APPROXIMATE LIMIT CYCLES FOR THE RAYLEIGH MODEL

Transcription

1 ROMAI J, 4, 228, 73 8 APPROXIMATE LIMIT CYCLES FOR THE RAYLEIGH MODEL Adelin Georgescu, Petre Băzăvn, Mihel Sterpu Acdemy of Romnin Scientists, Buchrest Deprtment of Mthemtics nd Computer Science, University of Criov Deprtment of Mthemtics nd Computer Science, University of Criov delingeorgescu@yhoocom, bzvn@yhoocom, mihels@centrlucvro Abstrct By using symptotic methods the pproximte limit cycle of the periodiclly forced Ryleigh oscilltor is deduced for g, where g is the mplitude of the forcing The limiting limit cycle s g is lso constructed The theoreticl results gree with the numericl ones deduced by mens of one of the uthors P B method Keywords: Ryleigh eqution, limit cycle, symptotic pproximtion 2 MSC: 34C5, 4A6 THE RAYLEIGH MODEL By the Ryleigh model we men the Cuchy problem x = x, x = y, for the generlized Ryleigh eqution x + 3 x 3 ẋ +x = g sin ωt 2 where x : R R, x = x t, is the unknown stte function, t is the time,,, g nd ω re positive prmeters nd the dot over quntities stnds for the differentition with respect to time The generlized Ryleigh eqution without forcing g = reds ẍ + 3ẋ3 ẋ + x = 3 For = =, the eqution 3 becomes the eqution introduced by Ryleigh in 883 to study the oscilltions of the violin chords The dynm- 73

2 74 Adelin Georgescu, Petre Băzăvn, Mihel Sterpu ics generted by it is closed to the dynmics generted by the Vn der Pol eqution ẍ + x 2 ẋ + x = 4 Indeed, by differentiting the originl Ryleigh eqution with respect to time nd by using the nottion ẋ = y, we obtin just the Vn der Pol eqution In its form 2 the Ryleigh eqution ws introduced in 979 by Diener in order to study the duck French cnrd phenomenon [6], [7] Subsequently, it ws found tht, for certin vlues of the prmeters, the dynmics generted by 2 cn be very complicted, presenting cscdes of bifurctions nd chotic regions In ddition, the model, 2, s well s the vn der Pol model, ws pplied to electronics nd physiology in series of ppers, eg [], [2], [5], [8], [], [2], [3], [5], [6], [7] Especilly it ws investigted numericlly nd theoreticlly the existence nd uniqueness of limit cycles corresponding to periodic oscilltions, the ttrctors the most importnt for pplictions In this pper, by using symptotic methods, [9], [], we extended the studies from [5], [6] concerning the construction of pproximte limit cycles for the Ryleigh model with or without forcing hrmonics s suggested in [4] We neglected the higher The eqution 2 ws trnsformed into the following system of two nonutonomous ordinry differentil equtions ODEs ẋ = y, ẏ = x + 5 y 3 y3 + g sin ωt, or to the following system of three utonomous ODEs in R 3 ẋ = y, ẏ = x + y 3 y3 + g sin z, ż = ω 6 First we used the first order symptotic pproximtion of 5 s g to derive the so-clled verged system ssocited with 5 x= y, y= x + y 3 y3, 7

3 Approximte limit cycles for the Ryleigh model 75 which coincides with the system 5 without forcing With system 7 two-dimensionl dynmicl system is ssocited while the dynmics generted by 5 or 6 is three-dimensionl In [6], it is proved tht the system 7 without forcing possesses limit cycle pssing through the point, 3 In Section 2 we derive n symptotic pproximtion of the limit cycle of 5 nd in Section 3 we give comprtive grphicl representtions of the exct or pproximte limit cycle 2 APPROXIMATE LIMIT CYCLE The unique limit cycle of 7 pssing through, 3 represents n pproximte limit cycle for 5 cycles of the form Another pproximtion corresponds to limit x t = γt + αt cos ωt + βt sin ωt, [ ] yt = γt + [ αt + ωβt] cos ωt + βt ωαt sin ωt Introducing 8 into 5, neglecting the higher hrmonics eg cos 2 ωt = cos 2ωt+ 2 2 nd the derivtives α nd β nd using the nottion γ = δ, we obtin γ= δ, δ + 3 δ3 + δ [ 2 A ] + γ =, ω2 β ωα + α + ωβ [ 4 A + γ2 ] + α =, ω2 α + ωβ + β ωα [ 4 A + γ2 ] + β g =, where A = α + ωβ 2 + β ωα 2 The equilibri of 9 correspond to periodic solutions of period 2π ω of 5, hence to the limit cycles of the ssocited dynmicl system These equilibri stisfy the lgebric system δ =, γ =, ω 2 α + ω [ 4 ω2 α 2 + β 2 ] β =, ω [ 4 ω2 α 2 + β 2 ] α + ω 2 β = g, 8 9

4 76 Adelin Georgescu, Petre Băzăvn, Mihel Sterpu which hs the solution α, β, γ, δ = α, β,, Since γ =, γ =, 8 becomes x t = α cos ωt + β sin ωt, y t = βω cos ωt αω sin ωt, where, with the nottion r 2 = α 2 +β 2, from we deduced α = ωg ω2 r 2, β = g ω 2, = ω ω 2 [ 4 ω2 r 2] 2 nd r is rel solution of eqution [ r 2 ω ω 2 4 ] 2 ω2 r 2 = g 2, 2 4 or, equivlently, r 2 = g 2 3 ie The eqution 2 ws deduced from = Cse g = In this cse 2 implies From we hve ω 2 r 2 = 4, ω 2 = 4 ω =, r = 2 5 α = x, β = y ω, 6 nd this reltion holds lso for the cse g > In ddition, we hve x 2 + y2 ω 2 = r 2 7 The simplest pproximte form for the ellipse corresponds to β = nd, hence, α = r = 2 nd it is x t = 2 cos t, y t = 2 sin t, 8 This is the only pproximte limit cycle considered in [6] It psses through the point 2, However, we do not know priori if this point belongs or not to the pproximte limit cycle In fct, the ellipse through 2, is only one mong the infinity of ellipses of the form x t = x cos t + y ω sin t, y t = y cos t ωx sin 9 t,

5 Approximte limit cycles for the Ryleigh model 77 where x nd y stisfy 7 From this set we must choose the pproximte limit cycle The philosophy under our choice is: the pproximte limit cycle from the cse g = is tht which is equl to the limit cycle from the cse g > s g Cse g > In order to deduce the limiting limit cycle s g we must derive symptotic expnsions for α, β, r, ω nd s g The min ide, suggested by 4, is ω 2 + g + g 2 +, ω 2 r 2 4 4rg 2 + 4rg 2 2 +, s g 2 The expnsion 2 shows tht ω = O s g nd 2 2 implies tht r = O s g too From 2 we hve s g ω 2 + g + g 2 +, g +, ω 2 r r 2 g + ω g g +, g 2 [ g + + g r 4 + 2r2 r2 g] + g 2 [ 2 + r4 + g r 4 + 2r2 r2 + 2 ] + [ + g r 2 ] + 2 Introducing 2 into 2 nd tking into ccount 2 it follows tht, up to terms of order g, we hve 4 ω + r 2 2 g g 2, ie 4 + rg 2 [ g g r 4 + g r 4 + 2r2 r 2 ] g 2 or, equivlently, r [ g 2 + r 4 r 2 + r 4 + 2r2 r 2 ] After the mtching, from 22 we obtin r 4 =, r 4 r 2 + r 4 + 2r2 r = 24 On the other hnd, tking into ccount 23, 24 nd the bove quoted expnsions for α nd β we hve, up to terms in g, s g, α 2 + r4 + r 2 g 2 + rg 2 [ + g r 2 ] 25

6 78 Adelin Georgescu, Petre Băzăvn, Mihel Sterpu [ 4 r 2 + g r 4 ] 2 r2 + r 2, β 2 + r4 4 [ + g [ + g + g r 2 ] ] + r Tking into ccount the initil conditions 6, then 25 nd 26 yield x = 4 r2 27 r 4 r r2 =, 28 y ω + r 2 2 The unique solution of 23, 24, 28 nd 3 reds 2 = 2 4 +, =, r 2 = = 4, 29 = 3 2 +, r2 = 8 +, 3 Numericl computtions reveled tht the pproprite route for is = 2 + Then α = 2 / +, β = 2 +, nd to the pproximte limit cycle for g x t = 2 / + cos g / 4 t + [ y t = 2 [ g g + This ellipse psses through the point this ellipse tends to the ellipse ω ] cos g / / 4 + g sin 4 + ] sin g / 4 + x t = 2 / + cos t t+ t g / 4 + t, , g As g 2 + sin t, y t = 2 + cos t sin t, 34

7 Approximte limit cycles for the Ryleigh model 79 Fig The pproximte limit cycle of 5 Cse g = : =, = 4; b = 3, = 2; c = 3, = 3 Cse g > : d = 3, = 3, ω = 985, g = 2; e = 3, = 3, ω = 99, g = ; f = 3, = 3, ω = 992, g = 5 which psses through the point x, y = 2 /, As expected, the ellipse 34 is of the form 9, where the initil point is 35, nd it is the pproximte limit cycle of 5 s g 3 NUMERICAL RESULTS In fig we give comprtive numericl results on the pproximte limit cycle of 5 In figs -c we represent the orbit through, 3 of 7 blck color nd the orbit 34 They show the good pproximtion relized In figs d-e we give numericl results on the pproximte limit cycle 33 for vrious vlues of the prmeters nd g The numericl method used ws tht from [4], [3] References [] Brnes, B, Grimshw, R, Anlyticl nd numericl studies of the Bonhoeffer vn der Pol system, Aust Mth Soc Series B, ,

8 8 Adelin Georgescu, Petre Băzăvn, Mihel Sterpu [2] Brnes, B, Grimshw, R, Numericl studies of the periodiclly forced Bonhoeffer vn der Pol oscilltor, Int J Bifurction nd Chos, 7, 2 997, [3] Băzăvn, P : The dynmicl system generted by vrible step-size lgorithm for Runge-Kutt methods, Int J Chos Theory Appl, 4, 4 999, 2-28 [4] Băzăvn, P, A vrible step-size lgorithm for Runge-Kutt methods, Rom J Inf Sch Tech, 3, 22, 5-2 [5] Băzăvn, P, Numericl study of the succession of ttrctors in the periodiclly forced Ryleigh system, Annls of the University of Criov, Mth Comp Sci Ser, 33 26, [6] Diener, M, Nessie et les cnrds, IRMA, Strsbourg, 979 [7] Diener, M, Quelques exmples de bifurctions et les cnrds, IRMA, Strsbourg, 979, [8] Flherty, J E, Hoppenstedt, F C, Frequency entrinemment of forced vn der Pol oscilltor, Studies Appl Mth, , 5-5 [9] Georgescu, A, Aproximţii simptotice, Ed Tehnică, Bucureşti, 989 [] Georgescu, A, Asymptotic tretment of differentil equtions, Chpmn & Hll, London, 995 [] Jordn, D W, Smith, P, Nonliner ordinry differentil equtions, Clrendon, Oxford, 989 [2] Mettin, R, Prlitz, U, Luterborn, W, Bifurction structure of the driven vn der Pol oscilltor, Int J Bifurction nd Chos, 3, 6 993, [3] Rjsekr, S, Lkshmnn, M, Period-doubling bifurctions, chos, phse-locking nd the devils stircse in the Bonhoeffer vn der Pol oscilltor, Physic D, , [4] Reithmeier, E, Periodic solutions of nonliner dynmicl systems, Springer Berlin, 99 [5] Sterpu, M, Băzăvn, P, Study on Ryleigh eqution, Bul St Univ Piteşti, Seri Mtemtică-Informtică, 3 999, [6] Sterpu, M, Georgescu, A, Băzăvn, P, Dynmics generted by the generlized Ryleigh eqution II Periodic solutions, Mthemticl Reports, 252, 32, [7] Wng, W, Bifurctions nd Chos of the Bonhoeffer vn der Pol model, J Phys A: Mth Gen, , L627-L632