Dynamics of nonlinear resonances in Hamiltonian systems

Dynamics of nonlinear resonances in Hamil tonian systems Miguel D. Bustama nt e † and Elena Ka rtashov a ∗ † Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK ∗ RISC, J.Kepler University, Li nz 4040, Aus tria It is w ell kn own that the dynamics of a H amiltonian system dep end s crucially on whether or not it p ossesses nonlinear resonances. In the generic case, the set of n on linear reso nances consists of indep endent clusters of resonantly interacting mo des, describ ed by a few low -dimensional dynamical systems. W e show that 1) most frequently met clusters are describ ed by in tegrable dynamical sy s- tems, and 2) construction of clusters can be u sed as the base for the Clipping metho d, su bstantial ly more effective for these systems than the Galerkin metho d. The results can b e used directly for system with cubic Hamiltonian. P A CS n um bers: 47.10.Df, 47.10.Fg, 02.70.Dh 1. In tro duction. A notion of r esonance runs through all our life. Without r e sonance we wouldn’t ha ve radio, television, music, etc. The gener al prop er ties of linear resonances are quite w ell-known; their nonlinea r coun ter- part is substan tia lly less studied though in ter est in un- derstanding nonlinear resonances is e normous. F amous exp eriments o f T esla sho w ho w disa s trous resonances can be: he studied exp erimentally vibrations of a n iron col- umn which ran down w ard in to the foundatio n of the building, and cause d sort o f a small ea rthquake in Man- hattan, with smashed windows a nd swa y ed buildings [1]. Another example is T acoma Nar rows Bridge which tore itself apa rt and co llapsed (in 1940 ) under a wind of only 42 mph, though designed for winds of 12 0 mph. Nonlinear resonances a re ubiquito us in ph y sics. Eu- ler equations, reg a rded with v arious boundar y conditions and sp ecific v alues of some par a meters, describ e a n eno r- mous nu m ber of nonlinear disp ers ive wa v e sy stems (cap- illary wa v es, surface water wav es , atmospheric planeta ry wa ves, drift wa ves in plasma, etc.) all po ssessing non- linear resonances [2]. Nonlinear resonances app ear in a great amount of t ypical mec hanica l systems such as a n infinite straight bar , a circula r ring, and a flat pla te [3]. The so -called “ nonlinear reso nance jump”, impor tant for the analysis of a turbine gov ernor p ositioning system of hydroelectric pow er plan ts, can cause s evere damage to the mec hanical, h ydraulic and electric a l systems [4]. It was rece ntly established that nonlinear resonance is the dominant mechanism b ehind outer ioniza tion and energy absorption in near infrared la ser-dr iven rare-gas or metal clusters [5 ]. The c haracteristic r esonant freq uencies o b- served in accr etion disks allow a stronomer s to determine whether the ob ject is a black hole, a neutron star, or a quark sta r [6]. Thermally induced v ariations o f the he- lium die le ctric p er mittivit y in super conductors are due to microw a ve nonlinear r esonances [7]. T empora l pro cessing in the central auditory nerv ous sys tem analy z es so unds using netw orks of no nlinear neural resonators [8]. The non-linear resonant resp onse of biological tissue to the action of an electroma gnetic field is used to in vestigate cases o f susp ected disease or cancer [9]. The v ery sp ecia l role of reso nant solutions of nonlin- ear ordinary differential equatio ns (ODEs) has bee n first inv estiga ted by Poincar´ e at the end of the 19th ce n tury . Poincar´ e prov ed that if a nonlinear ODE has no reso- nance solutions, then it can be linea rized by an in v ertible change of v ar iables (for details see [10] and refs. ther ein). This simplifies bo th a nalytical and n umer ical investiga- tions of the origina l nonlinea r equation, allows for the int ro duction of corresp o nding norma l forms of ODEs, etc. In the middle of the 2 0th century , Poincar´ e’s ap- proach has been genera lized to the c ase of nonlinear par- tial differential equations (PDEs ) yielding what is nowa- days known as K AM theory ([11]–[1 5] and o thers). This theory allows us to transform a nonlinea r disper s ive PDE int o a Ha miltonian equation of motion in F ourier spac e [16], i ˙ a k = ∂ H /∂ a ∗ k , (1) where a k is the a mplitude o f the F ourier mo de co rre- sp onding to the wa vevector k and the Hamiltonian H is represented as an expansion in p owers H j which ar e prop ortiona l to the pr o duct of j a mplitudes a k . In this Letter w e are go ing to consider ex pa nsions of Hamilto- nians up to th ird order in w ave a mplitude, i.e. a cubic Hamiltonian of the form H 3 = X k 1 , k 2 , k 3 V 1 23 a ∗ 1 a 2 a 3 δ 1 23 + complex conj. , where for brevity we in tro duced the notation a j ≡ a k j and δ 1 23 ≡ δ ( k 1 − k 2 − k 3 ) is the Kroneck er symbo l. If H 3 6 = 0, three-w av e pro ce s s is do minant and the main contribution to the nonlinear ev olution comes from the wa ves satisfying the following reso nance conditions : ( ω ( k 1 ) + ω ( k 2 ) − ω ( k 3 ) = Ω , k 1 + k 2 − k 3 = 0 , where ω ( k ) is a disper sion rela tion for the linear wa ve frequency and Ω ≥ 0 is called reso nance width. 2 If Ω > 0, the equa tion o f motion ( 1 ) turns into i ˙ A k = ω k A k + X k 1 , k 2 V k 12 A 1 A 2 δ k 12 + 2 V 1 ∗ k 2 A 1 A ∗ 2 δ 1 k 2 . (2) If Ω = 0, the equa tion o f motion ( 1 ) turns into i ˙ B k = P k 1 , k 2  V k 12 B 1 B 2 δ k 12 δ ( ω k − ω 1 − ω 2 ) +2 V 1 ∗ k 2 B 1 B ∗ 2 δ 1 k 3 δ ( ω 1 − ω k − ω 2 )  . (3) The co-ex istence of these t w o substantially differen t t yp e s of w av e interactions, describ ed by E qs.( 2 ) a nd ( 3 ), ha s been observed in n umerica l sim ulations [17] and prov e n analytica lly in the frame of the k inematic t wo-la y er mo del o f la minated tur bulence [18]. Dynamics of the la y er ( 2 ) is describ ed b y wav e kinetic equations and is well studied [1 6]. Dynamics of the lay er ( 3 ) is practically not studied though a lo t of preliminary results is already kno w n. Namely , the lay e r ( 3 ) is describ ed by a few indep endent wa v e clusters formed by the wav es which are in exa ct nonlinear r esonance [1 9]. The corres p o nding s o lutions of ( 2 ) can b e computed b y the sp ecially develop ed q-class metho d, pre sented in [20] and implemen ted in [21]. The g eneral form of dynamical systems describing resonant clusters can also b e found algorithmica lly [22], as well as co e fficie nts of dynamical systems [23]. Mo reov er, as it was demonstrated in [24] (n umerically) and in [25] (analytica lly), these cluster s “survive” for small enough but non-ze ro Ω . T he main goal o f this Letter is to study the dynamics of the most frequently met resonant clusters. 2. Clusters. In this Letter we present some analyti- cal and n umerica l results for the three most commonly met dynamica l systems c o rresp o nding to non-isomorphic clusters of nonlinear reso nances – a t riad , a kite , and a butterfly consisting of 3, 4 and 5 complex v ar iables cor- resp ondingly . The dynamical system for a triad has the form ˙ B 1 = Z B ∗ 2 B 3 , ˙ B 2 = Z B ∗ 1 B 3 , ˙ B 3 = − Z B 1 B 2 . (4) A kite consists of t w o triads a and b, with wa ve ampli- tudes B j a , B j b , j = 1 , 2 , 3, connected via t w o co mmon mo des. Analogo usly to [26], one ca n point out 4 types of kites according to the prope rties of connecting mo des. F or our consider ations, this is not imp ortant: the general metho d to study in teg r ability of kites will be the same. F or the co ncreteness o f pre s entation, in this Letter a kite with B 1 a = B 1 b and B 2 a = B 2 b has bee n chosen:      ˙ B 1 a = B ∗ 2 a ( Z a B 3 a + Z b B 3 b ) , ˙ B 2 a = B ∗ 1 a ( Z a B 3 a + Z b B 3 b ) , ˙ B 3 a = − Z a B 1 a B 2 a , ˙ B 3 b = − Z b B 1 a B 2 a . (5) A butterfly consists o f t wo tria ds a and b, with wav e amplitudes B j a , B j b , j = 1 , 2 , 3, co nnec ted via one common mo de. As it was shown in [26], there exist 3 different types of butterflies, a c c ording to the choice o f the connecting mo de. Let us take, for instance, B 1 a = B 1 b ( ≡ B 1 ) . The corresp o nding dyna mica l system is then as follows:      ˙ B 1 = Z a B ∗ 2 a B 3 a + Z b B ∗ 2 b B 3 b , ˙ B 2 a = Z a B ∗ 1 B 3 a , ˙ B 2 b = Z b B ∗ 1 B 3 b , ˙ B 3 a = − Z a B 1 B 2 a , ˙ B 3 b = − Z b B 1 B 2 b . (6) 3. In teg rabilit y of resonance clusters. F ro m here on, general notations a nd terminology will follo w O lver’s bo ok [27]. W e use hereafter Einstein conven tion on r e- pea ted indices and f ,i ≡ ∂ f /∂ x i . Co nsider a general N - dimensional system of autonomous evolution eq uations of the for m: dx i dt ( t ) = ∆ i ( x j ( t )) , i = 1 , . . . , N . (7) An y scalar function f ( x i , t ) that sa tisfies d dt  f ( x i ( t ) , t )  = ∂ ∂ t f + ∆ i f ,i = 0 is called a c on- servation law in [2 7]. It is easy to see that this definition gives us t w o t yp es of conserv ation la ws. The first t ype is the standar d notion used in classical physics: the conserv a tion law is of the for m f ( x i ), i.e. it does not depe nd explicitly on time. The second t ype lo o ks more like a mathema tical trick: it is of the form f ( x i , t ), where the time dependence is explicit. In this Letter w e will b e int erested in b oth t yp e s of co nserv atio n law. W e claim that they a re both physically imp or tant and to sho w that we present an illustrative exa mple from Mechanics. Let us regard the damp ed harmonic oscillato r. The equations of motion in non-dimensio nal form can be written as: ˙ q = p , ˙ p = − q − αp, (8) where α ≥ 0 is the damping co efficient. If this co effi- cient is equal to zero , α = 0, then the tota l energy of the system E ( q , p ) = 1 / 2  p 2 + q 2  is a co nserv ation law of the first type. If α > 0, then the system do es no t conserve the ene r gy anymore but one can still define a conserved quant it y which is a g eneralizatio n of E for the ca se α > 0. This new quantit y F ( q , p, t ) = exp( α t )  p 2 + q 2 + α p q  is a co nserv ation la w of the second t yp e. This means that for an arbitrary solution q ( t ) , p ( t ) of the system ( 8 ) w e hav e d dt  exp( α t )  p ( t ) 2 + q ( t ) 2 + α p ( t ) q ( t )  = 0 3 These tw o types o f c onserv a tio n law are very differen t but complementary . While the fir st t ype, E ( q , p ) in the case α = 0, defines wher e the motion ta kes place, the second type, F ( q , p, t ) in the case α > 0, defines how the motion ta kes place. In other words, the first t yp e defines orbits of the dynamical system ( 8 ) and the second type defines its motion within the orbit. T o k eep in mind the differe nc e b etw een thes e tw o t ypes of conserv atio n laws, we will call the fir st type just a conserv a tion law (CL), and we will call the seco nd type a dynamic al inva riant . W e say tha t Sys.( 7 ) is inte gr able if there ar e N functionally independent dynamical inv a riants. Ob- viously , if Sys.( 7 ) po ssesses ( N − 1 ) functionally independent CLs , then it is constrained to mo ve along a 1- dimens io nal manifold, and the wa y it mov es is dictated b y 1 dynamical in v ariant. This dynamical inv ariant can b e obtained from the knowledge of the ( N − 1) CLs and the explicit form of the Sys.( 7 ), i.e. Sys.( 7 ) is integrable then. It follows from the Theo- rem b elow that in many ca ses the knowledge of only ( N − 2 ) CLs is enoug h for the integrability of the Sys.( 7 ). Theorem. L et us assu me that the system ( 7 ) p os- sesses a standar d Liouvil le volume density ρ ( x i ) : ( ρ ∆ i ) ,i = 0 , and ( N − 2) functional ly indep en dent CLs, H 1 , . . . , H N − 2 . Then a new CL c an b e c onstructe d, which is functional ly indep endent of the original ones, and t her efor e the system is inte gr able. The (length y ) proof follows from the existence of a Poisson brack et for the original Sys.( 7 ) and is an extension of the general appro ach used in [28] for three dimensional first order autonomous equa tio ns. The pro of is cons tructive and allows us to find the e x plicit form of a new CL fo r dynamical systems of the fo rm ( 4 ),( 5 ),( 6 ), etc. In the examples below, we a lwa y s need to eliminate so- called slav e pha ses, whic h corr esp onds to the w ell-known order reduction in Hamiltonian systems [29]. The num- ber N used be low corresp onds to the effective num b er of degrees of freedom after this reduction has b een p er - formed. Int egrability of a triad , dynamical sy s tem ( 4 ), is a well- known fact (e.g. [3 0]) and its tw o conser v ation laws are I 23 = | B 2 | 2 + | B 3 | 2 , I 13 = | B 1 | 2 + | B 3 | 2 . Sys.( 4 ) has been used for a preliminary chec k of our metho d; in this case N = 4. The metho d can thus be applied and we obtain the following CL: I T = Im( B 1 B 2 B ∗ 3 ) , together with the time-dependent dynamical in v a riant of the form: S 0 = Z t − F  arcsin   R 3 − v R 3 − R 2  1 / 2  ,  R 3 − R 2 R 3 − R 1  1 / 2  2 1 / 2 ( R 3 − R 1 ) 1 / 2 ( I 2 13 − I 13 I 23 + I 2 23 ) 1 / 4 . Here F is the elliptic in tegral of the fir s t k ind, R 1 < R 2 < R 3 are the thre e real r o ots of the p o lynomial x 3 + x 2 = 2 / 2 7 − (27 I 2 T − ( I 13 + I 23 )( I 13 − 2 I 23 )( I 23 − 2 I 13 )) / 27( I 2 13 − I 13 I 23 + I 2 23 ) 3 / 2 and v = | B 1 | 2 − (2 I 13 − I 23 + ( I 2 13 − I 13 I 23 + I 2 23 ) 1 / 2 ) / 3 is a lwa ys within the interv al [ R 2 , R 3 ] ∋ 0 . A kite , dynamical sys tem ( 5 ), is als o an integrable s ys- tem. Indeed, after reduction of slav e v aria bles the s y s- tem cor r esp onds to N = 6 and has 5 CLs (2 linear, 2 quadratic, 1 cubic):          L R = Re( Z b B 3 a − Z a B 3 b ) , L I = Im( Z b B 3 a − Z a B 3 b ) , I 1 ab = | B 1 a | 2 + | B 3 a | 2 + | B 3 b | 2 , I 2 ab = | B 2 a | 2 + | B 3 a | 2 + | B 3 b | 2 , I K = Im( B 1 a B 2 a ( Z a B ∗ 3 a + Z b B ∗ 3 b )) , with a dynamical in v a riant that is essentially the same as for a tr iad, S 0 , after replacing Z = Z a + Z b , I T = I K ( Z 2 a + Z 2 b ) / Z 3 , I 13 = I 1 ab ( Z 2 a + Z 2 b ) / Z 2 − ( L 2 R + L 2 I ) / Z 2 , I 23 = I 2 ab ( Z 2 a + Z 2 b ) / Z 2 − ( L 2 R + L 2 I ) / Z 2 . The dynamics of a butterfly is go verned b y Eqs.( 6 ) and its 4 CLs (3 quadratic and 1 cubic) can ea sily b e o bta ined:      I 23 a = | B 2 a | 2 + | B 3 a | 2 , I 23 b = | B 2 b | 2 + | B 3 b | 2 , I ab = | B 1 | 2 + | B 3 a | 2 + | B 3 b | 2 , I 0 = Im( Z a B 1 B 2 a B ∗ 3 a + Z b B 1 B 2 b B ∗ 3 b ) , (9) while a Lio uville volume density is ρ = 1. Notice that all cubic CLs are canonical Hamiltonians for the resp ective triad , kite and butterfly sys tems. F rom now o n w e con- sider the butterfly case when no amplitude is identically zero; o therwise the system would b ecome integrable. The use of standar d amplitude-phase represe ntation B j = C j exp( iθ j ) of the complex amplitudes B j in terms of real amplitudes C j and phases θ j shows immediately that only tw o phase combinations are imp orta nt: ϕ a = θ 1 a + θ 2 a − θ 3 a , ϕ b = θ 1 b + θ 2 b − θ 3 b , a - a nd b -tria d phase s (with the r equirement θ 1 a = θ 1 b which co rresp ond to the chosen r esonance condition). 4 0.75 0.775 0.8 0.825 Α a H t L 1.5 1.55 1.6 1.65 j a H t L 1.2 1.4 1.6 1.8 2 j b H t L 0.75 0.775 0.8 0.825 Α a H t L 1.5 1.55 1.6 1.65 j a H t L 1.2 1.4 1.6 1.8 2 j b H t L 0.75 0.775 0.8 0.825 Α a H t L 1.5 1.55 1.6 1.65 j a H t L 1.2 1.4 1.6 1.8 2 j b H t L FIG. 1: Color online. T o f acilitate view, color h ue of the plot is a linear function of time t , v ary ing from 0 to 1 as t runs through one p erio d. Le ft pane l : Z a = Z b = 10 / 100( integ r abl ecase ) , I ab ≈ 2 . 1608. M iddle panel : Z a = 8 / 100 , Z b = 12 / 100 , I ab ≈ 2 . 2088. Right panel : Z a = 9 / 100 , Z b = 11 / 100 , I ab ≈ 2 . 1 846. This reduces five complex equations ( 6 ) to only four re al ones: dC 3 a dt = − Z a C 1 C 2 a cos ϕ a , (10) dC 3 b dt = − Z b C 1 C 2 b cos ϕ b , (11) dϕ a dt = Z a C 1  C 2 a C 3 a − C 3 a C 2 a  sin ϕ a − I 0 ( C 1 ) 2 , (12) dϕ b dt = Z b C 1  C 2 b C 3 b − C 3 b C 2 b  sin ϕ b − I 0 ( C 1 ) 2 . (13) The cubic CL reads I 0 = C 1 ( Z a C 2 a C 3 a sin ϕ a + Z b C 2 b C 3 b sin ϕ b ) (14) in terms of the a mplitudes and phases. This means that the dynamics o f a butterfly cluster is, in the g eneric case, confined to a 3 - dimensional manifold. Below we reg a rd a few particular cas e s in which Sys.( 6 ) is int egrable. Example 1 : Real ampli tudes, ϕ a = ϕ b = 0 . In this case the Hamiltonian I 0 bec omes identically zero while Liouville densit y in co or dina tes C 3 a , C 3 b is ρ ( C 3 a , C 3 b ) = 1 /C 1 C 2 a C 2 b . So in this case the e q uations for the un- known CL H ( C 3 a , C 3 b ) are: ρ ∆ C 3 a = − Z a C 2 b = ∂ ∂ C 3 b H, ρ ∆ C 3 b = − Z b C 2 a = − ∂ ∂ C 3 a H, and fro m Eqs.( 9 ) we rea dily obtain H ( C 3 a , C 3 b ) = Z b arctan ( C 3 a /C 2 a ) − Z a arctan ( C 3 b /C 2 b ) , i.e. Sys.( 6 ) is integrable in this case. Of course, this case is degenerate for I 0 ≡ 0 yields no constra in t on the re - maining indep endent v aria bles C 3 a , C 3 b satisfying equa - tions ( 10 ), ( 11 ). General change of coo rdinates . Going bac k to Eqs.( 10 )–( 14 ), we c a n jump from this deg enerate case to a more generic case by defining new coo rdinates whic h are sugg ested by H ( C 3 a , C 3 b ) . The new co o rdinates ar e to r eplace the amplitudes C 3 a , C 3 b : α a = arc ta n ( C 3 a /C 2 a ) , α b = arc ta n ( C 3 b /C 2 b ) . W e choose the inv er s e transfor mation to b e ( C 2 a = √ I 23 a cos( α a ) , C 3 a = √ I 23 a sin( α a ) , C 2 b = √ I 23 b cos( α b ) , C 3 b = √ I 23 b sin( α b ) , (15) so that the domain for the new v a riables is 0 < α a < π / 2 , 0 < α b < π / 2. F or these new co ordinates, the evolution equations simplify enormously :      dα a dt = − Z a C 1 cos ϕ a , dα b dt = − Z b C 1 cos ϕ b , dϕ a dt = Z a C 1 (cot α a − tan α a ) sin ϕ a − I 0 ( C 1 ) 2 , dϕ b dt = Z b C 1 (cot α b − tan α b ) s in ϕ b − I 0 ( C 1 ) 2 , (16) where the amplitude C 1 > 0 is obtained using eqs .( 9 ): C 1 = q I ab − I 23 a sin 2 α a − I 23 b sin 2 α b (17) and the c ubic CL is now I 0 = C 1 2 ( Z a I 23 a sin(2 α a ) s in( ϕ a ) + Z b I 23 b sin(2 α b ) s in( ϕ b )) . (18) Equations ( 16 )–( 18 ) r epresent the final form of our 3- dimensional general system. Example 2: Compl ex ampl itudes, I 0 = 0 . Here, we just impose the condition I 0 = 0 but the pha s es are otherwise ar bitrary: this case is therefore not degener - ate a nymore and w e have a 3-dimensional system whic h requires the exis tence of only 1 CL in order to be in- tegrable: a CL is A a = sin(2 α a ) s in( ϕ a ) , which can be deduced from E q s.( 16 ). Making use of the Theorem, one can find ano ther CL for this cas e: H new ( C 3 a , C 3 b ) = (1 + Z b / Z a ) a rccos  cos 2 α a √ 1 − A 2 a  − (1 + Z a / Z b ) a rccos  cos 2 α b √ 1 − A 2 b  . 5 0 2 4 6 8 10 Time - 4 - 2 0 2 4 6 Amplitudes FIG. 2: Color online. Chain of 4 triads, real amplitudes for all 9 mo des. Time and amplitudes in non-dimensioned units. Obviously A a and H new are functionally indep endent, i.e. the case I 0 = 0 is in tegrable. Example 3: Complex amplitudes, Z a = Z b . In this case a new CL has the form I 2 0 Z a E = C 2 2 a C 2 3 a + C 2 2 b C 2 3 b + 2 C 2 a C 3 a C 2 b C 3 b cos( ϕ a − ϕ b ) − C 2 1 ( C 2 1 − C 2 2 a + C 2 3 a − C 2 2 b + C 2 3 b ) , (19) which is functiona lly indep endent o f the other known constants of motion. Therefo re, according to the The- orem the case Z b = Z a is integrable. The numerical scheme is pro grammed in Mathematic a with stiffness-switching metho d in single pre c ision. F or arbitra ry Z a , Z b the scheme has b een chec k ed by computing I 0 from eq.( 18 ) at all conseq uent time steps; there is no noticea ble change o f I 0 up to machin e precision. 4. Numerical simulations. T o inv estigate the general b e havior of a butterfly cluster with Z a 6 = Z b , we int egrated directly Eqs.( 16 ),( 17 ) with I 0 computed from Eq.( 18 ) ev aluated at t = 0 and used to check nu merical sch eme afterwards. Some results o f the simulations with Sys.( 16 ) a re presented in Fig. 1 . Initial conditions α a (0) = 78 / 100 , α b (0) = 60 / 100 , ϕ a (0) = 147 / 1 00 , ϕ b (0) = 12 7 / 10 0 and v a lue s o f the constants of motion I 0 = 11 / 200 0 , I 23 a = 4 / 10 0 , I 23 b = 4 / 10 0 are the same for all three parts of Fig. 1 . 3D parametric plots are shown in space ( α a , ϕ a , ϕ b ) with color hue depe nding on time, so the plots ar e effectively 4D. The main g oal of this series o f numerical simulations w as to study changes in the dynamics of a butterfly cluster according to the magnitude of the ratio ζ = Z a / Z b . In Fig. 1 , left pa nel ζ l = 1 , middle panel ζ m = 2 / 3 and right pa nel ζ r = 9 / 11 . As it w a s shown above, the case ζ l = 1 is integrable, a nd one can see the closed tra jectory with p e rio d T l ≈ 2 1 . 7. Quite unexp ectedly , rational ζ m , ζ r pro duce what app ea r to b e p erio dic mo tions and closed tra jector ies with perio ds T m ≈ 53 and T r ≈ 215 corres p o ndingly . A few dozen of simulations made with different rational r atios ζ = Z a / Z b show that p erio dicit y depe nds - or is even defined b y - the co mmensurability of the co efficients Z a and Z b . Fig. 1 shows that ζ m = 2 / 3 gives 2 spikes in one dir ection and 3 spikes in the per p endicular direction, while ζ r = 9 / 1 1 g ives 9 and 11 spikes corres po ndingly; and so on. This does not mean, of course, that in the gener al ca se Z a 6 = Z b solutions of Sys.( 6 ) a re per io dic. But the opp os ite can o nly be prov en analytically for in numerical simulations any choice of the ratio ζ necessa r ily turns into a r ational nu m ber . Some preliminar y s e ries of sim ulations have bee n p erformed with the chains of triads with connection t yp e s as in ( 6 ), with maximum num b er of triads in a chain being 8 (co r resp onds to 17 modes). F or our nu merical simulations, re sonant clusters o f spherical Rossby w av es were taken, with initial (no n- dimensioned) energies of the order of measured atmospheric data, as in [31]. The dynamical system for computatio ns was taken in the original v ariables B j . The case of r eal amplitudes is shown in Fig. 2 : a ll re sonant mo des b ehav e (almost) p erio dically . Non-dimensioned units for time and amplitudes were chosen to illus trate clear ly the characteristic behavior of the amplitudes. 5. Galerkin metho d versus Clipping metho d. Spea king v ery generally , Galerk in metho d (GM) allows one to reduce infinite dimensional s ystems, des crib ed by PDEs , to low dimensional systems of ODEs. Three steps hav e to b e p erformed: a) c hoice of ans atz functions (mo de s ); b) choice o f num b er of modes N ; c) constr uc- tion of the resulting reduced system. While in the cas e of a line ar PDE, F ourier mo des is the natural choice of the ansatz functions, the num b er of mo des is mostly defined by the av ailable computer facilities and the c o nstruction of the corr esp onding ODE is o bvious, the case of a non- line ar PDE is much more involv ed [32]. The Clipping metho d (CM) has be e n introduced in [33] in order to deal with evolutionary disp ersive nonlin- ear PDEs. The idea o f the CM is very simple: under some physically relev ant co nditions, it is enough to re- gard the dynamics of the reso nantly interacting mo des. These r esonant mo des can b e found systematically us- ing the q-c lass metho d [20]. All other modes b ehave a s linear and can b e clipp ed out. Numerical evidence of the effectiveness of this approach was presented in [24], using a pseudo-spectr al model of the barotropic v ortic- it y equation. There , it w as obser ved that the most en- ergetically activ e mo des were the resonant modes (221 mo des were exc ited, among them 3 9 r esonant). These resonant modes appea r in clusters, i.e. a s mall num ber of modes , and the energy of each cluster is conserved (see [24], Fig.1 ). As for each o f the non- resonant mo de s , their energies a re approximately constant during ma ny p erio ds of ener gy exchange of the r esonant mo des ([24], Fig.3). Implemen tation of the q-cla ss metho d for v arious disp e rsion functions [21], together with explicit co n- struction o f the dy na mical s ystems corr esp onding to the exact r esonances [22], a llows one to eliminate the arbitrar iness o f the c hoices at a ll three s teps a), b) and 6 c) of the GM. Instead of one big sy s tem of ODEs of order N , we hav e a few independent systems of ODEs, of order ˜ N ≪ N , and these sy s tems are often in teg rable. Another example can b e found in [2 2]: o cean planetary motions, 128 r esonant mo des a mong 250 0 F ourier harmo nics in the chosen sp ectral doma in, all cluster s and their dynamical systems ar e written o ut explicitly . In doze ns of studied 3 -wa v e sy s tems, the amoun t o f resonant mo des do es not ex ceed 20% of all mo des in the sp ectral domain. 6. Con clusions. Our analysis a nd gener al mathemati- cal results [19] on resonant clusters are v alid for arbitra ry Hamiltonian H j , j ≥ 3 , though computation of clusters in the case j > 3 is mor e inv olved [21]. Clipping method has at lea st three adv antages com- pared to Gale r kin truncation: i) Numerica l schemes in CM can b e truncated at a substantially higher w a ven um- ber than in GM, dep ending not on the co mputer facilities but on s ome physically relev a nt parameters (say , dissipa- tion range of wav enum ber s). ii) Most o f the resulting dynamical systems cor resp onding to each resona nt clus- ter is ana lytically integrable. iii) The solutions obtained from the integrable ca ses c o uld b e used to parameterize the n umerical solutions of non-integrable systems found for bigger c lusters. This work is in pro gress. Last not least. Ev en for o ne specific PDE , it is a highly non-trivial task to prove that Galerkin truncation is a Hamilto nian system a nd to co ns truct additional conserved quantit y [34]. 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