7 Local bifurcation theory

Transcription

1 7 Locl bifrction theory In or liner tbility nlyi of the Eckh eqtion we w tht netrl crve i generted, ketched gin below. k σ<0 σ=0 n=1 σ>0 A the control prmeter R i moothly vried, the point t which σ = 0 (t ny fied k) define the tbility threhold or bifrction point t which the be flow witche from being linerly tble to linerly ntble with repect to pertrbtion of wvevector k. A we will how in Sec. 8, in the regime of liner intbility, nonliner term in the Eckh eqtion ct to retbilie the ytem omewht. Bece of thi, the finl nonliner tte i not too fr wy from the originl be tte. (Recll the digrm on pge 3.) To decribe the bifrction flly, thee nonliner effect mt clerly be tken into ccont. In thi ection, we introdce the generl theory of bifrction in the contet of ome impler eqtion. For or prpoe, ech of thee cn be viewed the implet tndrd eqtion to cptre given type of bifrction (ddlenode, trncriticl etc.). More fndmentlly, thogh, even the mot complicted eqtion cn be hown to redce to one of thee tndrd form when epnded bot bifrction point. In Sec. 8 below, for emple, we how tht the Eckh eqtion ehibit pitchfork bifrction t the R cm,k cm, decribed by imple eqtion of the form (30). With thee remrk in mind, we now introdce ech type of bifrction in trn. R 7.1 The ddlenode bifrction Conider the dynmicl ytem defined by d = 2, for, rel. (16) Here i control prmeter tht cn be tned eternlly. A tedy tte oltion (d/ = 0) i imply = B = ±. (17) Therefore, for < 0 we hve no rel oltion. > 0 we hve two rel oltion. We now conider ech of the two oltion for > 0, nd emine their liner tbility in the l wy. Firt, we dd mll pertrbtion: = B +. (18) 8

2 Sbtitting thi into the governing eqtion (16), we get d = ( 2 B ) 2 B 2. (19) The term in brcket on the RHS i trivilly zero, from (17). At firt order in the pertrbtion,, we therefore hve with oltion From thi, we ee tht for B = +, 0 t (liner tbility); d = 2 B, (20) (t) = Aep( 2 B t). (21) for B =, t (liner intbility). A ketched in the bifrction digrm below, therefore, the ddlenode bifrction t = 0 correpond to the cretion of two new oltion brnche. One of thee i linerly tble, the other linerly ntble. + 1/2 tble ntble 1/2 7.2 The trncriticl bifrction Conider the dynmicl ytem d = b2 for,,b rel. (22) Agin, nd b re control prmeter. We cn find two tedy tte (d/ = 0) to thi ytem = B1 = 0,b. = B2 = /b,b (b 0). We now emine the liner tbility of ech of thee tte in trn, following the l procedre. Strting with tte B1, we dd mll pertrbtion = B1 +. (23) 9

3 Thi give with the lineried form Thi h the oltion d = b 2 (24) d =. (25) (t) = Aep(t). (26) At liner order, therefore, pertrbtion grow for > 0 nd decy for < 0. So tte B1 = 0 i linerly ntble for > 0, nd tte B1 = 0 i linerly tble for < 0. Now conider the liner tbility of the econd tte B2. A l we write = B2 +. (27) Sbtitting thi into the eqtion of motion (22) nd lineriing, we get d = 2b B2 ( ) = 2b b =. (28) (Do the linerition n eercie.) Thi h the oltion (t) = Aep( t), (29) giving eponentil growth for < 0 nd decy for > 0. Th we ee tht tte B2 = /b i linerly ntble for < 0, nd tte B2 = /b i linerly tble for > 0. Thee finding re mmried in the following bifrction digrm. The bifrction t = 0 correpond to n echnge of tbilitie between the two oltion brnche. =0 tble ntble =/b 10

4 7.3 The pitchfork bifrction Conider the dynmicl ytem defined by d = b3 for,,b rel. (30) A l, nd b re eternl control prmeter. Stedy tte, for which d/ = 0, re follow: = B1 = 0, (31) = B2 = + /b for /b > 0, (32) = B3 = /b for /b > 0. (33) So tte B2 nd B3 only eit for > 0 if b > 0; nd for < 0 if b < 0. In drwing or bifrction digrm below, therefore, we will conider the ce b > 0 eprtely from the ce b < 0. A l, we now emine the liner tbility of ech of thee tedy tte in trn. (Thi cn be done for generl b.) Firt we write nd find the lineried eqtion with the oltion So we ee tht = B1 + (34) d tte B1 = 0 i linerly ntble for > 0, nd tte B1 = 0 i linerly tble for < 0. =, (35) = Aep(t). (36) The liner tbility of tte = B2 nd = B3 cn be conidered together. Setting = ± /b + (37) we get, t liner order in, the eqtion with the oltion in which d = 3b2 B (38) = Aep(t) (39) = 3b 2 B = 3b b = 2. (40) Th we ee tht tte B2 nd B3 re linerly tble for > 0, nd tte B2 nd B3 re linerly ntble for < 0. 11

5 We now collect thee relt in bifrction digrm in the plne. A noted bove, we will do thi eprtely for b > 0 nd b < 0. From (32) nd (33), we recll tht the tte = B2 nd = B3 only eit for /b > 0. So when b > 0, they only eit for > 0. Given thi, nd the tbility propertie dedced bove, we hve the percriticl pitchfork bifrction digrm ketched below. Nonlinerity h tbiliing inflence in thi ce. In the prticle-in--well nlogy, thi correpond to the bottom right ketch on pge 2. b>0 percriticl pitchfork bifrction tble ntble When b < 0, tte = B2 nd = B3 only eit for < 0. Given thi, nd the tbility propertie dedced bove, we hve the bcriticl pitchfork bifrction digrm ketched below. So nonlinerity h detbiliing inflence in thi ce. In the prticle-in--well nlogy, the left hnd prt of thi plot correpond to the bottom left ketch on pge 2. b<0 bcriticl pitchfork bifrction tble ntble A phyicl emple Pitchfork bifrction re common in phyicl ytem tht poe n nderlying ymmetry. Thi i intitively obvio, bece (30) i invrint nder the trnformtion. One phyicl emple i the o-clled Eler trt. Here we pply n increing lod to verticl trt, ntil it finlly bckle. Right nd 12

6 left bckling re eqivlent: the ymmetry pplie. A detiled nlyi of the problem how tht the ytem doe indeed ffer percriticl pitchfork bifrction t the point of bckling. We will dic ome other phyicl emple in Sec. 8 below. F F 7.4 The Hopf bifrction Conider the dynmicl ytem defined by the two eqtion d = y + ( 2 y 2 ) dy = + ( 2 y 2 )y. (41) for rel,y,. There i trivil tedy tte t = y = 0. To emine it liner tbility, we write = 0 +, y = 0 + ỹ. (42) Sbtitting thi into the defining eqtion (41), nd lineriing, we get d = ỹ +, dỹ = + ỹ. (43) The oltion of thee lineried eqtion h the form ( ) ( ) α = ep(t) + c.c. ỹ β (44) Sbtitting thi into (43), we find the eigenvle nd the eigenvector (α, β) to be determined by the following ytem of liner eqtion α = β + α β = α + β. (45) (Check thi n eercie.) Eliminting α nd β, we find the following eqtion for the eigenvle t ny : ( 2 + 1) = 0, (46) from which it i ey to how tht the eigenvle re = ± i. (47) (In principle, we cold btitte thee bck into (45) to find the correponding eigenvector (α,β), bt do not pre thi here.) Given (44) nd (47), we ee tht 13

7 if > 0 then R() > 0 nd o, ỹ (liner intbility); if < 0 then R() < 0 nd o, ỹ 0 (liner tbility). The fct tht i comple confer new dynmicl fetre not encontered in the previo emple: tht of temporl ocilltion. For < 0, for emple, the progre of nd ỹ in towrd the origin i vi dmped ocilltion, ketched in the left hnd plot, rther thn trightforwrd eponentil decy. y y <0 >0 A in the other bifrction emple, the lo of tbility t = 0 give rie to new oltion for > 0. In thi ce, the new oltion i periodic: = co(t + t 0 ), y = in(t + t 0 ). (48) The ytem orbit rond the limit cycle drwn by the dhed line in the right hnd ketch bove. The bifrction digrm i then follow. y Compring thi to the digrm on pge 12, yo will notice tht it look bit like higher dimenionl verion of percriticl pitchfork bifrction. Indeed, we cn gin clify Hopf bifrction percriticl or bcriticl, ccording to whether the nonlinerity i detbiliing or tbiliing repectively. 14

8 7.5 Imperfection theory / trctrl tbility A noted erlier, pitchfork bifrction re common in ytem tht poe n nderlying ymmetry: in the nottion ed here. In mny rel ittion, however, the ymmetry i only pproimte: imperfection led to light difference between left nd right (or whtever the relevnt oppoite generlied diplcement re). In thi ection, we re concerned with wht hppen when ch mll imperfection re preent. Conider lightly imperfect verion of (30), in which we chooe to et b = 1. d = 3 δ. (49) Here δ, which i med mll, i mere of the degree of imperfection preent. If δ = 0 we hve tedy tte t = 0 nd = ±, with pitchfork bifrction t = 0, conidered previoly. When δ 0, however, we hve tedy tte for = 2 + δ/, nd the bifrction digrm i modified follow: δ=0 δ>0 δ<0 Conider now lightly imperfect verion of (22), in which we chooe to et b = 1: d = 2 δ. (50) Agin, for δ = 0 we hve tedy tte t = 0, =, nd trncriticl [ bifrction t = 0. For δ 0, however, we hve tedy tte t = 1 2 ± ] 2 4δ. For mll δ, the bifrction digrm i modified follow. In prticlr, we note tht if δ > 0 then there i no tedy oltion for 2 < 4δ. δ=0 δ<0 δ>0 2δ 1/2 Both the pitchfork nd trncriticl bifrction re id to be trctrlly ntble, ince they ffer qlittive topologicl chnge when the governing eqtion i pertrbed lightly. 15

9 7.6 Bifrction in the Lorentz eqtion In thi ection, we conider the bifrction tht re ehibited by the Lorentz eqtion d = σ(y ), dy = r y z, dz dz = bz + y. (51) A l,,y,z re rel dynmicl vrible; nd σ,r,b re control prmeter, which we tke to be rel nd poitive. Throghot we will me σ,b to be fied, nd work with r the ingle control prmeter to be vried. The Lorentz eqtion rie in modelling convection in verticl tor, ketched below. We do not dic thi phyicl motivtion ny frther here: detil cn be fond in Phyicl Flid Dynmic by Tritton if yo re intereted. T=T 0+ Tz z In wht follow, or im will be firt to find ttionry tte of the Lorentz eqtion nd then to emine the liner tbility of thee tte. In doing o, we hll demontrte the eitence of percriticl pitchfork bifrction nd bcriticl Hopf bifrction in the model. Finlly, we will briefly dic the poible cenrio tht rie following the lo of tbility t bcriticl bifrction, in which there i no nerby nonliner tte to ettle to Sttionry tte By inpection, we cn eily ee tht there i trivil ttionry tte Another ttionry tte cn be fond follow ( B1,y B1,z B1 ) = (0,0,0). (52) d = 0 give = y, dy = 0 give (r 1) z = 0, dz = 0 give bz + 2 = 0. (53) From the econd of thee we get z = r 1. Ptting thi into the third, we get 2 = b(r 1). Combined with the firt, = y, we get finlly the ttionry tte ( B2,y B2,z B2 ) = (± b(r 1), ± b(r 1),r 1). (54) 16

10 Thee ttionry tte re collected on bifrction digrm follow. or y +[b(r 1)] 1/2 r r=1 [b(r 1)] 1/ Liner tbility We now emine the liner tbility of ech of thee ttionry tte. A l, we et = B +, y = y B + ỹ, z = z B + z nd linerie the eqtion in,ỹ, z. Thi give d dỹ d z = σ(ỹ ), = r ỹ B z z B, = b z + B ỹ + y B. (55) For the trivil be tte ( B1,y B1,z B1 ) = (0,0,0), thee redce to d = σ(ỹ ), dỹ = r ỹ, d z = b z. (56) The dynmic of z i trivil: the third eqtion give imple eponentil decy, z = γ ep( bt) where γ i contnt. The eqtion for nd ỹ re copled. We therefore eek oltion of the form = α ep(t) nd ỹ = β ep(t). In doing o, we obtin α = σ(β α), β = rα β. (57) Thi liner eigenvle problem h nontrivil oltion if nd o if + σ σ r + 1 = 0 (58) ( + σ)( + 1) σr = 0. (59) Solving thi qdrtic eqtion for give = 1 { } (σ + 1) ± (σ + 1) 2 2 4σ(1 r). (60) Thi give R < 0 (liner tbility) for r < 1 nd R > 0 (liner intbility) for r > 1. The bifrction t r = 1 i percriticl pitchfork. 17

11 We now nlye the liner tbility of the tte ( B2,y B2,z B2 ). For r jt greter thn 1, we epect thi to be linerly tble, conitent with the percriticl pitchfork bifrction tht we hve jt diced bove t r = 1. The following nlyi will confirm thi, bt will lo revel econdry intbility in the form of bcriticl Hopf bifrction t vle r = r crit > 1, to be determined. Inerting ( B2,y B2,z B2 ) into (55), nd eeking oltion to the relting eqtion et in the form = α ep(t), ỹ = β ep(t), z = γ ep(t), (61) we find the following polynomil eqtion for the eigenvle 3 + (σ + b + 1) 2 + b(σ + r) + 2bσ(r 1) = 0. (62) One cn how tht the only poibility in thi ce i Hopf bifrction: i.e. tht the eigenvle h non-zero imginry prt I 0 t the bifrction point where the rel prt chnge ign, R = 0. So we inert oltion in the form = iω for ω rel into (62). Tking rel nd imginry prt, we then get ω 3 + b(σ + r)ω = 0 (63) nd ω 2 (σ + b + 1) + 2bσ(r 1) = 0. (64) From (64) we get ω = ± 2bσ(r 1) σ + b + 1 for r > 1. (65) Combining thi with the reqirement from (63) tht (for ω 0) ω 2 = b(σ + r) (66) we get 2bσ(r 1) = b(σ + r)(σ + b + 1), (67) which cn be rerrnged to give r crit = σ(3 + b + σ)/(σ b 1) (68) Thi Hopf bifrction cn be hown to be bcriticl. Collecting ll the bove relt together, we get finlly the following bifrction digrm. or y = tble r = ntble r=1 r=rcrit 18

12 7.6.3 Dynmicl evoltion beyond bcriticl bifrction In the bifrction digrm ketched bove, we diced the eitence of bcriticl bifrction t r = r crit. Wht hppen for r > r crit, where tbility i lot nd there i no nerby nonliner tte to go to? In generl, everl cenrio re poible: Evoltion to infinity, typiclly indicting brekdown of the model. Evoltion to non-locl fied point. Evoltion to non-locl periodic or qi-periodic tte. Evoltion to trnge ttrctor, leding to chotic dynmic. In the Lorentz eqtion jt diced, the lt of thee cenrio occr for r > r crit. 19