Investigation of Transition to Chaos for a Lotka. Volterra System with the Seasonality Factor Using. the Dissipative Henon Map

Transcription

1 Applid Mathmatical Scics, Vol. 9, 05, o. 7, HIKARI Ltd, Ivstigatio of Trasitio to Chaos for a Lotka Voltrra Systm with th Sasoality Factor Usig th Dissipativ Ho Map Yu. V. Bibik Fdral Rsarch Ctr, Computr Scic ad Cotrol of Russia Acadmy of Scics, Vavilov str. 8/0, 9, Moscow, Russia Copyright 05 Yu. V. Bibik. This articl is distributd udr th Crativ Commos Attributio Lics, which prmits urstrictd us, distributio, ad rproductio i ay mdium, providd th origial work is proprly citd. Abstract Coditios udr which th classical Lotka Voltrra systm with a sasoality factor hibits chaotic bhavior ar ivstigatd. Th sasoality factor is itroducd i such a way that th systm ca b ivstigatd usig th Ho map. Th rductio of th systm to Ho s map allows o to calculat th priod doublig bifurcatios ad dtrmi th poit of th trasitio to chaos. Kywords: Lotka Voltrra systm, sasoality, Ho map, bifurcatios, trasitio to chaos. Itroductio Coditios of th mrgc of bifurcatios ad coditios for th trasitio to chaos i a classical mathmatical biology problm th Lotka Voltrra modl to which th author addd a sasoality factor ar ivstigatd. This factor is rprstd by a dpdc of th systm cofficits o tim ad maks it possibl accout for th ifluc of sasoal tmpratur variatios o th spcis populatio. It turd out that th sasoality factor chags th bhavior of th classical Lotka Voltrra systm, which acquirs th fudamtal proprtis of th uivrsal trasitio to chaos. Th systm with th sasoality factor is rducd to th dissipativ Ho map. A rormalizatio of th dissipativ Ho map usig a aalog of Hllma's mthod [ maks it possibl to fid th co-

2 580 Yu. V. Bibik ditios udr which priod doublig bifurcatios occur ad dtrmi th poit of th trasitio to chaos. Historical backgroud Th classical Lotka Voltrra systm of quatios was first proposd by th Amrica mathmaticia, statisticia, ad dmographr Lotka [5 (95 ad by th Italia mathmaticia Voltrra [ (96, [ (9. This systm givs a adquat dscriptio of th dyamics of two-spcis biological systms (prdator ad pry wh th spcious populatio is ot too larg. It ca b ivstigatd aalytically. Th classical Lotka Voltrra systm is Hamiltoia with o dgr of frdom, it is itgrabl by quadraturs, ad thrfor hibits o chaotic bhavior. Attmpts to mak mathmatical biology modls mor ralistic by takig ito accout additioal faturs ad phoma rsult i complicatios i th aalytical ivstigatio ad to th loss of itgrability. To aalyz th modl, w will mploy modr mthods that ar usd for th study of chaotic dyamics. Bgiig i th s, rsarchrs (physicists, mathmaticias, biologists, ad cologists paid atttio to chaotic phoma ad discovrd crtai ordr ad rgularitis i thm. A importat coclusio mad i th thory of chaos is th fact that v isigificat variatios i ay part of such a systm ca rsult i a radically diffrt dvlopmt of th systm. Chaotic phoma wr studid by may rsarchrs. Hr, I wat to mark out th rsarchrs who ca b cosidrd to b piors i th thory of chaos. It is rmarkabl that thir brilliat discovris wr mad litrally with th d of a p basd o th ituitio ad usig oly simpl calculators ad primitiv computrs. O of thos rsarchrs was th Amrica mtorologist ad mathmaticia E. Lorz. Whil studyig wathr prdictio, h proposd ad aalyzd a systm of thr coupld diffrtial quatios that spcify a flu i th thr-dimsioal spac. This systm could ot b aalyzd usig kow attractors, i... gomtric figurs that dscrib th bhavior of th systm i th phas spac. Th w attractor foud by Lorz, which was latr calld by his am, providd a ampl of a chaotic or strag attractor that has a mor compl structur tha th attractors kow arlir. Th computr modl of th atmosphr proposd by Lorz showd that v isigificat variatios i th atmosphr ca rsult i radical ad upctd cosqucs. It was Lorz who rvald i [, th mai faturs of th systm that provid a ky to th udrstadig of chaotic bhavior.

3 Ivstigatio of trasitio to chaos for a Lotka Voltrra systm 580 Th Frch mathmaticia Ho watd to costruct a simplr map tha Lorz's systm. H proposd a two-dimsioal attractor with th sam proprtis as Lorz's attractor. Ho's modl was asir to b aalyzd mathmatically, ad umrical computatios wr fastr ad mor accurat. Ho's map is a rfrc two-dimsioal map [9. A cosidrabl cotributio to th thory of chaos was mad by th brilliat works of th Amrica physicist М. Figbaum. I particular, h ivstigatd th logistic map. Figbaum showd that chaos ca mrg via bifurcatios. H rvald uivrsal laws of trasitio to chaos i th procss of priod doublig. Usig th rormalizatio group mthod, h cratd a thory that plaid th uivrsality of priod doublig. H also discovrd a w mathmatical costat Figbaum's costat that dscribs priod doublig bifurcatios for o-dimsioal maps. H foud out that th poits of priod doublig bifurcatios accumulat ar a crtai poit th thrshold of trasitio to chaos by a gomtric progrssio with th ratio.669. This ratio turd out to b uivrsal ad valid for othr maps ad various oliar dissipativ systms [,, 5. Figbaum's discovry was cofirmd primtally by th Frch physicist А. Libchabr, who foud a cascad of bifurcatios rsultig i chaos i dyamical systms [,. Th cosrvativ Ho map was rormalizd by th Dutch physicist Hllma. Th rormalizatio mad it possibl to dtrmi th squc of priod doublig bifurcatios ad obtai a w costat for th two-dimsioal cas with th ratio 9.09 (8.709 (th umbr i parthss is th bst valu of th costat obtaid so far. This costat is a aalog of Figbaum's costat for th o-dimsioal cas [7, 8. Rormalizatios of othr ara prsrvig maps wr dscribd i th works of th Amrica physicist MacKay [6. Th Frch ad Amrica mathmaticia Madlbrot is th foudr of fractal gomtry. Whil studyig various phoma, such as variatios i cotto prics ad oiss i lctroic circuits, h oticd that absolutly radom procsss bar idicatios of similarity. It was Madlbrot who dfid th cocpt of fractal. This cocpt bcam a global cocpt i th physics of chaos, ad it mad th pictur of rality mor clar [7. Th Russia physicist Chirikov publishd th papr [ i 979, whr h proposd a ovl approach to th ivstigatio of chaos i Hamiltoia systms usig th rsoac ovrlap mthod. Th modr viw of th thory of dtrmiistic chaos is prstd i th works by th Grma physicist Shustr [8, 9, Amrica physicist Tabor [0, Russia ad Amrica physicist Zaslavsky [5, British physicist Thompso [,, ad th Amrica rsarchrs Fishma ad Egolf [6. Th thory of strag attractors is prstd i th collctio of paprs [0 ditd by Hut, Li, Kdy, ad Nuss.

4 580 Yu. V. Bibik Th rsarch prstd i this papr is importat bcaus a ralistic systm of quatios is ivstigatd. It maks it possibl to tak ito accout th ifluc of sasoality o th dyamics. Th modr tchiqus of th thory of chaos allow us to fid th whol chai of priod doublig bifurcatios ad th poit of th trasitio to chaos. Ivstigatio tchiqus: - w itroduc ito th classical Lotka Voltrra systm th sasoality factor i such a way that th cotiuous map could b trasformd ito a discrt o, which facilitats th ivstigatio; - th scod stp is to limiat o of th variabls from th systm of discrt quatios i two variabls to simplify th problm; -th third stp is to rduc th rsultat discrt map with a sigl variabl to th dissipativ Ho map. This map is rormalizd usig a gralizd Hllma's mthod. Th map thus obtaid is usd to aalyz coditios for th mrgc of priod doublig bifurcatios ad coditios for th trasitio to chaos. Th papr is orgaizd as follows:. Itroductio.. Trasformatio of th classical Lotka Voltrra systm to a two-dimsioal discrt map by addig a sasoality factor.. Trasformatio of th systm of discrt quatios i two variabls to a discrt quatio i o variabl.. Rductio of th discrt quatio i o variabl to th dissipativ Ho map. 5. Rormalizatio of th dissipativ Ho map. 6. Coditios for th mrgc of th first ad scod priod doublig bifurcatios for th o-dimsioal discrt map. 7. Coditios for th mrgc of th t priod doublig bifurcatios ad coditios for th trasitio to chaos for th gralizatio of th Lotka Voltrra quatios with a sasoality factor. 8. Dscriptio ad aalysis of figurs. 9. Coclusios.. Trasformatio of th classical Lotka Voltrra systm to a twodimsioal discrt map by addig a sasoality factor Lt us add a sasoality factor to th classical Lotka Voltrra systm (., (.. To this d, w itroduc ito th quatios cofficits that dpd o tim, ad th trasform th rsultig quatios to a two-dimsioal discrt map. I th gral cas, th itroductio of tim-dpdt cofficits cosidrably complicats th systm's bhavior ad aalysis. Howvr, w itroduc th sasoality factor i such a way that v though th bhavior of th Lotka Voltrra systm bcoms mor compl, its aalysis bcoms simplr. Th diffr-

5 Ivstigatio of trasitio to chaos for a Lotka Voltrra systm 5805 tial quatios ar rducd to discrt quatios. This is achivd du to th us of dlta fuctios for modlig th dpdc of th systm's cofficits o tim. Th ffctivss of this mthod is largly plaid by th spcific algbraic structur of th Lotka Voltrra systm. Aftr discardig th liar trms cotaiig th dlta fuctios, th systm is asily itgratd. This is th first stp i th proposd mthod for obtaiig discrt quatios. At th scod stp, th ifluc of th dlta fuctios o th systm's dyamics is tak ito accout. Th oliar trms do ot play ay rol i this cas. Thus, th problm is solvd i two stps. To obtai diffrc quatios, th classical Lotka Voltrra quatios ar usd as th origial os. Thy hav th form d dt dy dt y, (. y y. (. Hr, th variabl is th pry populatio ad y is th prdator populatio. Th cofficits ad dtrmi th itsity of th spcis itractio with th viromt, ad ad dtrmi th itsity of itractio btw th spcis. Blow w assum that ad dpd o tim. Th cofficits ad ca b mad qual to uity by a appropriat chag of variabls. If w assum that th tim-dpdt cofficits ad hav a costat valu i witr ad i summr, th systm still rmais too complicatd to b aalyzd aalytically. For that raso, w mak furthr simplificatios. W assum that th icras of th pry biomass du to tral factors occurs at a poit i tim i th bgiig of summr, ad th dcras i th prdator biomass du to tral factors occurs at th sam tim, which ca b cosidrd as th d of witr. W dot this tim by t T, whr is th itgr umbr idicatig th umbr of cycls of icras ad dcras of th pry ad prdator biomasss, ad T is th yar duratio. Th, th tim-dpdt ad ca b writt as ( t T, (. ( t T. (. Hr ad ar th amplituds of th corrspodig dlta fuctios. Formulas (. ad (. show th way i which th sasoality factor is rprstd i th prst papr. Thy rprst th ifluc of sasoal tmpratur variatios o th dyamics of th two-spcis itractio. Thy ar a mathmatical rflctio of th fact that th icras i th biomass of pry ad th dcras i th biomass of

6 5806 Yu. V. Bibik prdators du to tral factors occur at th tims t T. Th cofficits ad ar rprstd by sums of dlta fuctios. Ths cofficits vaish vrywhr cpt for t T. Ths ar th poits i tim wh by our assumptio th witr is rplacd by summr. Thrfor, th quatios ar simplifid vrywhr cpt for ths poits. I th simplifid form, ths quatios ar valid vrywhr o th tim itrval from to T, whr is a ifiitsimal quatity. Th first stp to drivig diffrc quatios. At th first stp, w simplify th systm of quatios (., (. with th cofficits ad havig th form (. ad (.. To implmt this stp, w glct th sum of th dlta fuctios i (. ad (. o th itrval from to T. This simplifis th systm itgratio. Lt us mak th chag of variabls q =l(, p=l(y to rduc Eqs. (., (. to th form p q t T, (.5 t ( q p t T. (.6 t ( Th cofficits ad dfid by (. ad (., rspctivly, appar i ths quatios liarly. To obtai diffrc quatios, w should itgrat Eqs. (.5, (.6 o th itrval from to T. W solv this problm i two stps. First, w itgrat th diffrtial quatios (.5, (.6 o th itrval from to T ad th o th itrval from T to T. (Hr, is a small paramtr that allows us to dcompos th itgratio of Eqs. (.5, (.6 ito two stps. At th first stp, th ifluc of th dlta fuctios is glctd. At th first stp, w hav th quatios p q, (.7 t q pt. (.8 To solv ths quatios, it is covit to rtur to th origial variabls ad y. W hav t y, (.9 y t y. (.0 Nt, w rduc th two quatios (.9 ad (.0 i two variabls to o quatio i o variabl. For this purpos, w us a cosrvatio law. It is implid by Eqs. (.9, (.0 that th variabl qual to th sum of spcis populatios is prsrvd:

7 Ivstigatio of trasitio to chaos for a Lotka Voltrra systm 5807 ( y 0. (. t This cosrvatio law abls us to rplac th ivstigatio of th populatio of two spcis to th ivstigatio of th populatio of o spcis, which is cosidrably simplr. W hav As a rsult, oly th variabl rmais i Eq. (.9: t y. (. t ( a ( a a X. (. For th covic of calculatios, w itroduc th w variabl X dfid by X a, (. a. (.5 I trms of th w variabls, Eq. (.9 bcoms vry simpl: X t X a. (.6 This quatio is solvd by sparatio of variabls as dx X a d l dt. (.7 X a a X a Th lft-had sid is itgrabl i lmtary fuctios. Upo th itgratio of th lft-had sid w obtai l X X a a l X X 0 0 a a at. (.8 Lt us trasform Eq. (.8 to a quatio i, y, 0, ad y 0. To this d, w us th rlatioships btw th variabls X 0, 0, y 0, a, ad : X a, (.9 0 y 0 X a, ( X a, (.

8 5808 Yu. V. Bibik X a. (. Usig ths rlatios, w obtai from (.8 a quatio coctig th variabls, y, 0, ad y 0 To simplify Eq. (., w gt rid of th logarithm: y0 l l T. (. ( 0 y 0 T. (. Now w ca asily fid quatios for ad y, which ar th populatios of th spcis at th tim t T 0 0 T, (.5 y 0 0 y. (.6 Thus, w hav trasformd th origial diffrtial quatios ito th prlimiary diffrc quatios (.5, (.6. Th dlta fuctios wr ot usd for this purpos. Formulas (.5 ad (.6 rlat th valus of th variabls at th tim with thir valus at t T. Thrfor, w hav itgratd th simplifid systm of quatios o th tim itrval from to T. At th scod stp, i ordr to obtai th fial diffrc quatios, w tak ito accout th dlta fuctios whil itgratig o th tim itrval from T to T, which yilds diffrc quatios istad of Eqs. (.5, (.6. Scod stp: Drivatio of th ultimat diffrc quatios. To obtai th fial form of th diffrc quatios, w itgrat Eqs. (.5, (.6 o th tim itrval from T to T. Up to, w obtai q( T q( T O(, (.7 p( T p( T O(. (.8 Th prlimiary diffrc quatios ar writt for th variabls ad y. For that raso, upo drivig th rlatios btw th variabls q ( T ad p ( T with th variabls q ( T, p ( T, w rtur to th variabls, y.

9 Ivstigatio of trasitio to chaos for a Lotka Voltrra systm 5809 Calculat th potial fuctio of th lft ad right-had sids of Eqs. (.7 ad (.8 to obtai ( T ( T O(, (.9 y( T y( T O(. (.0 Hr, ad ar th potial fuctios of th dlta fuctio amplituds:, (.. (. By combiig (.5, (.6 ad (.9, (.0, w obtai th fial form of th diffrc quatios: [ ( y, (. y p(( y T [( ( y y y. (. y p(( y T Hr, ad y ar th populatios of spcis at th tim t ( T, ad T is th yar duratio. Thus, w hav rducd th diffrtial quatios (.5, (.6 to th diffrc quatios (., (.. Thrfor, th classical Lotka Voltrra systm has b rducd to a two-dimsioal discrt map by addig th sasoality factor.. Trasformatio of th systm of discrt quatios i two variabls to a discrt quatio i o variabl Th diffrc quatios (., (. ar still too complicatd to b aalyzd aalytically. W simplify thm whil prsrvig thir basic proprtis. Itroduc th w variabls T, (. y Ty. (. I trms of ths variabls, Eqs. (. ad (. tak th form

10 580 Yu. V. Bibik [ ( y, (. y p( y [( ( y y y. (. y p( y Now thy ar idpdt of th yar duratio T ad ca b rducd to form (., (. for ay T. Nt, w trasform (., (. to mak thm mor covit for th aalysis. To this d, w itroduc th variabls u, (.5 y z y. (.6 Th, Eqs. (., (. tak th form or [ zu, (.7 u p( z [ zu p( z. (.8 u p( ( z For y w hav or [ zu p( z y z (.9 u p( ( z [ z y. (.0 u p( ( z Lt us limiat th variabls, y o th lft-had sids of Eqs. (.8 ad (.0. To this d, w divid Eq. (.8 by (.0 to obtai u u p( ( z. (.

11 Ivstigatio of trasitio to chaos for a Lotka Voltrra systm 58 By summig Eqs. (.8 ad (.0 w obtai p( ( zu z z z. (. u p ( z To simplify th dpdc o th variabls u ad z, w tak th logarithm of both sids of Eq. (.: l( u l( l( u z. (. Now, th dpdc of th logarithm of u o th logarithm of u ad o z is simpl. To simplify Eq. (. v furthr, w itroduc th w variabls q l( u, (. a l( q z. (.5 Th, Eq. (. taks th form l( u a. (.6 q Usig (.5, w obtai for z th formula z l( a a. (.7 Fially, usig (.6 ad (.7, w ca obtai a quatio i o variabl a. To this d, w plug (.6 ad (.7 ito (. to obtai l( a a (l( a a ( a (l( a a. a (.8 Rarrag th trms ad rvrs th sigs o th right- ad lft-had sids to obtai

12 58 Yu. V. Bibik a ( l( ( a - a - ( a (l( a a -. a (.9 Thus, th systm of two discrt quatios (., (. i two ukows has b trasformd to th discrt quatio (.9 i a sigl ukow. I th t sctio, w cotiu th trasformatio of Eq. (.9 to rduc it to th dissipativ Ho map. This will abl us to dtrmi coditios for th trasitio to chaos.. Rductio of th discrt quatio i o variabl to th dissipativ Ho map To fid th coditios udr which th Lotka Voltrra systm with th sasoality factor bgis to hibit chaotic bhavior, w rduc Eq. (.9 to th dissipativ Ho map. Th Ho map is a typical systm i which it ca b s how dtrmiistic chaos mrgs. This map is giv by th quatios y a. y, (.a (.b Lt us itroduc th otatio ; a 0; A a, (. a Usig otatio (., w trasform Eqs. (.a, (.b to th form A y. y, (. (. Nt, Eqs. (., (. i two variabls ar trasformd to a quatio i o variabl: A. (.5 To simplify Eq. (.9 ad rduc it to Ho's quatio (.5 w should ovrcom som difficultis.

13 Ivstigatio of trasitio to chaos for a Lotka Voltrra systm 58 Th first difficulty is that th right-had sid of Eq. (.9 icluds a strogr liarity tha th quadratic oliarity i Ho's map. To ovrcom this difficulty, w approimat this oliarity by a quadratic trm i th viciity of th fid poit of priod o. Map (.9 producs a squc of poits. Typically, it covrgs to o or svral poits. Wh it covrgs to a sigl poit, w dal with a fid poit of priod o. Th scod difficulty is that, i additio to a strogr oliarity, th right-had sid icluds th trm a alog with th trm a. Th trm a appars o th right-had sid of Eq. (.9 as th diffrc with a. W should limiat it bcaus it complicats th trasitio to Ho's map. Th ida of rplacig a with th fid poit of priod o sms to b quit rasoabl. Th first stp i th costructio of th simplifid map is as follows: rplac th trm (l( a a F ( a i Eq. (.9 with th simplifid trm a (.6 F a [l( w(, ( a a (. (.7 a Hr a is th fid poit of priod o ad F is th oliar part of map (.9. I tur, F is th simplifid oliar part map (.9. Th choic of F maks th fid poit of priod o ivariat. Th t stp is to rduc Eq. (.9 with th trm F to Ho's map. To this d, w chag to th variabl : a a. (.8 Now, th oliar trm F taks th form a [l( w(, ( F (. (.9 a

14 58 Yu. V. Bibik Hr, w is rplacd with w for covic. Th w trm F cotais th product of th liar trm i with a oliar trm, which is dotd by Q. Th factor Q has th form a a Q. (.0 a a To mak furthr simplificatios, w rtai i Q th dgrs of ot gratr tha two. Th rsultig oliar factor will b dotd by Q : Q [ ( ( ; (. hr, a. (. a Now, w ca rplac F with th additioally simplifid trm F F ( Q. (. [l( w(, To mak furthr simplificatios, w fid th fid poit a of map (.9. For this purpos, w substitut i Eq. (.9 a for th trms a, a, ad a. Th, w obtai th quatio ( l( ( l(. (. a a

15 Ivstigatio of trasitio to chaos for a Lotka Voltrra systm 585 Nt, it is covit to calculat bcaus th liar trms i aothr i (.9. W hav a cacl o (. (.5 ( Now, it rmais to calculat F : F [l( ( ( w(, [ ( Nglctig th cubic trms i (.6, w obtai. (.6 F ( (l( [ l( (l( ( w ( w(. (.7 Havig i (.7 th oliar trm F, which is quadratic i, w ca trasform (.9 to Ho's map (.5. I trms of th w otatio, it has th form ( (l( ( [(l( ( w( By rarragig ad ramig th trms, w obtai. ( w (.8 A P ; (.9 hr A ( ( (l( ( w (.0 or, takig ito accout formula (.5 for,

16 586 Yu. V. Bibik A ( l( ( ( ( w, (. P ( (l( ( w(. (. W rduc Eq. (.9 to a mor usual form. Lt us trasform th variabl : by multiplyig it by th calibratig factor P : This yilds th quatio P. (. A. (. Upo drivig Eq. (., th rductio of Eq. (.9 i o variabl to th dissipativ Ho map is almost compltd. It oly rmais to dtrmi th ukow fuctio w. I Eq. (. it appars i th paramtr A. Th sam fuctio also appars o th right-had sid of Eq. (.. It is a fittig fuctio. Usig it, w will b abl ot oly to rval bifurcatios i th systm udr amiatio but also to choos th valu of w such that th first two bifurcatios of Ho's map corrspod to th first two bifurcatios of Eq. (.9. This abls us to obtai a good approimatio of Eq. (.9 by Eq. (. upo which th approimatio of Eq. (.9 will b compltd. Not that Eq. (. is actually a lik btw Eq. (.9 i o variabl ad th dissipativ Ho map (.. Th lft-had sid of this quatio cotais th paramtr A, which also appars i (.. Th right-had sid of Eq. (. cotais th paramtrs ad. Wh a bifurcatio occurs, ths paramtrs bcom rlatd by a fuctioal rlatioship (, whr i is th ( i bifurcatio id. Ths paramtrs ad also appar i (.9. W will us Eq. (. to dtrmi bifurcatios of Eq. (.9 giv th bifurcatios of Eq. (.. Th fuctio w will b dtrmid i th t sctio i trms of th fuctios w ad w. Formulas for dtrmiig th fittig fuctios w ad w ar drivd i th t sctio (formulas (5.9 ad (5.0. Ths fuctios ar dtrmid i trms of th fuctios ad, which ar obtaid i Sctio 6 (formulas (6.0 ad ( Rormalizatio of th dissipativ Ho map Lt us brifly discuss th purpos of rormalizatio. Th itroductio of th sasoality factor ito th origial quatios (., (. rsults i th mrgc

17 Ivstigatio of trasitio to chaos for a Lotka Voltrra systm 587 of chaos i th Lotka Voltrra systm. I this papr, th rormalizatio mthod is usd to dtct th chaotic bhavior. It was show that th trasitio of Ho's map to chaos occurs through a chai of. Sic Eq. (.9 is rducd to th dissipativ Ho map (., it also has th basic proprtis of Ho's map. To dtrmi th poits of priod doublig bifurcatios, fid poits of th corrspodig priods should b dtrmid ad th aalyzd for stability. I th bst cas, this yilds polyomial quatios of larg dgrs, which ar difficult to solv. Th rormalizatio procdur maks it possibl to avoid ths opratios. It rlats th form of quatios o diffrt scals ad thus provids rcurrt formulas for rlatig bifurcatio valus. Rormalizatios hav spcific faturs dpdig o th quatios big rormalizd. Th rormalizatio of th dissipativ Ho map (. also has som spcific faturs compard with th rormalizatio of th cosrvativ Ho map prformd usig Hllma's tchiqu [7, 8. Cosidr th basic ida udrlyig th rormalizatio. Th rormalizatio of th dissipativ Ho map prformd i [ is basd o a simpl ida that th assumptio of th possibility of rormalizatio maks it possibl to actually prform th rormalizatio. Lt us plai this i mor dtail. Assum that Eq. (. is alrady rormalizd. For covic, w chag th otatio i Eq. (. by rplacig with X. Th, Eq. (. taks th form X X AX X. (5. It has two fid poits of priod o, which w dot by fid poits of priod two, which w dot by Lt ad. ad, ad two X, (5. X. (5. Th rormalizability of a systm implis that th followig quatio is satisfid: С P. (5. It must follow from Eq. (5.. O th othr had, Eq. (5. ca b padd i th viciity of th fid poits ad. Th, w hav th quatios, (5.5. (5.6

18 588 Yu. V. Bibik Thy ar cosqucs of Eq. (5.., (5.7 A. (5.8 A By combiig thm, w obtai (, (5.9 (. (5.0 Th lft-had sids of Eqs. (5. ad (5.9 ar idtical. It is clar that w ca driv from thm quatios that rlat,, ad. Howvr, w alrady hav othr quatios rlatig ths variabls ths ar Eqs. (5.5 ad (5.6. Th combid us of Eqs. (5., (5.9, (5.5, ad (5.6 maks it possibl to limiat o of th variabls,, or ad stablish a rlatioship btw th two othr variabls. For dfiitss, w limiat (ad i (5.0 usig Eqs. (5., (5.5, (5.9. As a rsult, w obtai f(, (5. f (. (5. Blow w will cosidr th cass whr approimatios ar small; for that raso, w us th, (5.. (5. Equatios (5. ad (5. ar cosqucs of th assumptio that Eq. (. ca b rormalizd. Thy allow us to rstor th symmtry of Eq. (. wh is rplacd with ad th us Hllma's rormalizatio tchiqu. Usig th rsults obtaid abov, w procd to th rormalizatio of Eq. (.. Dfi th paramtr. Equatio (. is th writt as X X X AX X. (5.5 W pad this quatio about th poits of priod two, which ar dotd by ad. This yilds

19 Ivstigatio of trasitio to chaos for a Lotka Voltrra systm 589 ( A, (5.6 ( A. (5.7 To mak th rormalizatio, w us rlatios (5. ad (5.:, (5.8. (5.9 Usig ths formulas, w rduc Eqs. (5.6 ad (5.7 to th form, (5.0, (5. whr ( ( ( A, (5. ( ( ( A, (5. (, (5. (. (5.5 Equatios (5.0 ad (5. hav th sam form as th pasio of th cosrvativ Ho map about th fid poits of priod two; thrfor, thy ca b rormalizd usig Hllma's tchiqu. To this d, w combi Eqs. (5.0 ad (5. for ad : [ [. (5.6 Nt, w limiat th sum of ad ad th sum of ad from Eq. (5.6. W mak this usig Eq. (5.0 for : ( [ [ [ [. (5.7

20 580 Yu. V. Bibik Upo ths trasformatios, Eq. (5.7 bcam similar to Eq. (.. It rmais to trasform th cosrvativ quatio (5.7 ito a dissipativ quatio. To this d, w rwrit it as ( [ [ [ [ (. (5.8 Now, w rprst i trms of usig (5. ad (5.. W obtai ( ( [ ( [ [ [. (5.9 It rmais to calculat th costats, ad,. Equatio (5.9 has th sam form as Eq. (5.5. Rpatig th rasoig usd aftr formula (5.5, w obtai quatios for, ad,. This complts th rormalizatio of Eq. (.. Nt, w us this rormalizatio to driv a rcurrc rlatig th coditios for th mrgc of priod doublig bifurcatios (th quatitis i A. Sic Eq. (5.9 o th doubld scal has th sam form as Eq. (., th rol of th costat A i (. is ow playd by th prssio (. Thrfor, if th prcdig priod doublig bifurcatio occurrd at A, th t priod doublig bifurcatio will occur at A dtrmid from th quatio A A A A A ( ( ( (. (5.0 Aftr writig th prssios for ad plicitly, w fially obtai th rcurrc for dtrmiig th priod doublig bifurcatios for Eq. (.: ( [ [ (5 q A A q q A. (5. I this cas, q A,, q ad.

21 Ivstigatio of trasitio to chaos for a Lotka Voltrra systm 58 Th fial rcurrc (5. allows us to dtrmi priod doublig bifurcatios for th origial quatio (.9 with th sasoality factor usig Eqs. (. ad (.. To sur th bst match btw Eqs. (.9 ad (., w should dtrmi th fittig paramtr w from Eq. (.. Formula (5. dtrmis th lft-had sid of Eq. (.. Thrfor, kowig rcurrc (5., w ca fid w from (.. To this d, w procd as follows. O th o had, th rormalizatio allows us to calculat for Ho's map (. th paramtrs A ( ad A (, which appar o th lft-had sid of Eq. (.. At ths paramtrs, w hav th first two priod doubligs. O th othr had, w ca dfi th fuctios ( ad (, which appar o th right-had sid of Eq. (.. W dfi thm i such a way as to sur that th first priod doublig i (.9 occurs at, ad th ( scod priod doublig occurs at (. Nt, w choos w such that th first priod doublig for (.9 corrspods to th first priod doublig for Ho's map (.. A o th lft-had sid of ( Eq. (. is associatd with th fuctio ( o th right-had sid of Eq. (.. For th scod priod doublig, w choos w so as to sur that th scod priod doublig for (.9 corrspods to th scod priod doublig for Ho's map (.. I othr words, th paramtr A o th lft-had sid of Eq. ( (. is associatd with th fuctio ( o th right-had sid of Eq. (.. Hc, w obtai th quatios ( ( A ( l( ( ( ( w, ( ( ( A ( l( ( ( ( w. ( (5. (5. W rwrit ths quatios as whr w (, P (, (5. ( w (, P (, (5.5 (

22 58 Yu. V. Bibik (A ( ( ( P l( (, (5.6 ( ( (A ( ( ( P l( (. (5.7 ( ( To dtrmi w, w will sk it as a combiatio of two ukow simplr fuctios w w [ f ( f ( ( w (. (5.8 Th, for w ad w, w obtai th quatios ( w ( P, (5.9 [ P w ( w (. (5.0 [ f ( ( f ( ( I this papr, th fuctio f is tak i th form f ( l(. Formulas (5.9 ad (5.0 iclud th ukow fuctios ( ad (. Thy will b foud usig a computr (formulas (6.0 ad (6.59. Nt, th fial valus of th fittig cofficits w ad w (5.9 ad (5.0 will b obtaid. Substitut thm ito (. to obtai a rady-to-us formula for dtrmiig th coditios for th mrgc of priod doublig bifurcatios ad trasitio to chaos. 6. Coditios for th mrgc of th first ad scod priod doublig bifurcatios for th o-dimsioal discrt map (.9 Sic th coditios of th mrgc of priod doublig bifurcatios for Ho's map ar alrady foud (formula (5., it rmais to dtrmi th coditios for th mrgc of priod doublig bifurcatios for Eq. (.9. This also yilds th ukow fuctios ( ad (. To fid th coditios for th first ad scod priod doublig for Eq. (.9, w rwrit it i a slightly diffrt form. Mak th chag of variabls Th, l( ( a a. (6.

23 Ivstigatio of trasitio to chaos for a Lotka Voltrra systm 58 a a ( ( a - a - ( - ( a- - a [, (6. a l(. (6. Equatio (6. is quivalt to a two-dimsioal diffrc quatio that ivolvs oly o tim stp. This quatio has th form a ( ( a - b - ( - ( b - a a [ a, (6. b a. (6.5 Equatio (6. has a fid poit of priod o ( a, b ( a, a ad two fid poits of priod two ( a, z ad ( z, a. First, w dtrmi th fid poits of priod two for Eq. (6., (6.5. Th, w fid th fid poit of priod o as a spcial cas for a z. To dtrmi th fid poits of priod two, w should fid a ad z. Equatio (6. implis that z a ( ( z - a - ( - ( a - z[, (6.6 z a z ( ( a - z - ( - ( z - a[. (6.7 a W rarrag ths quatios as follows: z z [( ( [ a [( ( [ z z [( ( [, z a a [( ( [ z [( ( [ a a [( ( [. a z z a a (6.8 (6.9 Usig (6.8 ad (6.9, w fid a as

24 58 Yu. V. Bibik z [( ( [ z a z. (6.0 z [( ( [ z Multiply th umrator ad domiator by I (6.,,,, ad hav th valus Similarly, w fid z i th form az to obtai z [ a z. (6. z [, (6., (6., (6., (6.5 a [ z a. (6.6 a [ Equatios (6.9 ad (6. iclud two variabls a ad z. W simplify (6.9 ad (6. by itroducig a w variabl P. This allows us to rduc th quatios i two ukows to a quatio with o ukow P. Dfi Usig (6. ad (6.7, w obtai a [ P. (6.7 a [ [ P [ a P a P 0. (6.8 It is clar that

25 Ivstigatio of trasitio to chaos for a Lotka Voltrra systm 585 a P [ P. (6.9 [ P O th othr had, (6.7 implis a [ P. (6.0 [ P Thrfor, P [ P[ P. (6. [ P[ P Thus, w hav obtaid formula (6.9, which cotais oly o variabl P. Aftr calculatig its valu usig (6.7, w obtai a. Now w us (6.6 to fid z. Lt us aalyz th stability of th fid poits of priod o ad two. Th loss of stability of ths fid poits idicats th mrgc of priod doublig bifurcatios. Th stability of fid poits is dtrmid by th igvalus of th Jacobia matri. First w aalyz th stability of th fid poits of priod o. Th corrspodig Jacobia matri is L ( ( a [ ( a ( z [[ a [ a a a, (6. L y a ( [, (6. a L, (6. y L 0. (6.5 yy It is covit to rwrit ths formulas i trms of th paramtr P bcaus it is this paramtr that is foud from Eq. (6.9. This allows us to skip th itrmdiat calculatios dd for fidig th paramtrs a ad z. Formula (6.7 implis [ a ( a [- P [- P [( ( P ( ( P [ ( P. ( P (6.6

26 586 Yu. V. Bibik Th dtrmiat D of th Jacobia matri (6. (6.5 is qual to w calculat this dtrmiat. Usig (6.6, w obtai Ly. First, L y ( P ( P ( P ( [ [. ( ( P ( P ( P (6.7 Formula (6.6 implis that th fid poit ( a, a of priod o corrspods to P 0; thrfor th absolut valu of th matri dtrmiat is D at this poit, hc, this poit is lliptic. This cosidrably simplifis th aalysis of stability of this fid poit bcaus th igvalus of th Jacobia matri i this cas ar, [ spl spl. (6.8 Nt, w fid th trac of th Jacobia matri (6. (6.5, which appars i (6.8: spl L ( ( ( [ ( (. ( [ ( ( ( ( [[ ( ( (6.9 Formula (6.8 implis that th lliptic fid poit bcoms ustabl udr th coditio spl. (6.0 This is a quatio for dtrmiig th first priod doublig. Th poit ( a, a bcoms ustabl actly udr this coditio. Equatios (6.9 ad (6.0 yild th fuctio (, which was arlir dotd by (. Procd to th aalysis of stability of th fid poits ( a, z ad ( z, a of priod two. For this purpos, w fid th dtrmiat of th Jacobia matri for th twostp map (twic cosisttly applid map (6., (6.5. It quals th product of th dtrmiats of o-stp maps (6., (6.5: D ( D D D( P D( P. (6. ( a ( z

27 Ivstigatio of trasitio to chaos for a Lotka Voltrra systm 587 Th fid poit ( a, z is associatd with P ; thrfor, th fid poit ( z, a is associatd with P. Hc, ( ( P ( P D ( (. (6. ( P ( P Th fid poits of th two-stp map ar lliptic, ad thy ar foud by th formula ( ( (, [ spl ( spl. (6. Ths lliptic fid poits bcom ustabl udr th coditio ( spl. (6. This quatio dtrmis th scod priod doublig. Lt us writ out this quatio i mor dtail. Th Jacobia matri of th two-stp map is L ( L ( y L ( P L ( P L ( P, (6.5 y L ( P L ( P, (6.6 y L ( y L ( P, (6.7 Its trac has th form L ( yy L ( P. (6.8 y spl ( L ( P L ( P L ( P L ( P. (6.9 y y Th quatio dtrmiig th coditios for th scod priod doublig is writt as L ( P L ( P Ly ( P Ly ( P. (6.0 Lt us fid L (P :

28 588 Yu. V. Bibik a L ( P ( ( [ a a a ( ( z a[[ [ a [ P ( ( ( ( P ( [ P [ P ( P [. ( ( P ( ( P a (6. Upo collctig similar trms, w obtai L [ P ( P [ P ( P ( ( P [[( ( P [ ( P. ( ( P (6. Now w ca fid th product L ( P L( P : L ( P L [[( ( P ( P ( ( P ( P. (6. W us th fact ( P ( ( ( (, (6. P ( Th, formula (6. taks th form (. (6.5 L ( P L ( ( P [[( P ( (( ( ( ( P (( P. (6.6 Now w ca fid L y (P :

29 Ivstigatio of trasitio to chaos for a Lotka Voltrra systm 589 L y ( P ( [ P. ( P a [ a Fially, w fid th sum L ( P L ( P : н y ( [ P ( ( P (6.7 L y ( P L y ( P [ P P [ P ( P [ P [ P ( P ( P P [ P P ( P [ P [ P Usig ths rsults, w rwrit (6.0 as. P (6.8 L ( P L ( P L ( ( P y ( P L y ( P [[( ( P ( (( ( (( P [ P [ P. (6.9 This formula dtrmis th coditios for th scod priod doublig. Multiply th lft- ad right-had sids of (6.9 by ( ( to obtai P [[( (( ( ( ( P ( P [ P ( ( [ P 0. (6.50 This quatio ca b writt as a P a P a 0, (6.5 a ( (, (6.5

30 580 Yu. V. Bibik a ( ( [( ( ( ( ( (, (6.5 For P w hav a [( ( ( ( (. (6.5 P [ a a a. (6.55 a,,, a Itroduc th otatio a a ( [( ( ( [ ( ( ( ( ( (, (6.56 a a ( [[( ( ( ( ( ( (, (6.57 P,,, [. (6.58 Now w ca plug (6.58 ito Eq. (6., which dtrmis P, ad obtai coditios for th mrgc of th scod priod doublig: P,,, [ P,,, [ P,,,. (6.59 [ P [ P,,,,,, Thus, formulas (6.59 ad (6.0 dtrmi coditios for th first ad scod priod doublig for th o-dimsioal discrt map. Equatios (6.57 ad (6.59 ca b usd to fid th fuctio (, which was arlir dotd by (. Amog th solutios to Eq. (6.59, th o that is th closst to th solutio to Eq. (6.0 should b chos. Havig formulas dtrmiig th coditios for th first

31 Ivstigatio of trasitio to chaos for a Lotka Voltrra systm 58 ad scod priod doublig, w will dtrmi th t priod doublig i th followig sctio. 7. Coditios for th mrgc of th t priod doublig bifurcatios ad coditios for th trasitio to chaos for th gralizatio of th Lotka Voltrra quatios with a sasoality factor I th prcdig sctios, w cosidrd idividual parts of th study, which w will combi ito th whol i this sctio. Ths partial ivstigatios providd us with th data that ow abl us to aalyz th gralizatio of th Lotka Voltrra systm with th sasoality factor (.9 from th viwpoit of th mrgc of bifurcatios ad coditios for th trasitio to chaos. I Sctio, th origial Lotka Voltrra quatio with sasoality factor (.5, (.6 i two variabls was rducd to Eq. (.9 i o variabl. I Sctio, th Lotka Voltrra quatio (.9 with sasoality factor was basically approimatd by th dissipativ Ho map ad trasformd to th approimatig quatio (.. This abld us to coclud that Eq. (.9 has th proprtis of Ho's map, icludig th coditios for th mrgc of bifurcatios ad trasitio to chaos. I Sctio 5, th dissipativ Ho map was rormalizd, ad a rcurrc for dtrmiig th priod doublig bifurcatios of Ho's map (. was obtaid. This rcurrc has th form A q q [ [ A (5 ( q A. (7. Hr, A q, q, ad. I Sctio Eq. (.9 was approimatd oly accurat to a arbitrary fuctio w, ad this fuctio was th foud usig th rsults obtaid i Sctios 5 ad 6. Th fuctio w provids a lik btw th approimatig quatio (. ad th basic Lotka Voltrra quatio (.9 with th sasoality factor. Usig this fuctio ad formula (., w ca stablish a rlatioship btw th paramtr A of Ho's map (. ad th paramtrs ad of th Lotka Voltrra quatio (.9 with th sasoality factor. Thrfor, giv th coditios for th mrgc of bifurcatios for th paramtr A of Ho's map (., w ca fid coditios for th mrgc of bifurcatios for th paramtrs

32 58 Yu. V. Bibik ad, which appar i th Lotka Voltrra quatio (.9 with th sasoality factor. I Sctio 6, coditios for th mrgc of th two first priod doublig bifurcatios for th Lotka Voltrra quatio (.9 with th sasoality factor wr foud. I th prst sctio, w fid coditios for th t priod doublig bifurcatios for Eq. (.9. Th valus of th paramtr A i at which th priod doublig bifurcatios of Ho's map (. occur wr foud i Sctio 5 (formulas (5. ad (7.. I additio A i ca b rprstd i trms of th paramtrs ad i (.. Th lft-had sid of this formula icluds th kow paramtr A i, ad th righthad sid icluds th kow fuctio w ad th paramtrs ad. Aftr A i dtrmiig th mrgc of priod doublig bifurcatios of Ho's map usig formula (. has b foud, th rlatioship btw th paramtrs ad ca b stablishd. It is clar that priod doublig bifurcatios for Eq. (.9 occur at crtai combiatios of th paramtrs ad. Th rlatios btw thos paramtrs at which such bifurcatios occur dtrmis th coditios for th mrgc of th third ad all subsqut bifurcatios for th Lotka Voltrra quatios with th sasoality factor (.9. Th corrspodig formula is i ( i ( A ( l( ( [( ( ( i i w. (7. Dot th rlatio by th fuctio. For ach idividual ( bifurcatio, this fuctio will b dotd by. Formulas (. ad (7. i ( ca b usd to fid i ( (. This yilds th fuctios i ( for th third ad th subsqut priod doublig bifurcatios for Eq. (.9, startig from i. Th first two fuctios ad for th first two priod doublig bifurcatios wr arlir foud i Sctio 6 (formulas (6.0 ad (6.59. Du to th complity of formulas (. ad (7., th valus of wr ( i ( obtaid o a computr. Th plots of as a fuctio of ar dpictd i Fig.. To dtrmi th poit of th trasitio to chaos for th Lotka Voltrra quatios with th sasoality factor, Eq. (7. is also usd. By pluggig th paramtr A ito this quatio, w fid th fuctio. This fuctio dtrmis th coditios for th trasitio to chaos of systm (.9. Thus, th aalysis of th coditios for th mrgc of priod doublig bifurcatios ad th trasitio to chaos for th Lotka Voltrra quatios with th sasoality factor (.9 is compltd.

33 Ivstigatio of trasitio to chaos for a Lotka Voltrra systm Dscriptio ad aalysis of figurs Figur shows th curvs of th first thr priod doublig bifurcatios. Th paramtr, whr is th amplitud of sasoal variatios i th prdator populatio dcras cofficit, is plottd o th horizotal ais. Th paramtr (, whr is th amplitud of sasoal variatios i th pry populatio icras cofficit, is plottd o th vrtical ais. Ths paramtrs idirctly dtrmi th populatio of prdators ad pry. Th thr bifurcatio curvs ar costructd basd o th data obtaid usig formula (7.. It is s i Fig. that th curv corrspodig to th t bifurcatio lis abov th curv corrspodig to th prcdig bifurcatio. Thrfor, for ach fid, th trasitio to th t bifurcatio rquirs th paramtr to b icrasd. Th paramtr idirctly affcts th populatio of pry. Th iflow of biomass ito th systm i th form of pry populatio rsults i th systm

34 58 Yu. V. Bibik citatio, th mrgc of w bifurcatios, ad ultimatly i th trasitio to chaos. O th othr had, th curvs rprstig th bifurcatios icras with icrasig. Th icras i idirctly rsults i th growth of prdator populatio. Thrfor, as th umbr of prdators icrass, bifurcatios ca mrg oly if th populatio of pry icrass. It is s from Fig. that, i ordr for bifurcatios i th viciity of to appar, th paramtr must cost cost grow as. Figur dpicts th curv of trasitio to chaos for th origial systm. Th paramtr is plottd o th horizotal ais, whil ( is plottd o th vrtical ais. It is s from Fig. that th curv of trasitio to chaos lis abov all th priod doublig bifurcatio curvs show i Fig.. Thrfor, at a fid, th curv of th trasitio to chaos is attaid at th maimum valu of th paramtr. Furthrmor, as i Fig., th icras i is associatd with th icras i. As icrass, additioal iflow of biomass i th form of pry populatio is dd for th trasitio to chaos. As, w hav cost cost. 9. Coclusios Rct studis of oliar systms showd that v a small modificatio of simpl modls towards mor ralistic os rsults i th mrgc of chaos ad compl dyamical bhavior, which ar charactristic of ral lif. Th Lotka Voltrra systm with a sasoality factor studid i th prst papr cofirms this fact. Itroductio of th sasoality factor rsultd i th mrgc of chaos i th Lotka Voltrra systm at crtai valus of th paramtrs. Takig ito accout th complity of th problm, th aalysis was prformd i svral phass. First, th origial systm of diffrtial quatios (.5, (.6 was rplacd with th discrt map (., (.. This complicatd th problm du to th itroductio of th sasoality factor. Howvr strag it may sm, this simplifid th ivstigatio mthod bcaus th diffrc quatios (., (. wr obtaid. Nt, ths quatios wr rducd to a form that is mor covit for aalysis; mor prcisly, Eqs. (., (. wr rducd to a simplr quatio (.9. Nt, Eq. (.9 was approimatd by th dissipativ Ho map (.. This suggstd th coclusio that th Lotka Voltrra systm with th sasoality factor has th basic proprtis of Ho's map, icludig priod doublig bifurcatios ad trasitio to chaos. Th, th rormalizatio group tchiqu was applid to th dissipativ Ho map (., which abld us to obtai a rcurrc for dtrmiig th coditios for th mrgc of

35 Ivstigatio of trasitio to chaos for a Lotka Voltrra systm 585 priod doublig bifurcatios. Th rormalizatio of th dissipativ Ho map usig a aalog of Hllma's mthod [ abld us to fid priod doublig bifurcatios ad dtrmi th poit of th trasitio to chaos for systm (.9. Thus, th multistp aalysis ad modr tchiqus of th thory of chaos abld us to fid th tir chai of priod doublig poits ad th poit of trasitio to chaos for th systm udr amiatio. Rfrcs [ Yu.V. Bibik, D.A. Saracha, A Mthod of th Dissipativ Ho Map Rormalizatio, Computatioal Mathmatics ad Mathmatical Physics, 50 (00, o., [ B.V. Chirikov, A uivrsal istability of may-dimsioal oscillator systms, Physics Rports, 5 (979, [ M.J. Figbaum, Quatitativ Uivrsality for a Class of Noliar Trasformatios, Joural of Statistical Physics, 9 (978, o., [ M.J. Figbaum, Th Uivrsal Mtric Proprtis of Noliar Trasformatios, Joural of Statistical Physics, (979, o. 6, [5 M.J. Figbaum, J.M. Gr, R.S. MacKay, F. Vivaldi, Uivrsal Bhaviour i Familis of Ara-prsrvig Maps, Physica D: Noliar Phoma, (98, o., [6 M.P. Fishma ad D.A. Egolf, Rvalig th Buildig Block of Spatiotmporal Chaos: Dviatios from Etsivity, Joural Physical Rviw Lttrs, 96 (006, [7 R.H.G. Hllma, Slf Gratd Chaotic Bhaviour i Noliar Mchaics, I E.D.G. Coh (ds: Fudamtal Problms i Statistical Mchaics, Vol. 5, North Hollad, Amstrdam, 980. [8 R. H. G. Hllma, G. Iooss, R. Stora, Chaotic Bhaviour of Dtrmiistic Systms, Rport Ls Houchs Sssio, Vol. 6, 98, North Hollad, Amstrdam (Nthrlads, 6. Ls Houchs Summr School, Ls Houchs (Frac, 9 Ju - Jul, 98.

36 586 Yu. V. Bibik [9 M. Ho, A Two-dimsioal Mapig with a Srag Attractor, Commuicatios i Mathmatical Physics, 50 (976, o., [0 B.R. Hut, T.Y. Li, J.A. Kdy, H.E. Nuss, Th Thory of Chaotic Attractors, Sprigr, Nw York, [ A. Libchabr, C. Laroch, S. Fauv, Priod Doublig Cascad i Mrcury, a Quatitativ Masurmt, Joural d Physiqu Lttrs, (98, o. 7, [ A. Libchabr, C. Laroch, S. Fauv, Two-Paramtr Study of th Routs to Chaos, Physica D: Noliar Phoma, 7 (98, o. -, [ E.N. Lorz, Dtrmiistic Nopriodic Flow, Joural of th Atmosphric Scics, 0 (96, o., [ E.N. Lorz, Th prdictability of a flow which prosssss may scals of motio, Tllus, (969, o., [5 A.J. Lotka, Elmts of Physical Biology, Williams ad Wilkis, Baltimor, Md., 95. [6 R.S. MacKay, Rormalizatio i Ara-Prsrvig Maps, Ph. D. Thsis, (Uiv. Microfilms Ic. A Arbor, MI, Pricto, 98. [7 B.B. Madlbrot, Th Fractal Gomtry of Natur, Frma, Sa Fracisco, 98. [8 H.G. Schustr, Dtrmiistic Chaos: A Itroductio, Physic-Vrlag, Wihim, 98. [9 H.G. Schustr ad W. Just, Rmarks ad Rfrcs, i Dtrmiistic Chaos: A Itroductio, Wily VCH Vrlag GmbH &Co KGaA, Wihim, FRG, [0 M. Tabor, Chaos ad Itgrability i Noliar Dyamics: A Itroductio, Joh Wily&Sos Ic., Nw York, 989. [ J.M.T. Thompso ad H.B. Stwart, Noliar Dyamics ad Chaos, Joh Wily, Chichstr.

37 Ivstigatio of trasitio to chaos for a Lotka Voltrra systm 587 [ J.M.T. Thompso, R. Ghaffari, Chaos Aftr Priod-doublig Bifurcatios i th Rsoac of a Impact Oscillator, Physics Lttrs A, 9 (98, o., [ V. Voltrra, Variazioi Fluttuazioi di Numro d Idividui i Spci Aimali Covivti, 96, Mmori dlla Rgia Accadmia Nazioal di Lici, (96, -, Eglish traslatio i Chapma, R.N., Aimal Ecology, McGraw Hill, Nw York, 9. [ V. Voltrra, Lcos sur la Thori Mathmatiqu d la Lutt pour la Vi, Gauthr-Villars, Paris, 9. [5 G.M. Zaslavsky, Hamiltoia Chaos ad Fractioal Dyamics, Oford Uiv. Prss, Oford, 008. Rcivd: August, 05; Publishd: Sptmbr 8, 05