Iraioal Joural of horical ad Mahmaical Physics 4, 4(6: 5-9 DOI: 59/jijmp446 Rig of arg Numbr Muually Coupld Oscillaors Priodic Soluios Vasil G Aglov,*, Dafika z Aglova Dparm Nam of Mahmaics, Uivrsiy of Miig ad Gology S I Rilski, Sofia, Bulgaria Dparm of Mchaics ad Mahmaics, Uivrsiy of Srucural Egirig & Archicur (VSU Karavlov, Sofia, Bulgaria Absrac h papr is dvod o h ivsigaio of priodic rgims of a rig of a larg umbr muually coupld oscillaors wih aciv lms characrizd by a simpl symmric cubic oliariy I coras o h usually accpd approach ladig o Va dr Pol quaios w rduc h origial sysm cribig h rig oscillaor o a firs ordr igro-diffrial o Iroducig a suiabl fucio spac w dfi a opraor acig i his spac ad by fixd poi mhod w prov h basic rsul: a xisc-uiquss of -priodic soluio of h obaid igro-diffrial sysm Fially w giv a umrical xampl Kywor Oscillaors, Coupld oscillaors, Rig of oscillaors, Va dr Pol diffrial quaios, Igro-diffrial sysm, Priodic soluios, Fixd-poi horm Iroducio Oscillaory bhavior is ubiquious i all physical sysms i diffr disciplis ragig from biology ad chmisry o girig ad physics, spcially i lcroic ad opical sysms I radio frqucy ad lighwav commuicaio sysms, oscillaors ar usd for rasformaio sigals ad chal slcio Oscillaors ar also prs i all digial lcroic sysms, which rquir a im rfrc Coupld oscillaors ar oscillaors cocd i such a way ha rgy ca b rasfrrd bw hm Sic abou 96, mahmaical biologiss hav b sudyig simplifid modls of coupld oscillaors ha rai h ssc of hir biological prooyps: pacmakr clls i h har, isuli-scrig clls i h pacras; ad ural works i h brai ad spial cord ha corol such rhyhmic bhavior as brahig, ruig ad chwig Rig of coupld oscillaors is cascadd combiaio of dlay sags, cocd i a clos loop chai ad i has a umbr of applicaios i commuicaio sysms, spcially i aaomic orgas such as h har, isi ad urr cosis of may cllular oscillaors coupld oghr A idal oscillaor would provid a prfc im rfrc, i, a priodic sigal Howvr all physical oscillaors ar corrupd by uird prurbaio ois Hc sigals, grad by physical oscillaors, ar o prfcly priodic Mahmaical basis of coupld oscillaors has b sablishd i [], [] ad muual sychroizaio of a larg * Corrspodig auhor: aglov@mgubg (Vasil G Aglov Publishd oli a hp://jouralsapuborg/ijmp Copyrigh 4 Sciific & Acadmic Publishig All Righs Rsrvd umbr of oscillaors has b ivsigad i [] Vry irsig applicaios ca b foud i [4] ad [5] I [6] h auhors hav ivsigad a rig array of va dr Pol oscillaors ad clarifid ach mod srucur hir rsuls ar basd o h mhod of quival liarizaio of h oliar rms usig Krylov-Bogoliubov mhod A aciv lm is characrizd by a simpl symmric cubic oliariy, ha is, wih V-I characrisic ( ( = I ( u g u g u, ( ( ( g, g > ( =,, N A rig oscillaor is cribd by h followig igro-diffrial sysm: di ( ( ( u u = d ( du ( ( ( i i = C u( s I( u(, d h usually accpd approach (cf [6] is o xclud h curr fucios afr diffriaio ad o obai a scod ordr sysm of va dr Pol diffrial quaios ( ( d u( g g du( u ( ( u d C g d C C u ( u ( = ( =,,, N C C
6 Vasil G Aglov al: Rig of arg Numbr Muually Coupld Oscillaors Priodic Soluios Hr w cosidr h origial igro-diffrial sysm ( ad xclud curr fucios wihou diffriaio So du( ( w rach h followig firs ordr (isad of scod ordr igro-diffrial sysm: ( ( u ( s u ( s u ( s C u ( s g u ( g u (, = d ( =,,, N or du ( ( ( = u ( s u( s u ( s g u( g u( d C Uu (, u, u ( = N (,,, I viw of h boudary codiios for h rig cocio un( = u(, un ( = u( h abov sysm yil h followig sysm of N quaios for N ukow fucios du ( ( ( = un ( s u( s u( s g u( g u( d C Uu (, u, u du ( ( ( = u ( s u( s u ( s g u( g u( d C Uu (, u, u dun ( ( N ( N = un( s un( s u( s g un( g un( d C Uu (, u, u N N saisfyig h iiial codiios u( = u( = = u N ( = W dfi a opraor acig o suiabl fucio spac ad is fixd poi is a priodic soluio of h abov sysm h advaags of our mhod is is simplr chiqu ad obaiig of succssiv approximaios bgiig wih simpl iiial fucios W formula a priodic problm: o fid a priodic soluio ( u (,, un ( o [, ] of h sysm ( o solv h priodic problm w us h mhod from [7] By C [, ] w ma h spac of all diffriabl -priodic fucios wih coiuous drivaivs Firs w iroduc h ss (assumig = m : MU = u( C [, ]: u( d ( k,,,,, m, = = k (
Iraioal Joural of horical ad Mahmaical Physics 4, 4(6: 5-9 7 h s U { ( k ( U : ( }, [, ( ] M = u M u U k k ( k =,,,,, m M* = MU MU M U urs ou io a compl mric spac wih rspc o h mric: N ims { ( ( ( ( u u un u N u u un u N = k u u k u u k un un k u N u N ρ ((,,,,,(,,,, max ρ (,, ρ (,,, ρ (,, ρ (, : k =,,,,, m } whr ( k ρ ( k ρ { ( k } { ( k } ( u, u = max u ( u (: [ k, ], ( u, u = max u ( u (: [ k, ] ( k ( k I is asy o s ( k ρ ( uu, ( uu, ( uu, ρ ρ Iroduc h opraor B as a -upl (,, N( = ( (,, N(, (,, N(,, (,, N( ( B ( u, u, u (, B ( u, u, u (,, B ( u, u, u ( B u u B u u B u u B u u = N N N [, ]( u = u, u = u dfid o vry irval whr Uu (, u, u N N [ k, ( k ], ( k =,,, m by h xprssios u( = u( k = ; ( k k (,, ( : (,, ( k B u u u = U u u u s U ( u, u, u ( s U ( u, u, u ( s d, k k k [ k, ( k ], ( k =,,, m, ( =,,, N, ( ( g g k k k = u ( s u ( s u ( s u ( u ( C C C C C Prlimiary Rsuls Now w formula som usful prlimiary assrios for our ivsigaio mma If ( u,, un M * h F( = u( τ dτ,, N Proof: Idd, w hav F ( = un( τ dτ ar -priodic fucios
8 Vasil G Aglov al: Rig of arg Numbr Muually Coupld Oscillaors Priodic Soluios F ( = u ( τ dτ = u ( τ dτ u ( τ dτ k k k = F ( u ( τ dτ u ( τ dτ u ( τ dτ k k = F ( u ( τ dτ u ( θ = F ( k mma is hus provd mma If ( u,, un M * h Uu (, u, u ( ar -priodic fucios h proof is sraighforward basd o h prvious mma mma For vry i follows s ( u,, un MU MU M U N ims U ( u, u, u ( θ = U ( u, u, u ( θ ( k =,,, k k Proof: W oic ha i viw of mma h fucios Uu (, u, u ( ar -priodic us dfi h fucios s s Γ k( s = Uu (, u, u ( θ ad Γ k ( s = Uu (, u, u ( θ k ad rwri hm i h form η k Γ ( η = Uu (, u, u ( θ, Γ ( η = Uu (, u, u ( θ k k s η k Chagig h variabl τ = θ w obai η k η Γ k ( = Uu (, u, u ( d = Uu (, u, u ( d k Cosquly η θ θ τ τ k η = Uu (, u, u ( τ dτ = Γ ( η k k Γ ( η dη = Γ ( η dη ( k =,,, k k mma is hus provd mma shows ha opraor fucio Bu (,, un ( is - priodic o k mma 4 h iiial valu problm ( has a soluio ( u,, un M * iff h opraor B has a fixd poi ( u,, un M *, ha is,
Iraioal Joural of horical ad Mahmaical Physics 4, 4(6: 5-9 9 ( ( u,, u = B ( u,, u, B ( u,, u,, B ( u,, u N N N N Proof: ( u,, un M * b a priodic soluio of ( h igraig ( w hav: Bu hrfor u ( = U( u, u, u ( d k = u (( k = U ( u, u, u ( d k k U ( u, u, u ( d = k U ( u, u, u ( s d = u ( d = k k k u ( = U( u, u, u ( s is quival o k = k k u ( U( u, u, u ( s U( u, u, u ( s k k U ( u, u, u ( s d ad h h soluio ( u,, u N of ( is a fixd poi of B or Covrsly, l ( u,, un M * b a fixd poi B, ha is, hrfor by dfiiio of B w obai W show ( k ( = (,, N(, u k B u u k ( k N u = B ( u,, u, [ k,( k ] k k k k k = u ( k = U ( u, u, u ( s U ( u, u, u ( s k U ( u, u, u ( s d = U ( u, u, u ( s k k k k k U ( u, u, u ( s d U ( u, u, u ( s =
Vasil G Aglov al: Rig of arg Numbr Muually Coupld Oscillaors Priodic Soluios which implis Idd, pu = = cos ad h w hav U ( u, u, u ( s d = k k ( k ( k ( k U ( u, u, u ( d u ( s d u ( s d C C k k k k k C k k u ( ( g ( g k ( s d u( d u( d C C k k ( k ( k ( sk ( s U d U d C k k k k ( k ( sk ( ( k U d ( ( k g U d g ( U d k k k k ( k ( k U ( k U d C U d d k k k ( ( ( g U g U U ( k U ( k d d C k k ( ( ( g U g U U 4 g g ( U C ( ( U 4 = ( C Obviously M ( as ha implis hrfor h opraor quaio ( ( g g U M ( U ( u, u, u ( d = k ( k = N u B ( u,, u, [ k,( k ]
Iraioal Joural of horical ad Mahmaical Physics 4, 4(6: 5-9 bcoms u ( = U( u, u, u ( s k Diffriaig h las qualiis w obai ( mma 4 is hus provd Mai Rsul horm h followig assumpios b valid: u ( = = u N ( = ; For sufficily larg > ( ( ( ( 4 g g U ; C ( ( 4 g U g ; C < ( ( ( g g U 4 < C h hr xiss a uiqu -priodic soluio of ( Proof: Firs w show ha Idd W hav * ( B u u B u u B u u M (,,, (,,,, (,, ( N N N N ( k k k B ( u, u, u ( U( u, u, u ( s U( u, u, u ( s U ( u, u, u ( s d V V V k k s V = U ( u, u, u ( s u ( θ C k k k s s ( ( u( θ u ( θ g u( s g u( s k k k k k k s s θ ( k ( θ θ ( k U U U C k k k k k k
Vasil G Aglov al: Rig of arg Numbr Muually Coupld Oscillaors Priodic Soluios ad g U g U ( ( ( ( ( ( s sk s k U k C k k k ( sk ( sk U k k U ( k ( k ( ( ( g U g U U ( sk U ( sk C k k ( k ( k ( ( ( g gu g U ( k U 4 ( ( g g ( U C ( k U 4 ( ( g g ( U C Usig h simas from mma w obai V U ( u, u, u ( d = ( k U 4 g g U C k k k k ( ( V = U ( u, u, u ( s d U ( u, u ( s d ( ( ( k k U 4 g g U C d U 4 ( ( g g ( U C hrfor for sufficily larg w hav ( U 4 B u u u g g ( U ( k ( k ( ( (,, ( C U 4 ( ( ( ( g g U C
Iraioal Joural of horical ad Mahmaical Physics 4, 4(6: 5-9 U 4 g g ( U C ( ( ( ( ( ( U 4 g g U C U ( k ( k I rmais o show ha Idd, B= ( B, B, B N is a coraciv opraor o M ( k ( k B ( u, u, u ( B ( u, u, u ( k ( (,, ( (,, ( Uu u u s Uu u u s k ( Uu (, u, u ( s Uu (, u, u ( s k k k ( U ( u, u, u ( s U ( u, u, u ( s d P P P h ( (,, ( (,, ( P Uu u u s Uu u u s k s s u( θ u( θ u( θ u( θ C k k k k s ( ( u ( θ u ( θ g u( s u( s g u ( s u ( s k k k k ( k s s ρ ( u, u θ ( k ( k θ ( k ρ ( u, u C k k k k ( k s ( u, u θ ( k ( ( k ( sk (, θ ρ k k k ρ d g u u ( ( k ( sk g ρ ( u, u ( u( s u( s u( s u( s k
4 Vasil G Aglov al: Rig of arg Numbr Muually Coupld Oscillaors Priodic Soluios ( k ( ( ρ ( u, s k s k u ( k ρ ( u, u C k k ( k u ( ( u s k ( ( k k g ( u, u ρ k ρ (, U g ( u, u ( k ( ( ( ( (, s k k s k u u ρ ρ C k k ( k ( ( k (, ( ( ρ u s k, s k u u u ρ k k ( k ( ( ( ( (, k k ( (, k ρ u u ρ u u g U g ( k ρ (( u, u,( u, u N N N N g U g C ( ( k = k ; ( (,, ( (,, ( P Uu u u s Uu u u s ( k ( s k s u( θ u( θ u( θ u( θ C k k k k s u ( θ u ( θ k k ( ( g u( s u( s g u ( s u ( s k k ( k ( (, s k ρ s u u θ ( k ( k θ ( k ρ ( u, u C k k k k ρ ( k (, s u u θ ( k k k
Iraioal Joural of horical ad Mahmaical Physics 4, 4(6: 5-9 5 ( ( k ( sk ( ( k ρ ρ k g ( u, u g ( u, u ( sk ( u( s u( s u( s u( s k ad ( k ( k ( ( ρ u, s k u ( k ( ρ ( u, s k u C k k ( k ρ ( u (, s k u k ( k ( ( (, k ( (, ρ u u ρ u u g U g ρ (( u, u,( u, u N N N N ( ( k k C ( k ( ( g U g ( k ρ (( u, u,( u, u N N N N ( ( ( ( 4 g U g C P U u u u s U u u u s d ( (,, ( (,, ( k k ρ (( u, u,( u, u N N N N ( 4 g U g k d C k ρ (( u, u,( u, u N N N N 4 g U g C I follows ( ( ( (
6 Vasil G Aglov al: Rig of arg Numbr Muually Coupld Oscillaors Priodic Soluios B ( u, u, u ( B ( u, u, u ( (( u, u,, u, u,( u, u ( k ( k ( k ρ N N N N ( ( 4 g U g C ( ( ( ( 4 g U g C 4 g U g C ρ (( u, u,( u, u N N N N ( ( ( k ( ( 4 g U g C ( k KU u u un un u u un u N hus ρ ((,,,,,(,,,, ( k ( k ( ρ B ( u, u, B ( u, u K ρ (( u, u,( u, u N N N N U N N N N For h drivaiv w obai B ( u, u, u ( B ( u, u, u ( U( u, u, u ( U( u, u, u ( ( k ( k ( Uu (, u, u ( s Uu (, u, u ( s Q Q k W hav Q u( s u ( s u( s u( s u ( s u ( s C k k k g u u g u u u u k ρ ( ( ( ( ( s k ( ( ( (, C k ( k ( sk ( k ( sk ρ (, ρ (, k k u u u u k (, ρ (, ( ( k ( k ( ( k ( ρ g u u g U u u
Iraioal Joural of horical ad Mahmaical Physics 4, 4(6: 5-9 7 k ( u, k u ρ ( u, u ( ( ( ( k ρ k C ( u, ( k u ( u, u ( u, u ( k ( k ( k ( ( k ( ( k g g U ρ ρ ρ ( k 4 ( ( ρ (( u, u,, un, u N,( u, u,, un, u N g g U ; C Q Uu u u s Uu u u s ( (,, ( (,, ( k 4 g U g C ρ (( u, u,( u, u hus N N N N ( ( B ( u, u, u ( B ( u, u, u ( ( k ( k ( k ρ (( u, u,( u, u N N N N ( ( ( g g U 4 C ( k K u u u u u, u Uρ ((,,, N, N,( h abov iqualiis imply ρ ( B ( u,, u, B ( u,, u K ρ (( u, u,( u, u,, u, u N N U N N N N ad h ρ (( ( B, B, B, B, B, B,, B, B Kρ (( u, u,( u, u N N N N N N N N whr K max { KU, KU} = horm is hus provd 4 Numrical Exampl h iqualiis guaraig a xisc-uiquss of a priodic soluio ar N N ( ( ( ( 4 g g U ; C
8 Vasil G Aglov al: Rig of arg Numbr Muually Coupld Oscillaors Priodic Soluios ( ( 4 g U g ; C < ( ( ( g g U 4 < C us cosidr h cas wh = 44 mh, = mh, C = F W assum h aciv lms hav characrisics I ( u =,u, 4 u ( =,,, N h ( 4 6,, 4 U ; ; 44 ( U 4,, 6 ; < 44 4,, U 6 < 44 If for isac 6 = = sc, ad U =, h w hav o choos f Hz 4, 44,6 4, 44,5 ad hc 6 iqualiy ca b disrgardd h = = ha implis (( ( K = 4 / / 44,,U,4 < 6 = ad h scod 5 Coclusios Succssiv approximaios o h soluio ca b obaid bgiig wih h followig iiial fucios ( π u ( = Usi = Usiω I is asy o calcula h x approximaio from h righ-had sid of h abov sysm: ( ( k ( ( ( ( ( ( = = k u ( B ( u, u, u ( : U( u, u, u ( s k ( ( ( ( ( ( k k k U ( u, u, u ( s U ( u, u, u ( s d U si k k k k U = ( ωθ si ( ωθ C C ( ( U g U g si si si k k k k U ( ωθ ( ωs ( ωs C C C
Iraioal Joural of horical ad Mahmaical Physics 4, 4(6: 5-9 9 k U U si ( ωθ si ( ωθ C k C k k k ( ( U U ( g si U ( g d si s si ( s C ωθ θ C ω ω C k k k k ( k s ( k s U U si ( ωθ d si ( ωθ d C k C k k k k k U C s k k k si ( ωθ d ( ( U g U ( g si ω s d si ( ωs d C C k k k k Pricipal Rmark Oc w hav foud h volags i viw of ( w solv h quaios wih rspc o i ( : di ( whr u(, u ( ar alrady kow fucios h currs ar priodic fucios oo: ( ( ( ( ( [ ] = u u i = u u d, [ ] [ u( u ( ] [ u( u ( ] i ( = u ( u ( = [ ( (] [ ( (] [ ] = u u u u = u( u ( = i( 974 REFERENCES [] R J P Figuirdo ad C Y Chag, O h bouddss of soluios of classs of muli-dimsioal oliar auoomous sysms, SIAM J Appl Mah, vol 7, pp 67-68, July 969 [] J K Aggawal ad C G Richi, O coupld va dr Pol oscillaors, IEEE ras Circui hory, vol C-, pp 465-466, Dc 966 [] D A iks, Aalyical soluio of larg umbrs of muually-coupld arly-siusoidal oscillaors, IEEE ras Circuis ad Sysms, vol CAS-, pp 94-, March [4] Robrso-Du ad D A iks, A mahmaical modl of h slow-wav lcrical aciviy of h huma small isi, J Md Biol Eg, pp 75-757, Nov 974 [5] D A iks, Mahmaical modl of h colorcal myolcrical aciviy i humas IEEE ras Biomd Eg, vol BME-, pp -, March 976 [6] Edo ad S Mori, Mod aalysis of a rig of a larg umbr of muually coupld va dr Pol oscillaors IEEE ras Circuis ad Sysms, vol CAS-5, No, pp 7-8, Ja 978 [7] V G Aglov, A Mhod for Aalysis of rasmissio is rmiad by Noliar oa, Nova Scic, Nw York, 4