]. Indeed, it follows from (20a) that r(θ) q=t(θ) q. (24) · · Since t(θ) = q = 1 it follows from the Schwartz inequality that 1 | | | | − ≤ t(θ) q 1. Now, for given p, q and r we have 0 · ≤ A t(θ) q A , 1 A , A 1. (25) 1 2 1 2 ≤ · ≤ − ≤ ≤ The constants A depending on the parameters p, q and the initial condition 1,2 r can be computed from (20b) by elementary means 0 A1,2 =c qb2+(a c)2, (26a) ∓ − 5 where a=r q, b=(r p) q, c=(p r )(p q). (26b) 0 0 0 · × · · · It thus follows that the motion on the sphere r(θ) = 1 is bounded by two | | parallels: A r(θ) q A . 1 2 ≤ · ≤ 4 Differential equation Themap(13),asfollowsfromthesolutionshowninEq.(20),isthestroboscopic map with strobe time α of a differential equation which will be deduced from the solution (21 ) where θ is a continuous variable (it can be derived from the form (22) as well). Differentiating Eq.(21) with respect to θ and using (14) we get (see Exercise 41.3 in Ref.[21] for similar computations) dσ r(θ) · =i[σ u(θ), σ r(θ)], (27) dθ · · df where [A, B]=AB BA and − u(θ)=q+λp(θ), λ= β , (28a) 2α σ p(θ)=exp(iθσ q) σ p exp( iθσ q). (28b) · · · − · It follows that the sequence r ,r ,r ,... , generated from Eq.(20), samples 0 2 4 the flow of Eq.(27) exactly. As it was remarked in Subsection 3.2 the vectors r and r are very close on a unit sphere for very small α and fixed λ, 2K 2(K+1) cf. Eq.(20). Therefore for decreasing α, β and fixed λ = β/(2α) the sequence of points on the unit sphere given by r , r , r ,... can be arbitrarily dense. 0 2 4 Equations (27), (28) can be written in explicit form. Using Eqs.(17), (18) we get p(θ)=cos(2θ)p+sin(2θ)p q+(1 cos(2θ))(q p)q, (29) × − · and hence dr(θ) =2r(θ) u(θ), (30) dθ × with u(θ) given by Eqs.(28a), (29). Obviously the length of the vector r is a conserved quantity, and we shall put r(θ) =1. The angle θ can be treated as | | dθ increasing with angular velocity ω =const, =ω [21], and time variable can dt be introduced to obtain finally dr(t) =2r(t) u(t), (31a) dt × u(t)=ω [1+λ λcos(2ωt)(q p)]q+λcos(2ωt)p+λsin(2ωt)p q . { − · × } (31b) Let us note that Eq.(31) is the Bloch equation [16]. Equation (31) has two invariants: r(t) = const, u(t) dr(t) = 0 which follow from the structure of | | · dt (31a). 6 5 Computational results We have performed several computations for the Bloch equation (31) and the discrete-timedynamicalsystem(13),parameterizedasdescribedinSection3.2, to show dynamics of the Bloch equation and to demonstrate how the map (13) samples the flow of Eq.(31). Exact solutions of the map (13) as well as of the Bloch equation (31) are given by (20) and (22), respectively. Since ω determines time scale only we put ω = 1. In all computations described below vectors p, q are orthogonal, p = [1, 0, 0], q = [0, 0, 1] and the initial condition is r = [0.6, 0, 0.8]. Motion on the sphere is bounded by 0 parallels A = max r(θ) q given by (26). For the present choice of p, q 1,2 ∓ | · | , r we have A = 0.8. In all figures below the vector q, parallels A and 0 1,2 1,2 ∓ theequatorareplotted,wherethindashedlines indicatepointsinvisibleforthe observer. Thesolution(20),ofdiscrete-timedynamicalsystem(13)withQ,P givenby (15) has been plotted in Fig. 1 for α=0.01, β =0.06 (λ=β/(2α)=3) where angles α, β have been given in degrees. The value of α is so small that points r , r , r , ... lie so close one to another that a seemingly continuous curve, 0 2 4 sampling the flow of the Bloch equation (31) very densely, has been obtained. Thedynamicshasbeenalsogenerateddirectlyfrom(13)withR =QPR−1Q−1 1 0 (the angle χ = 0 and arbitrary otherwise) to the same effect, the unit vectors 0 6 r have been renormalizedafter each iteration to avoidnumerical instabilities. N The whole trajectory has three-fold symmetry with respect to the q axis. q Figure 1: The Bloch equation (31) and discrete-time dynamical system (13), α=0.01, β =0.06, λ=3. In Fig. 2 dynamics of vectors r obtained from (20) has been plotted for N α= 2, β = 8 (λ = β/(2α)= 2). We thus obtain forty five points marked with dots. The solution (20) for α= 0.01, β =0.04 (λ =β/(2α)= 2) sampling the 7 Bloch equation densely has been also plotted. The closed curve has two-fold symmetry with respect to the q axis. q Figure 2: The Bloch equation (31) and discrete-time dynamical system (13), α=2, β =8 (dots) and α=0.01, β =0.04 (solid and dashed lines), λ=2. q Figure 3: The Bloch equation (31) and discrete-time dynamical system (13), α=0.01, β =0.0205, λ=1.025. In Fig. 3 initial stage of dynamics of vectors r (20 ) has been plotted for N α = 0.01, β = 0.0205 (λ = β/(2α) = 1.025), dot on the upper parallel marks the initial vector r . Had the value of λ be equal to one exactly the trajectory 0 8 would consist of one closed loop. Since λ = 1.025 = 41/40 the whole closed curve consists of forty one loops (in Fig. 3 seven such loops have been shown) andsamplesdenselytheBlochequation. Inthecaseofclosebutirrationalvalue of λ, e.g. λ = √2 0.389 = 1.0252... , the trajectory is ergodic on the whole − spherical sector bounded by two parallels A = 0.8. 1,2 ∓ 6 Summary and discussion We have introduced in Subsection 3 a class of discrete-time invertible maps (7) on an arbitrary group and the exact solution (10) of this map has been G found. Maps of form (7), parameterized on a Lie group, generate points in the parameter space which sample a trajectory in this space. This curve can be generated forward as well as backward from a given initial condition. This suggests that the group action (7) may correspond to a flow of a differential equation. Theresultsdescribedabovegeneralizesignificantlyourearlierfindings [17]. In Section 3.2 a simplified dynamics (13) has been parameterized on the unit sphere,i.e. for =SU(2), andithas been showndirectly by constructing G the solution (14a), (20) and (22) that the maps samples the flow of the Bloch equation (31) exactly and arbitrarily densely. It can be thus stated that the Bloch equation (31) is a unique suspension of the map (13). In the special case p q=0 we recover the solution obtained by H.K. Kim · and S.P. Kim [22], see also Kobayashi papers where several methods to solve the Blochequationweredescribedearlier[23,24,25]. Itseems thatthe present formulation leads to some progress in understanding the Bloch equation. The role of the parameter λ has been elucidated in Subsection 3.2. More exactly, it follows from the condition (16) that for fixed rational value of parameter λ=β/(2α)theflowofEq.(31)generatesclosedtrajectoriesconsistingofafinite set of points. For α,β 0 and fixed value of λ points obtained according to → (20)sampletrajectoriesof(31)exactlyandarbitrarilydensely. Moreover,ithas beenshowninSection 3.3that dynamicalsystem(13)has rotationalsymmetry with respect to the q axis from which dynamical restrictions for the quantity r(θ) qhavebeenderived,see(26). Itfollowsfrom(22)thatforλ=m/n,with · m,nrelativelyprime,thecurver(θ)hasm-foldsymmetrywithrespecttothe q axis. All these results have been visualized in Section 5 where computational results have been presented. The figures have been produced by code written in the MetaPost picture-drawing language [26]. The results described in the present paper can be generalized in several di- rections. Firstofallitshouldbedeterminedwhenthegeneralgroupdynamical system (7) samples a continuous curve. Whenever this is the case it should be possible to construct from a solution (10), computed for some Q ’s, an ODE N forwhichthe groupactionisastroboscopicmap. Ofcourse,uniquenessofsuch construction should be investigated. A very simple choice of Q ’s is Q =S, N 2K Q = T, K = 0,1,... . We found in our early computations that for 2K+1 ST = TS dynamics of Eq.(7) was very complicated [27], yet there is a closed 6 9 form solution (10). Finally, group actions on other groups such as SL (R) can 2 be considered. Acknowledgement It is a pleasure to thank Andrzej Lenarcik for introducing the author to the graphics programming language MetaPost. References [1] A. Okninski, Physica D 55, 358 (1992). [2] A. Okninski, Int. J. Bifurcation and Chaos 4 , 209 (1994). [3] E. Ahmed, A.E.M. El-Misiery, Int. J. Theor. Phys. 33, 1681 (1994). [4] A. Okninski, A. Kowalska, J. Tech. Phys. 37 , 395 (1996). [5] H. Shimizu, Ann. of Math. 77, 33 (1963). [6] A. Leutbecher, Math. Zeit. 100, 183 (1967). [7] R. Brooks, J.P. Matelski, in Riemann Surfaces and Related Topics: Pro- ceedings of the 1978 Stony Brook Conference, edited by I.Kra, B. Maskit, Ann. Math. Studies 97 (Princeton Univ. Press, Princeton, N.J., 1981), p. 65. [8] F.W. Gehring, G.J. Martin, Bull. Am. Math. Soc. 21, 57 (1989). [9] D.Mumford,C.Series,D.Wright,Indra’s pearls: The vision of Felix Klein (Cambridge University Press, Cambridge, 2002). [10] K.Kowalski,J.Rembielinski,PhysicaD99,237(1996);chao-dyn/9801019. [11] K. Kowalski, J. Rembielinski, Chaos, Solitons and Fractals 9, 437 (1998); chao-dyn/9801020. [12] R. Aldrovandi, L.P. Freitas, J. Math. Phys. 39, 5324 (1998). [13] P. Gralewicz, K. Kowalski, Chaos, Solitons and Fractals 14, 563 (2002); math-ph/0002044. [14] N.B. Tufillaro,T. Abbott, J. Reilly, An Experimental Approach to Nonlin- ear Dynamics and Chaos (Addison-Wesley, Redwood City, 1992). [15] J.deVries,ElementsofTopologicalDynamics (Kluwer,Dordrecht,Boston, London, 1993). [16] L. Allen, J.M. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975). 10