Locating periodic orbits by Topological Degree theory

Transcription

1 Locatg perodc orbts by Topologcal Degree theory C.Polymls, G. Servz, Ch. Skokos 3,4, G. Turchett ad M.N. Vrahats 5 Departmet o Physcs, Uversty o Athes, Paepstmopols, GR-5784, Zograos, Athes, Greece Departmet o Physcs, Bologa Uversty, Va Irero 46, I-406 Bologa, Italy ad I.N.F.N. Sezoe d Bologa, Va Irero 46, I-406 Bologa, Italy 3 Departmet o Mathematcs, Dvso o Appled Mathematcs ad Ceter o Research ad Applcatos o Nolear Systems (CRANS), Uversty o Patras, GR- 6500, Patras, Greece 4 Research Ceter or Astroomy, Academy o Athes, 4 Aagostopoulou str., GR- 0673, Athes, Greece 5 Departmet o Mathematcs, Uversty o Patras, GR-60 Patras, Greece ad Uversty o Patras Artcal Itellgece Research Ceter (UPAIRC), Uversty o Patras, GR-60 Patras, Greece ABSTRACT We cosder methods based o the topologcal degree theory to compute perodc orbts o area preservg maps. Numercal appromatos to the Kroecker tegral gve the umber o ed pots o the map provded that the tegrato step s small ''eough''. Sce ay eghborhood o a ed pot the map gets our deret combato o ts algebrac sgs we use pots o a lattce to detect the caddate ed pots by selectg boes whose corers show all combatos o sg. Ths method ad the Kroecker tegral ca be appled to bouded cotuous maps such as the beam-beam map. O the other had they ca ot be appled to maps deed o the torus, such as the stadard map whch has dscotuty curves propagatg by terato, or ubouded maps such as the Héo map. However, the systematc use o the bsecto method talzed o the lattce, eve though uable to detect all ed pots o a gve order, allows us to d a sucet umber o them to provde a clear pcture o the dyamcs, eve or maps o the torus because the dscotuty curves have measure zero. e-mal: hskokos@cc.uoa.gr

2 . The topologcal degree (TD) ad ts computato We cosder the problem o dg the solutos o a system o olear equatos o the orm F () = Θ, where F = (,,, ): D s a ucto rom a doma D to, Θ = (0, 0,,0) Τ ad = (,,, ) T. The above system s equvalet to (,,, ) = 0, (,,, ) = 0, (,,, ) = 0. The topologcal degree (TD) theory gves us ormato o the estece o solutos o the above system, ther umber ad ther ature. Kroecker troduced the cocept o the TD 869, whle Pcard 89 succeeded provdg a theorem or computg the eact umber o solutos. For detals about the TD theory ad ts applcatos we reer the reader to the ollowg papers ad books: Cro (964), Lloyd (978), Vrahats (989, 995), Vrahats et al. (996, 997) ad Mourra et al. (00). Deto. Cosder the ucto F = (,,, ) : D, whch s cotuous o the closure D o D, such that F () Θ or o the boudary b(d ) o D. We also cosder the solutos o equato F () = Θ (where Θ = (0, 0,,0) Τ ), to be smple.e. the determat o the correspodg Jacoba matr (J F ) to be deret rom zero. The the topologcal degree o F at Θ relatve to D s deed as: deg[f, D, Θ ] = - F ( Θ ) sg(det J F ()) = N + - N - () where det J F s the determat o the Jacoba matr o F, sg s the well-kow sg ucto, N + the umber o roots wth detj F >0 ad Ν - the umber o roots wth detj F <0.

3 3 It s evdet that a ozero value o deg[f, D, Θ ] s obtaed the there est at least oe soluto o system F () = Θ wth D (Kroecker s estece theorem). Kroecker s tegral Uder the assumptos o the above deto the deg[f, D, Θ ] ca be computed by: deg[f, D, Θ ] = ) ( d d d A d π ) Γ( ) b(d / - / = + where (-) A (-) = " " " " " ad Γ() s the gamma ucto. Pcard s theorem We cosder the assumptos o the deto o TD. We also cosder the ucto F + = (,,,, + ) : D where

4 4 + = y detj F, + :,,,, y ad D + s the product o D wth a real terval o the y-as cotag y=0. The the eact umber N o the solutos o equato F () = Θ s Ν= deg[f +, D +, Θ + ]. Number o roots or a system o equatos By applyg Pcard s theorem ad Kroecker s the case o a set o two equatos: (, ) = 0, (, ) = 0, () we d that the umber N o roots the doma D = [a,b] [c,d] s gve by: ε Q d d N = (3) π 3/ ε J ) (P d + P d ) + π ( + + b(d ) D where ε a arbtrary postve value, ad P = ( + ) ( - ε J + + ε J ) /, =, Q = J J J where J deotes the determat o the Jacoba matr o F =(, ) Steger s method (Steger 975). Steger s theorem allows us to compute the TD o F at a doma D we kow the sgs o uctos,,, a sucet set o pots o the boudary b(d ) o D..

5 5. The characterstc bsecto method The characterstc bsecto method s based o the characterstc polyhedro cocept or the computato o roots o the equato F () = Θ. The costructo o a sutable -polyhedro, called the characterstc polyhedro, ca be doe as ollows. Let M be the " matr whose rows are ormed by all possble combatos o - ad. Cosder ow a oreted -polyhedro Π, wth vertces V k, k=,...,. I the " matr o sgs assocated wth F ad Π, whose etres are the vectors sgf (V k ) = (sg (V k ), sg (V k ),..., sg (V k )), (4) s detcal to M, possbly ater some permutatos o these rows, the Π s called the characterstc polyhedro relatve to F. I F s cotuous, the, ater some sutable assumptos o the boudary o Π we have: deg[f, Π, Θ + ] = ± 0. (5) So, by applyg Kroeker s estece theorem we coclude that there s at least oe soluto o the system F () = Θ wth Π. To clary the characterstc polyhedro cocept we cosder a ucto F =(, ). Each ucto, =,, separates the space to a umber o deret regos, accordg to ts sg, or some regos <0 ad or the rest >0, =,. Thus, gure (a) we dstgush betwee the regos where <0 ad <0, <0 ad >0, >0 ad >0, >0 ad <0. Clearly, the ollowg combatos o sgs are possble: (-,-), (-,+), (+,+) ad (+,-). Pckg a pot, close to the soluto, rom each rego we costruct a characterstc polyhedro. I ths gure we ca perceve a characterstc ad a ocharacterstc polyhedro Π. For a polyhedro Π to be characterstc all the above combatos o sgs must appear at ts vertces. Based o ths crtero, polyhedro ABDC does ot qualy as a characterstc polyhedro, whereas AEDC does. Net, we descrbe the characterstc bsecto method. Ths method smply amouts to costructg aother reed characterstc polyhedro, by bsectg a kow oe, say Π, order to determe the soluto wth the desred accuracy. We

6 6 compute the mdpot M o a oe-dmesoal edge o Π, e.g. <V,V j >. The edpots o ths oe-dmesoal le segmet are vertces o Π, or whch the correspodg coordates o the vectors, sg F (V ) ad sg F (V j ) der rom each other oly oe etry. To obta aother characterstc polyhedro Π * we compare the sg o F (M) wth that o F (V ) ad F (V j ) ad substtute M or that verte or whch the sgs are detcal. Subsequetly, we reapply the aoremetoed techque to a deret edge (or detals we reer to Vrahats 988a;b, 995). Fgure. (a) The polyhedro ABDC s ocharacterstc whle the polyhedro AEDC s characterstc, (b) Applcato o the characterstc bsecto method to the characterstc polyhedro AEDC, gvg rse to the polyhedra GEDC ad HEDC, whch are also characterstc. To ully comprehed the characterstc bsecto method we llustrate gure (b), ts repettve operato o a characterstc polyhedro Π. Startg rom the edge AE we d ts mdpot G ad the calculate ts vector o sgs, whch s (-,-). Thus, verte G replaces A ad the ew reed polyhedro GEDC, s also characterstc. Applyg the same procedure, we urther ree the polyhedro by cosderg the mdpot H o GC ad checkg the vector o sgs at ths pot. I ths case, ts vector o sgs s (-,-), so that verte G ca be replaced by verte H. Cosequetly, the ew reed polyhedro HEDC s also characterstc. Ths procedure cotues up to the pot that the mdpot o the logest dagoal o the reed polyhedro appromates the root wth a predetermed accuracy.

7 7 3. Applcatos We cosder methods based o the topologcal degree theory to compute perodc orbts o the ollowg area preservg maps: Stadard map (map o the torus T) SM : k = + y - s(π) π k y' = y - s(π) π mod(),, y [-0.5,0.5) (6) Héo map (ubouded map o ) = cos(πω) + (y + ) s(πω) HM : y' = s(πω) + (y + ) cos(πω) (7) Beam-beam map (bouded map o ) - = cos(πω) + (y + - e ) s(πω) BM : - y' = s(πω) + (y + - e ) cos(πω) (8) The perodc orbts o the beam-beam map have bee studed by Polymls et al. (997, 00). Gve a dyamcal map M: {'=g (,y), y'=g (,y)}, the perodc pots o perod p are ed pots o M p ad the zeroes o the ucto: F = M p I = where I s the detty matr. = g = g p p (, y) - (, y) - y (9)

8 8 Color map Oe ca use a color map to spect the geometry o ucto F (9) ad to locate ts zeroes. The color map s created by choosg a lattce o N N pots ad by assocatg to each pot a color chose accordg to the sgs o the uctos, : red or (+,+), gree or (+,-), yellow or (-,+), blue or (-,-) as show gure. A smple algorthm allows to detect the cells, ormed by the lattce o N"N pots, whose vertces have deret colors. A cell s a caddate to have a zero ts teror the correspodg topologcal degree s oud to be deret rom zero. I gures 3 ad 4 we costruct the color map ad apply the above metoed algorthm or locatg perodc orbts o perod 3 or the SM (6) ad o perod 5 or the BM (8). The red crcles deote the posto o the oud perodc orbts. We see that or both maps some perodc orbts were ot oud because some o the our color domas close to the ed pot were very th. O the other had, due to the dscotuty o F, some zeros that do ot correspod to real perodc orbts were oud or the SM (gure 3). Fgure. Sketch o the domas where uctos ad (equato 9) have a dete sg.

9 9 Fgure 3. Stadard map (6) or k=0.9: color map or p=3 teratos o the map computed o a square o N N pots or N=5 (let pael); phase plot o the map (rght pael). The red crcles deote the posto o the zeros o the correspodg ucto (9). Fgure 4. Beam-beam map (8) or ω=0.: color map or p=5 teratos o the map computed o a square o N N pots or N=5 (let pael); phase plot o the map (rght pael). The red crcles deote the posto o the zeros o the correspodg ucto (9).

10 0 Dscotuty curves For maps deed o the torus lke the SM (6), the computato o the TD usg Steger s method or the Kroecker tegral (3) aces a problem due to the presece o dscotuty curves. Ideed the above tegral s deed o a doma where F (9) s cotuous. For the SM the dscotuty curves are the les =-0.5 ad y=-0.5, plotted red ad blue color respectvely at the let pael o gure 5. By applyg the SM map M oce these les are mapped o the red ad blue curves see the rght pael o gure 5. O the tal phase space there est also the dscotuty curves that wll be mapped ater oe terato to the les =-0.5 ad y=-0.5. These curves are plotted black ad gree color respectvely gure 5. These curves ca be produced by applyg the verse SM to the dscotuty les =-0.5 ad y=-0.5. So the dscotuty curves dvde the tal phase space ve cotuous regos marked as I, II, III, IV ad V gure 5. I each rego the computato o the TD ca be perormed accurately by Steger's method or by Kroecker s tegral evaluato. I, however, the boudary o the doma where these procedures are appled, cross a dscotuty curve the results we get are ot correct (gure 6). Fgure 5. The dscotuty curves o the stadard map M (6) dvde the phase space ve cotuous regos (I, II, III, IV, V). I each rego the computato o the TD ca be perormed accurately.

11 Fgure 6. (a) Number o ed pots N evaluated or the SM (6) wth k=0.9 usg the Kroecker s tegral (3), a rectagular doma whose topsde moves. The rectagle ad the dscotuty les are show (b). For the varous rectagles, N should be equal to sce they cota oly ed pot o perod, pot (0,0). The two pots marked by arrows (a) where N devates rom the correct value N=, correspod to y ad y respectvely, where the upper-sde o the rectagular crosses the two dscotuty curves (b). Roots ear the boudary We cosder the smple map F*=(, ) : (,y)=y- 3 /3+, (,y)=y. The les =0, =0 are plotted gure 7(a). The above system o equatos has three roots. The determat o the correspodg Jacoba matr (detj F* ) s postve or root (0,0) ad egatve or roots (- 3,0) ad ( 3,0). I order to study the depedece o the procedure or dg the TD a rego D, wth respect to the dstace o a root rom the boudary o D, we cosder a rectagular o the orm [-a,]"[-,] wth a> 3, show the gure 7(a). Sce ths doma cotas the three roots o system the value o TD s -. We let a= 3+ε wth ε>0 so that the boudary approaches the root as ε 0, as show by the arrow gure 7(a). We compute the TD or deret values o ε by Steger's method, by usg the same umber o pots N o every sde o the rectagle. We deote by gp =4N the smallest umber o grd pots eeded to compute the TD wth certaty. I gure 6(b) we plot log-log scale, gp wth respect to ε (dashed le). The slope o the curve s almost - so that N ε -. The

12 same result holds or ay map whe a root approaches the boudary (the sold le gure 7(b) s obtaed or a smlar eample or the SM (6)). (a) (b) ε e Fgure 7. (a) Plot o the curves y- 3 /3+=0, y=0. (b) Depedece o the umber o terato pots gp, eeded or computg the correct value o the TD a doma, o the dstace ε o a root rom the boudary o the doma, or the set o equatos o (a) (dashed le) ad the SM (cotuous le). Perodc orbts Usg the characterstc bsecto method we were able to compute a sucet umber o the perodc orbts wth perod up to 40 or the BM (gure 8) ad the SM (gure 9). Fgure 8. Perodc orbts up to perod p=40 or the BM (8) or ω=0.4. The ellptc perodc orbts are blue ad the hyperbolc oes are red.

13 3 Fgure 8. Perodc orbts up to perod p=40 or the SM (8) or k=0.9. Deret colors deote deret kd o stablty: the ellptc perodc orbts are blue, the hyperbolc perodc orbts are red ad the hyperbolc wth relecto perodc orbts are pk. The margally stable perodc orbts, havg detj F < 0-6, are gree. 4. Syopss We have studed the applcablty o varous umercal methods, based o the topologcal degree theory, or locatg hgh perod perodc orbts o D area preservg mappgs. I partcular we have used the Kroecker s tegral ad appled the Steger s method or dg the TD a bouded rego o the phase space. I the TD has a o-zero value we kow that there est at least oe perodc orbt the correspodg rego. The computato o the TD or a approprate set o equatos allows us to d the eact umber o perodc orbts. We also appled the characterstc bsecto method o a mesh the phase space or locatg the varous ed pots. The ma advatage o all these methods s that they are ot aected by accuracy problems computg the eact values o the varous uctos used, sce, the oly computable ormato eeded s the algebrac sgs o these values. We have appled the above-metoed methods to D symplectc mappgs deed o R ad o the torus T. The methods or computg the TD are appled to cotuous regos o the phase space, so ther use or maps o the torus s lmted to

14 4 regos where o dscotuty curves est. O the other had the characterstc bsecto method proved to be very ecet or all deret types o mappgs, sce, t allowed us to compute a bg racto o the real ed pots o perod up to 40 reasoable computatoal tmes. Fally we beleve that ths method ca be eteded also to hgher dmesoal maps. Reereces. Cro J. (964), Fed pots ad topologcal degree olear aalyss, Mathematcal Surveys No., Amer. Math. Soc., Provdece, Rhode Islad.. Lloyd N. G. (978), Degree Theory, Cambrdge Uversty Press, Cambrdge. 3. Mourra B., Vrahats M. N. & Yakoubsoh J. C. (00), J. Complety, 8, Polymls C., Servz G. & Skokos Ch. (997), Cel. Mech. Dy. Astro., 66, Polymls C., Skokos Ch., Kollas G., Servz G. & Turchett G. (000), J. Phys. A, 33, Steger F. (975), Numer. Math., 5, Vrahats M.N. (988a), ACM Tras. Math. Sotware, 4, Vrahats M.N. (988b), ACM Tras. Math. Sotware, 4, Vrahats M. N. (989), Proc. Amer. Math. Soc., 07, Vrahats M. N. (995), J. Comp. Phys., 9, 05.. Vrahats M. N., Bouts T. C. & Kollma M. (996), Iter. J. Burc. Chaos, 6, 45.. Vrahats M.N., Islker H. & Bouts T. C. (997), Iter. J. Burc. Chaos, 7, 707.