Global organization of spiral structures in bi-parameter space of dissipative systems with Shilnikov saddle-foci

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1 Globl orgniztion of spirl structures in bi-prmeter spce of dissiptive systems with Shilnikov sddle-foci Roberto Brrio,, Fernndo Bles, 2 Sergio Serrno, nd Andrey Shilnikov 3 Deprtmento de Mtemátic Aplicd nd IUMA. University of Zrgoz. E-9. Spin. 2 Deprtmento de Físic Aplicd. University of Zrgoz. E-9. Spin. 3 Neuroscience Institute nd Deprtment of Mthemtics nd Sttistics, Georgi Stte University, Atlnt 333, USA (Dted: Mrch 4, 2) We revel nd give theoreticl explntion for spirl-like structures of periodicity hubs in the bi-prmeter spce of generic dissiptive system. We show tht orgnizing centers for shrimp - shped connection regions in the spirl structure re due to the existence of L. Shilnikov homoclinics ner codimension-2 bifurction of sddle-foci. PACS numbers:.4.ac,.4.pq Over recent yers, gret del of experimentl studies nd modeling simultions hve been directed towrd the identifiction of vrious dynmicl nd structurl invrints to serve s the key signtures uniting often diverse nonliner systems in single clss. One such clss of low order dissiptive systems hs been identified to possess one common, esily recognizble pttern involving spirl structures clled the periodicity hub long with shrimp-shped domins in biprmetric phse spce [, 2]. Such ptterns turn out to be ubiquitously like in both time-discrete [3, 4] nd time-continuous systems [ 7], s well s in experiments [, 8]. Despite the overwhelming number of studies reporting the occurrence of spirl structures, there is still little known bout the fine construction detils nd underling bifurction scenrios for these ptterns. In this Letter we study the genesis of the spirl structures in two exemplry, low order systems nd revel the generlity of underlying globl bifurctions. We will demonstrte tht such prmetric ptterns long with shrimp-shped zones re the key feture of systems with homoclinic connections involving sddle-foci meeting the single Shilnikov condition [9]. The occurrence of this bifurction cusing complex dynmics is common for plethor of dissiptive systems, describing (electro)chemicl rections [], popultion dynmics [], electronic circuits nd nonliner optics [2, 8, 2]. The first prdigmtic exmple is the cnonicl Rössler system [3]: ẋ = α (y + z), ẏ = x + y, ż = b + z(x c), () with two bifurction prmeters nd c (we fix b =.2). This clssicl model exhibits the spirl nd screw chotic ttrctors fter period doubling cscde followed by the Shilnikov bifurction of the sddle-focus. For c 2 > 4b the model hs two equilibrium sttes, P,2 ( p, p, p ), where p = ( c c 2 4b)/2. The second exmple is the Rosenzweig-McArthur tritrophic food chin model [, 4]: ẋ = x [ r ( x/k) y/( + 3z) ], ẏ = y [ y/( + 3x) z/( + 2y).4 ], ż = z [ y/( + 2y). ], for food chin composed of logistic prey, x, Holling type II predtor, y, nd top-predtor, z; two bifurction prmeters K nd r controls the regrowth rtes of the prey []. Bi-prmetric screening the Rössler (pnels A-B) nd food chin (pnels C-D) models unveils stunning universlity of the periodicity hubs in the bifurction digrms shown in Fig. of both models. Ech is built on dense grid of points in the prmeter plne. Solutions of the models were integrted using high precision ODE solver TIDES []. The color brs on the right in Fig. yield spectrum of the Lypunov exponents. Pnels (A) nd (C) of the figure revel the chrcteristic spirl ptterns, where drk nd light colors discriminte the regions of regulr nd chotic dynmics corresponding to zero nd positive Lypunov exponent λ, respectively. Pnels (B) nd (D) show the enhnced fine structures of the bifurction digrms of the models, due to vritions of both Lypunov exponents λ nd λ 2. The white stripes expose shrimps-shped res (within red-boxes) on the drk bckground of the regulr (λ = ) region, s well s in the multicolored region corresponding to complex dynmics (λ > ). Pnels re overlid with the (blue) curves, obtined using [6], corresponding to sddle-node (or fold) bifurctions of periodic orbits. The curves on-side demrcte the stbility windows from chotic regions within the spirl structure either vi the intermittency of type I boundry crisis [6], or due to clssicl period doubling cscde. In the cse of the Rössler model, the sddle-node curves spirl onto single T(erminl) point. This T-point lso termintes bifurction curve (blck) corresponding to formtion of homoclinic loop of sddle-focus, P 2 of the Rössler model. Besides, nother curve (green) psses though the T-point: crossing over it mkes the chotic (2)

2 ROSSLER MODEL prmeter c 2 prmeter c λ 2 (λ =) λ > A B FOOD CHAIN MODEL prmeter r. * *.. C.. prmeter K prmeter r.... D... prmeter K λ2 (λ =) λ > Figure. (Color online) Spirls nd shrimps in biprmetric bifurction digrms for the Rössler (A-B) nd tritrophic food chin (C-D) models. The T-point of the hub is locted t (, c) = (.798,.384) nd (K, r ) = (.87, ) (resp.). The color brs for the Lypunov exponent rnge, red for positive nd blck for negtive vlues, identify the regions of chotic nd regulr dynmics (resp.). For visibility the prmeter plne of the food chin model is untwisted by trnsformtion r = r +.(K )/ Left monochrome pnels re superimposed with bifurction curves: blue for sddle-nodes, nd blck for homoclinic bifurctions of sddle-foci. The green boundry determines chnge in the topologicl structure of chotic ttrctors from spirl (t ) to screw-shped (t ). ttrctor in the phse spce of the model chnge the topologicl structure from spirl to screw-shped. The topologicl structure of the Rössler ttrctor is described in terms of pper-sheet model, clled templte mde of norml nd twisted, like Möbius bnd, stripes. The topologicl model cn be quntified by set of linking numbers the locl torsions. The torsions re, loclly, the crossings number of the stripes in the templte, i.e. the number of twists of the lyering grph between ny two unstble periodic orbits in the chotic ttrctor. The locl torsions determine the linking mtrices nd hence the templte of the ttrctor [7]. Prcticlly, the templte my be derived using Poincré return mp defined on successive locl mxim, y(i), of trjectories in the chotic ttrctor nd studying the unstble periodic orbits foliting the ttrctor. So, the spirl ttrctor of the Rössler model t =.4 (lbeled by in Fig. (A,C)) genertes D unimodl mp s one shown in Fig. 2) The single criticl determines the boundry between the norml nd twisted (resp.) stripes. This lets symbolic description be nturlly introduced using two symbols, nd, for ech brnch or stripe. In the cse of screw ttrctor t =.8 (lbeled by in Fig. (A,C)), the corresponding mp in Fig. 2 hs bimodl grph, i.e. three monotone brnches divided by two criticl points. Hence, the symbolic dynmics here is defined on set of three symbols: {,, 2}. Addition of the second criticl point in the mp is direct indiction tht the spirl ttrctor chnges its topology. This criteri ws used to locte the corresponding boundry (green) tht seprtes the existence regions of the ttrctors of both types in bifurction digrm in Fig.. Notice tht this boundry goes through the T-point. The linking mtrices, which contin ll topologicl in-

3 3 2 = y(i+) y(i) =.8 2 within the prmeter rnge in the digrms. Mgnifiction of the corresponding bifurction curve the digrm revels tht wht ppers to be s single bifurction curve (blck), indeed is mde of two brnches (Fig. 3). This curve hs U-shpe with cusp t the T-point. To exmine the U-shpe we plot the bifurction curve in terms the L 2 -norm of the homoclinic orbit ginst the bifurction vlues of the. Fig. 3 shows tht the T-point termintes two brnches of homoclinic loops: bottom one corresponding to the primry loop, while the top one correspond to secondry loop with two rounds..8.6 =.8 homoclinic bif. curve 2 y(i+) y(i) L =. =. =. =. =.8 Figure 2. (Color online) Poincré return mppings in left pnels for spirl nd screw-shped (resp.) chotic ttrctors in the Rössler model () t =.4 nd.8 with c =. Right pnels show the corresponding topologicl templtes =.8 formtion for the spirl nd screw ttrctors, re given by [8]: ( ) M sp =, M sc = 2. (3) 2 2 The digonl elements in ech mtrix re the sum of the signed hlf-twists in ech brnch, i.e. the self-torsion. The off-digonl elements re the sum of the oriented crossings between the brnches. Thus, in the spirl ttrctor we hve vlue implying tht the right brnch () hs no torsion, nd the middle brnch () (left brnch (2)) hs hlf-twist (the vlue (or 2)). The other vlues indicte tht two brnches cross once, s shown in the topologicl templte. The (blck) bifurction curve in Fig. (A) corresponds to formtion of the primry homoclinic orbit to the sddle-focus, P 2, of topologicl type (,2), i.e. D stble nd 2D unstble mnifolds, in the model (). Depending on the mgnitudes of the eigenvlues of the sddlefocus, the homoclinic bifurction cn give rise to the onset of either rich complex or trivil dynmics in the system [9, 9]. The cses under considertions meet the Shilnikov conditions nd hence the existence of single homoclinic orbit implies chotic dynmics in the models Figure 3. (Color online) Trnsformtion of homoclinic orbit to the sddle-focus P 2 in the Rössler system: its L 2-length is plotted ginst the bifurction. T-point termintes two brnches: bottom corresponding to the primry homoclinic loop, while the top one corresponds to the secondry loop with two rounds. Homoclinic orbits (shown blue) re smpled t the indicted points. Fig. 4 outlines the bifurction unfolding round the spirl hub [7]. Inset (A) for the Rössler model depicts number of identified, sddle-node bifurction curves originting from codimension-2 point, lbeled s B(elykov), towrd the spirl hub. At this B-point, the equilibrium stte, being sddle with double rel positive eigenvlues becomes sddle-focus for smller vlues of the. The unfolding of this bifurction is known [2, 2] to contin bundles of countble mny curves corresponding to sddle-node nd period doubling bifurctions of periodic orbits, s well s vrious homoclinic ones. Indeed, both B- nd T- points together globlly determine the structure of the (c, )-bifurction portrit of the Rösler model. Fig.4(B) sketches phenomenologicl scheme of the formtion of the spirl hub long with the shrimps. All sddlenode bifurction curves originting from the B-point determine the boundries of different shrimps ner the spirl hub. Indeed, the spirl structure cn generte n infinite chin of shrimps [2, ]. Fig.4(C) presents few

4 4 visible shrimps, S 2j nd S 2j+, singled out by solid red curves, lso corresponding to sddle-node orbits, which fold round the T-point in the region of spirl chotic ttrctor. Menwhile, the cusp-shped (blue) sddle-node bifurction curves join the successive S 2j -th nd S 2j -th in the region of the screw-type ttrctor. These folded nd cusp-shped bifurction curves of sddle-node periodic orbits shpe the structure of the spirl hub nd of the shrimps. The shrimps serve s connection centers tht form the chrcteristic spirl structure in the prmeter plne. We hve presented generic scenrio for the formtion of the spirl structures nd shrimps in the biprmeter spce of systems with Shilnikov sddle-foci. The skeleton of the structure is formed by bifurction curves of sddlenode periodic orbits tht ccompny the homoclinics of the sddle-focus. These bifurction curves distinctively shpe the shrimp zones in the vicinity of the spirl hub. In the Rössler model, these bifurction curves originte from codimension-2 Belykov point corresponding to the trnsition to the sddle-focus from simple sddle. The feture of the spirl hub in the Rössler nd the tritrophic food chin models is tht the T-point gives rise to the lterntion of the topology structure of the chotic ttrctor trnsitioning between the spirl nd screw-like types. The findings let us hypothesize bout universlity of the structure of the spirl hubs in similr systems with chotic ttrctors due to homoclinics of the Shilnikov sddle-focus. The uthors R.B. nd S.S. cknowledge the support from the Spnish Reserch project MTM29-767, F.B. from the Spnish Reserch project AYA28-72, nd A.S. cknowledges NSF grnt DMS-99 nd RFFI Grnts No c c 2 2 Homoclinic bif. B point n-brnches (n+)-brnches chotic Rossler b=.2 regulr S 2j- S 2j+ S 2j S 2j- Sddle-node bif. Rossler b=.2 Homoclinic bif. S 2j+ B point S 2j A B C Corresponding uthor: rbrrio@unizr.es [] C. Bontto, J. C. Grreu, nd J. A. C. Glls, Phys. Rev. Lett. 9, 439 (2). [2] C. Bontto nd J. A. C. Glls, Phys. Rev. Lett., 4 (28). [3] J. A. C. Glls, Phys. Rev. Lett. 7, 274 (993). [4] Y. Zou, M. Thiel, M. C. Romno, J. Kurths, nd Q. Bi, Internt. J. Bifur. Chos Appl. Sci. Engrg. 6, 367 (26); E. N. Lorenz, Phys. D 237, 689 (28). [] J. A. C. Glls, Internt. J. Bifur. Chos Appl. Sci. Engrg. 2, 97 (2). [6] R. Brrio, F. Bles, nd S. Serrno, Phys. D 238, 87 (29). [7] P. Gsprd, R. Kprl, nd G. Nicolis, Journl of Sttisticl Physics 3, 697 (984). [8] R. Stoop, P. Benner, nd Y. Uwte, Phys. Rev. Lett., 742 (2); V. Cstro, M. Monti, W. B. Prdo, J. A. Wlkenstein, nd E. Ros, Jr., Internt. J. Bifur. Chos Appl. Sci. Engrg. 7, 96 (27). [9] L. P. Shilnikov, Sov. Mth. Dokl. 6, 63 (96); L. P. Shilnikov, A. L. Shilnikov, D. Turev, nd L. O. Chu, Figure 4. (Color online) Outline of the spirl structures: (A) two kinds, folded nd cusp-shped of sddle-node bifurction curves for the Rössler model originting from the cod-2 homoclinic B-point. (B) phenomenologicl sketch of the spirl hub formed by the shrimps. (C) Mgnifiction of the bifurction portrit of the spirl hub, overlid with principl folded (red) nd cusp-shped (blue) bifurction curves setting the boundries for lrgest shrimps in the Rössler model t b = 2. Methods of qulittive theory in nonliner dynmics. Prt II (World Scientific Publishing Co. Inc., 2). [] M. A. Nscimento, J. A. C. Glls, nd H. Vrel, Phys. Chem. Chem. Phys. 3, 44 (2). [] A. Hstings nd T. Powell, Ecol. 72, 896 (99); Y. A. Kuznetsov, O. De Feo, nd S. Rinldi, SIAM J. Appl. Mth. 62, 462 (2). [2] J. G. Freire nd J. A. C. Glls, Phys. Rev. E 82, 3722 (2). [3] O. E. Rössler, Phys. Lett. A 7, 397 (976).

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