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Handbook of Applications of Chaos Theory

Christos H. Skiadas, Charilaos Skiadas

On the Transition to Phase Synchronized Chaos

Publication details https://www.routledgehandbooks.com/doi/10.1201/b20232-5 Erik Mosekilde, Jakob Lund Laugesen, Zhanybai T. Zhusubaliyev Published online on: 01 Jun 2016

How to cite :- Erik Mosekilde, Jakob Lund Laugesen, Zhanybai T. Zhusubaliyev. 01 Jun 2016, On the Transition to Phase Synchronized Chaos from: Handbook of Applications of Chaos Theory CRC Press Accessed on: 01 Oct 2021 https://www.routledgehandbooks.com/doi/10.1201/b20232-5

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The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The publisher shall not be liable for an loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. Downloaded By: 10.3.98.104 At: 17:44 01 Oct 2021; For: 9781466590441, chapter3, 10.1201/b20232-5 . Introduction 3.1 Zhusubaliyev T. Zhanybai and Laugesen, Lund Jakob Mosekilde, Erik ayivsiaost osdrdfeetfrso h ocle rjciesnhoiainb hc the which by synchronization projective so-called the of forms different consider to investigators many by instance, periodic for demonstrated, all been has form: [48]. oscillators al. Rössler unusual this et identical Yanchuck of rather application coupled The two a are of [13]. stable system takes to that transversely a trajectories be for criterion attracted must condition of state are stability chaotic set synchronized the average dense the the conditions, in a embedded on these orbits contain Under may trajectories state it. the this from although of repelled neighborhood which, any in state, situation these phe- synchronized All a the [29]. from bifurcations derived phenomena and blow-out these are [2,21], bubbling, nomena attraction attractor of sta- [36], basins its intermittency riddled loses on–off globally state range include and synchronized broad also locally the the so-called as of the observed because Besides be and [27]. can bility [4,44] that communication phenomena secure dynamic and nonlinear [42,46] interesting control of chaos as such areas in the an oscilla- from chaotic in identical distinguished two vary between clearly interaction to the is [30,34]. continues in tors synchronization observe amplitude can chaotic oscillator’s one synchronization or of the full signal, or form forcing complete while This external oscillator, manner. an chaotic of uncorrelated rhythm another essentially the of to dynamics dynamics internal the its to of frequencies chaotic the a which adjusts in synchronization attractor of form interesting an denotes [3,11,22] synchronization phase Chaotic © 2016 Group, by Taylor LLC & Francis nrcn er,teitrs napiaino ho ycrnzto o euecmuiainhsled has communication secure for synchronization chaos of application in interest the years, recent In application potential its of because both attention significant attracted has synchronization Complete References...... 59 Conclusions...... 56 Acknowledgments ...... 59 Bifurcations...... 54 Saddle–Node of Set Accumulating 3.8 ...... 53 Window Period-3 the in Structure Bifurcation 3.7 3.6 Multi-Layered of Reconstruction and Formation ...... 46 Torus Ergodic the of Doubling Period 3.5 Bifurcation...... 45 Fold-Flip a Near Structure Bifurcation 3.4 Synchronization 1:1 3.3 the of Structure Bifurcation Main Introduction...... 39 3.2 3.1 oPaeSynchronized Phase to eoac oi...... 50 Tori Resonance ...... 41 Zone nteTransition the On Chaos 3 39 Downloaded By: 10.3.98.104 At: 17:44 01 Oct 2021; For: 9781466590441, chapter3, 10.1201/b20232-5 hoiainrgm eaet h eidduldegdctr htaekont xs usd h resonance the outside exist to known unexplored. are largely that remained syn- tori have the the [5,43], ergodic how in zone of exist period-doubled that the details tori to the resonance relate however, the regime far, how mediated chronization and So organized, crises manifold. are boundary they branched a how a arise, as bifurcations on synchronization saddle–node bifurcation phase pair chaotic to unstable–unstable transition an the by described have [39] al. et o h anbhvoa oe ftehroial ocdRslrsse.Mroe,b pligalift 2 a by applying separated by Moreover, points system. phase that Rössler (such forced variable harmonically existence phase the of the of regions syn- to the modes phase to of behavioral overview main preliminary the transition a the provided of for have structure with [45] nested associated al. a et families Vadivasova large described attractor and a chronization, have different that [37] for suggested al. regions [32] et synchronized For Postnov al. synchronization. phase oscillators, phase et chaotic period-doubling Pikovsky to coupled transition oscillator, of the cycle systems edge at place the limit take along forced collisions bifurcation attractor–repellor a tangent of for the number zone with cycles, analogy periodic synchronization the unstable the on different of Drawing the state. for chaotic existence the the of constitute of range which form the delineate and that size bifurcations the saddle–node in changes through [45]. and state attracting oscillations, the the with in characteristic of associated section exponents, distribution Poincaré the Lyapunov of spectral and the the variation synchronized of from specific phase changes the in Apart between [15]. transition reflected sources the is different distribution, chaos many area nonsynchronized spectral from this the in series works in time through changes experimental above-mentioned developed interpret methods to the used and been concepts have the and [35,41,42], interest translational theoretical time this breaks signal forcing a [33]. of phases and application amplitu forcing The small relatively oscillator. for even of unforced and, instability symmetry an inherent of the phase between the balance for a by controlled is thesystemononesideandnonlinearrestrainingmechanismsontheother,thereisnopreferredvalue starts oscillations again endogenous oscillator the chaotic of the of amplitude forcing locks frequency the mean frequency the with interval, increases spectral this typically toslideawayfromtheforcingfrequency.Thistypeofbehaviorisassociatedwiththefactthat,whilethe mean leaves interval system the synchronization the the where as of and, synchronization amplitude width phase The until frequency. frequency chaotic forcing the forcing of the of region to the variation the enters by system unaffected relatively the remains spectrum frequency [7]. the neurons how interacting bursting of for networks [27], scale-free reactors for and microbiological [23], of kidney chains the for of units instances, functional for biological performed, multidimensional in been synchronization have phase systems chaotic of oscilla- analyses cellular bifurcation coherent of nearly Detailed [12]. a and state seemingly tions of states state into a concentrations from functional calcium transition intracellular a of involves of wall range fluctuations arteriolar random the wide in cells that a muscle instance, to smooth for the well-established, adapt of is activation It can the dynamics. that coherent almost structure to un-synchronized a from into switch rapidly translate to likely is blood chronization the in [14]. dynamics the kidney oscillatory reac- of and electrochemical units [26], functional coupled cells neighboring [40], of nerve regulation generator interacting flow wave of behaviors low-amplitude sub-threshold a [50], by tors [9], paced lasers chemical, tubes chaotic physical, interacting discharge different of arrays of plasma [3], variety oscillators electronic wide coupled a including systems, in biological occurs of and synchronization properties of the synchronization form This oscillators. to study chaotic effort [31,47]. significant been initiated a has time, different oscillators same for chaotic order allow the fractional to At different reformulated [25]. be can scaling oscillators of interacting forms the of identity complete of requirement 40 © 2016 Group, by Taylor LLC & Francis o prltp ho,i skonta h deo h ycrnzto oecnit fadnestof set dense a of consists zone synchronization the of edge the that known is it chaos, spiral-type For significant attracted has synchronization phase chaotic of phenomenon the observation, first its From [6,27,33] observe can one oscillator, chaotic driven periodically single a with experiments several In syn- phase chaotic with associated modes oscillatory of coordination complex the world, animate the In nonidentical interacting for characteristic is chapter this in described as chaos, synchronized Phase e,tecatcoclao ilajs t frequencies its adjust will oscillator chaotic the des, adoko plctoso ho Theory Chaos of Applications of Handbook π olne r osdrdtesm) Rosa same), the considered are longer no Downloaded By: 10.3.98.104 At: 17:44 01 Oct 2021; For: 9781466590441, chapter3, 10.1201/b20232-5 hw cnars h anrgo f11paesnhoiain[,73] ee ehv lte h ratio the plotted have we Here, frequency [6,27,33]. spectral synchronization mean phase the 1:1 between of region main the across scan a shows develops region this of structure chaos. the to transition how doubling examine to is parameter analysis the current as the synchronization prominent of most purpose the produces the generally and resonance regime, 1:1 synchro- The oscillations forcing. periodic external simple generated the internally of with the dynamics where nize regime (ergodic) zones resonance quasiperiodic the of two-mode set in dense of a applied regions by interrupted displays is system forcing Rössler periodic the oscillations, external periodic an When bifurcations. period-doubling of sda iucto aaees o h bv aaee aus h nocdRslrsse undergoes system Rössler unforced the values, parameter above aHopfbifurcationat the For parameters. bifurcation as used h values the parameters The forcing. external the resents hthsas evda eil o h netgto fcatcpaesnhoiaini ayprevious many in synchronization phase chaotic of investigation the for vehicle Here a [35,41,42]. studies as served also has that oeaeognzdadilsrtstetasto rmmlilyrdrsnnetrst period-doubled to resonance torus the resonance of multi-layered edge from the transition along the curves torus. ergodic illustrates bifurcation and resonance saddle–node organized multilayered the are how complicated zone explains increasingly of also chapter breakdown This final tori. and illustrates chapter present reconstruction, the cascade formation, period-doubling the the through system Rössler forced periodically [24]. a curves in bifurcation saddle–node the between holes bifurca- the global) close and/or to required (local bifurcation therefore, Additional saddle–node the are, bifurcation. tions to period-doubling preceding are tangent the in curves produced period-doubling the curves fold- so-called where period-doubling the [20] in subsequent not points of but bifurcation to, pairs flip close to arise relates curves bifurcation scaling charac- saddle–node the the new causing where the Moreover, alternately thereby transitions. transition cycles. emanate and the curve resonance curves of period-doubling structure emerging bifurcation the cyclic of the saddle–node branches teristic for unstable these the existence zone, and of stable synchronized range the the from the of side delineate to particular order a For in zone synchronized the of [52–54]. produc- tori thereby resonance torus, the multilayered resonance so-called both period-1 the of original of the bifurcations system the to a transverse of period-doubling ing direction dynamics of a in the cascades cycles how saddle through demonstrated and develop node have system we way, Rössler an of this forced displays analysis In complete periodically tongue more [24,51]. these a involved resonance by perform structure a to Inspired bifurcation [10,19] of bifurcations. the techniques continuation edge period-doubling applied recently of the have pairs along we results, subsequent chaos involving to behavior, demon- has transition scaling work unusual this period-doubling particular, In the [16–18]. al. that et Kuznetsov strated by developed criticality C-type) called (also fteapidfriga ucino h ocn rqec.Tesnhoiainzn lal tnsotas out stands clearly zone synchronization The which frequency. in forcing frequencies forcing the of of interval an function a as forcing applied the of e scnie h eidclyfre öse system Zone Rössler forced periodically Synchronization the consider 1:1 us the Let of Structure Bifurcation Main 3.2 Chaos Synchronized Phase to Transition the On © 2016 Group, by Taylor LLC & Francis Fgr 3.1a Figure chaos, synchronized phase to transition the of features characteristic the of some illustrate To yfloigtedvlpeto h hs otatfrtevrossal n ntbersnnecycles resonance unstable and stable various the for portrait phase the of development the following By sides both on born are curves bifurcation saddle–node new bifurcation, period-doubling each After cyclic of theory the is picture detailed more a of development the toward contribution important An a = b = c nrae hog h ag,weeteufre öse ytmudrosisperiod- its undergoes system Rössler unforced the where range, the through increases . and 0.2 c = x ˙ x , =− . n,fricesn ausof values increasing for and, 0.4 y A ,and = y .,weestenniert parameter nonlinearity the whereas 0.1, − z z r h yai aibe fteufre siltr and oscillator, unforced the of variables dynamic the are + ω A 1 sin( ftefre öse ytmi qain31adtefrequency the and 3.1 Equation in system Rössler forced the of ω 1 ωt a onie with coincides ); and y b ˙ = n h ocn amplitude forcing the and x + ay c o ocn rqece eo hsinterval, this below frequencies forcing For ω. h ytmehbt egnamcascade Feigenbaum a exhibits system the , ; z ˙ = b + c z n h ocn frequency forcing the and (x − c A (3.1) ) ilb etcntn at constant kept be will A sin( ωt rep- ) ω are 41 ω Downloaded By: 10.3.98.104 At: 17:44 01 Oct 2021; For: 9781466590441, chapter3, 10.1201/b20232-5 poie noeve ftefrtfu eidduln iuctosi h : eoac oge[24]. tongue 1:1 resonance the in Fig- in bifurcations period-doubling diagram four bifurcation first The two-dimensional the of overview cascade. an provides period-doubling 3.2 the ure of steps first its takes tem n h rniin htocrfrlwrvle ftenniert parameter nonlinearity the of values lower for occur that transitions the ing etergo hr h siltrrpdyapoce t qiiru on ln h tbemnfl (i.e., manifold stable the along point equilibrium its approaches the rapidly fortheunsyn- oscillator even the that, where region note the to be is alsointeresting It state. is points chronizedattractor,thedensityofstroboscopicpointsisverylowinaparticularregion.Thisislikelyto synchronized the the of phase the for spread in distribution the point while smaller attractor, the synchro- whole significantly how phase the Note, across of points. uniformly range nearly 1000 the spreads of state in synchronization, order unsynchronized distribution the phase point involves of distribution characteristic region Each the chaos. of the nized example outside an observed shows distribution 3.1c point Figure obtained and typical points of the syste shows distribution the a 3.1b of with Figure position system the Rössler of forced marking of periodically stroboscopic projection the by a for superimposed have trajectory we Here space frequency. forcing phase the the in shift a to response in chaos chronized h enseta frequency spectral mean the 3.1 FIGURE 42 h enseta rqec xed h ocn rqec n,we h ocn rqec rse the crosses frequency forcing the when interval, and, synchronization the frequency of forcing edge the upper exceeds frequency spectral mean the hs ycrnzto.Poetoso h ocdRslroclao r hw sbcgon o h toocpcpoints stroboscopic the for background as shown (c). chaotic are and of oscillator range (b) the Rössler in inside forced (c) the and of outside Projections (b) system synchronization. Rössler phase forced the for positions space phase sampled boscopically © 2016 Group, by Taylor LLC & Francis obte nesadwa apn ntetasto opaesnhoie ho e ssatb examin- by start us let chaos synchronized phase to transition the in happens what understand better To iue31 n rvd natraieilsrto ftetasto rmusnhoie opaesyn- phase to unsynchronized from transition the of illustration alternative an provide c and 3.1b Figure z -axis). (b) y –11.0 11.0 anfaue ftetasto opaesnhoie ho.()Snhoiaincaatrsi:Rtoof Ratio characteristic: Synchronization (a) chaos. synchronized phase to transition the of features Main –11.0 ω =1.04 ω 1 otefrigfrequency forcing the to ω /ω (a) 1 0.96 0.98 1.02 1.04 x 1.0 .410 1.08 1.06 1.04 c =4.8 ω 1 11.0 olne olw h aito of variation the follows longer no ω fe ahcmltdpro ftefrigsignal. forcing the of period completed each after m safnto ftefrigfeuny itiuino stro- of Distribution frequency. forcing the of function a as (c) y –11.0 adoko plctoso ho Theory Chaos of Applications of Handbook ω 11.0 1. 11.0 –11.0 ω =1.08 .01.12 1.10 c steufre öse sys- Rössler unforced the as x ω. c =4.8 Downloaded By: 10.3.98.104 At: 17:44 01 Oct 2021; For: 9781466590441, chapter3, 10.1201/b20232-5 iucto uvsoiiaefo onso h eidduln curve period-doubling the on points from originate curves bifurcation r eurdt opeetebre ftersnnezn o h eid2cycles. period-2 the for zone bifurcations resonance global the and the of local between gap border additional a the of is number complete a to see, There required shall points. are we bifurcation as fold-flip and, the curves from bifurcation precisely saddle–node emanate not do curves cation ipasapi fsdl n obyusal adepro- ylstgte ihapi fsdl and saddle of pair a with together cycles period-1 saddle unstable cycles. doubly period-2 node and stable saddle of pair a displays aldfl-lpbfrainpit 2]weetepro-obigcrei agn otesaddle–node the to is tangent curve period-doubling the curves where bifurcation [20] points bifurcation fold-flip called htrpeet h eoac ou.Aogtelwrbranch lower the Along torus. resonance the represents that n ih ytesdl–oebfraincurves bifurcation saddle–node the by right and smnindaoe h iucto aaeesaetefrigfrequency forcing the eter are parameters bifurcation the above, mentioned As frequency forcing the are parameters Bifurcation (1). in oscillator 3.2 FIGURE Chaos Synchronized Phase to Transition the On osrce ymaso ovninlcniuto ehd 1]uigsfwr aeaalbeb .J odle al. et Doedel 2002.) Technology, J. of E. Institute report, by Technical homcont). California (with available equations differential made ordinary for software software bifurcation using and Continuation [19] 2000: Auto methods continuation conventional of means by bifurcation constructed period-doubling each cycles, resonance emerging the of givesrisetoanewpairofsaddle–nodebifurcationcurvesalongtheedgeoftheresonancetongue.(Thefigurewas existence of range the delineate To system. Rössler obigbfrain cu iutnosy bv h curve the Above simultaneously. occur bifurcations doubling tbepro- yl nege t is eidduln iucto hl h orsodn adecycle saddle corresponding the while at doubles bifurcation period period-doubling first its undergoes cycle period-1 stable ontgte ntesdl–oebifurcation saddle–node the in together born iucto cn ob icse ntefollowing. the in arrows discussed the be and to zone, scans bifurcation resonance the zone. outside resonance exist the that of tori sides the along and respectively, observed cycles, resonance (saddle) unstable and öse ytmdsly tbe ycrnzdpro- cycle period-1 synchronized stable, a displays system Rössler eid2cce snttesm sta o h eid1cce,adanwsto adend bifurcation saddle–node of new set the new for a and zone cycles, period-1 synchronization the the for However, that as curves dynamics. same the period-1 not resonant is cycles of period-2 region the delineate to © 2016 Group, by Taylor LLC & Francis h adend iucto curves bifurcation saddle–node The eo h is eidduln iucto curve bifurcation period-doubling first the Below c o h öse oscillator. Rössler the for SN 2 L and w-iesoa iucto iga o h : eoac oeo h eidclyfre Rössler forced periodically the of zone resonance 1:1 the for diagram bifurcation Two-dimensional SN PD SN 2 R 1 U r eurdt eiet h eino eid2dnmc.Teenwsaddle–node new These dynamics. period-2 of region the delineate to required are 1 L tteeg ftersnnezn,tetopro- ylsmre n h period- the and merge, cycles period-1 two the zone, resonance the of edge the At . and SN 2.0 5.0 c PD 1 R oee,a lsriseto hw 2] h e adend bifur- saddle–node new the [24], shows inspection closer as However, . 1.055 TD TD S and B A 2 L 1 L SN PD 1 L SN U SN and eoepro-obigbfraincre o tbe(node) stable for curves bifurcation period-doubling denote SN 1 L 2 L C 1 L SN TD PD (or SN PD 1 PD R U 2 ω oae ou-obigbfrain o h ergodic the for bifurcations torus-doubling locates 1 L SN otneu ln h deo h eoac zone resonance the of edge the along up continue PD PD 2 S PD 4 S and SN 1 R 1 S U 1 S 1 )andbothsituatedonaclosedinvariantcurve ,theresonancezoneisdelineatedtotheleft A ω SN eest h adend iucto curves bifurcation saddle–node the to refers PD n h olnaiyparameter nonlinearity the and → PD 1 R N 1 U A 1 S epciey nti ein h forced the region, this In respectively. , 1 nFgr .,tefre öse system Rössler forced the 3.2, Figure in ftefrtpro-obigcre the curve, period-doubling first the of and n orsodn adecycle saddle corresponding a and SN SN TD TD 1.12 R 1 B R 2 PD R 2 R 1 → ω 1 n h olnaiyparam- nonlinearity the and B (PD eaet one-dimensional to relate 1 S or PD 1 U c )neartheso- fteunforced the of S 43 1 , Downloaded By: 10.3.98.104 At: 17:44 01 Oct 2021; For: 9781466590441, chapter3, 10.1201/b20232-5 thezoneedgefortheperiod-1cycles.Thesaddl h cl ftefgr) stevleof value the As figure). the of scale the ht tlati h ideo h eoac oe hstasto cusbfr h adeslto has torus- solution the saddle that the suggests before also occurs 3.2 transition Figure bifurcations this of zone, doubling Inspection resonance bifurcation. period-doubling the second of shows its middle 3.2 undergone Figure the of in Inspection chaos. least synchronized at phase to that, transition a undergoes system the until again h is eidduln iuctosfrte11rsnnend n adecce at cycles saddle and node resonance 1:1 the for bifurcations period-doubling first the obigat doubling obigcre,adcoet h onso agnyanwpi fsdl–oebfraincurves bifurcation saddle–node of pair new a tangency of points the to close and curves, doubling and 3.3 FIGURE 44 eoac ylsmreaddsperin disappear and merge cycles resonance oe otesdl–oebifurcation saddle–node the to node) eid2ivrattrs hstrssbeunl nege ou-obigat torus-doubling a undergoes subsequently torus This torus. invariant period-2 eiddulsat doubles period necnetdpro-obigpoesscniu otelf ntefgr.Hr ecnlct h period- the locate can we Here figure. bifurcation the doubling in left the to continue processes period-doubling interconnected ntbend ouin.Ntc o h tbe11slto hteit nteuprrgtcre ftefig- the of corner right 3.2. upper the Figure in at in exists doubly period-doubling that A the a solution curves direction undergoes 1:1 dotted ure stable the and the cycles, along how saddle Notice place the solutions. take curves node dashed unstable that cycles, transitions periodic stable the represent shows curves Full 3.3 Figure in diagram cation tongue. the in cycles resonance the of bifurcations period-doubling the with coupled are tongue iebrht eidduldegdctrs ial,we rsigtetrsduln iucto curve bifurcation torus-doubling the crossing when Finally, torus. TD ergodic period-doubled a to birth give wyfo h eoac oe h roi ou trst odadi ial nege ou destruction torus undergoes finally it and fold further to moves system starts the torus node As point ergodic torus. unstable the the ergodic doubly at zone, period-4 and saddle–node stable resonance saddle a the the period-1 of in from region The ends away the born. torus into is exist resonance torus to period-4 continue period-4 cycles the resonance ergodic 2:2 that an the observe and which 1:1 we in the bifurcation Hence, both and destroyed. stable, is been node have period-4 tori the only but cycles, period-4 and period-2, subsequently zone. can synchronization bifurcations their these demarcates that in bifurcation born saddle–node cycles the node to unstable followed doubly be and saddle of pair Each cycle. dle © 2016 Group, by Taylor LLC & Francis h tbepro- ouinudrosanwpro obigat doubling period new a undergoes solution period-2 stable The sosasmlroedmninlbfraindarmfrtedirection the for diagram bifurcation one-dimensional similar a shows 3.4 Figure h n-iesoa bifur- one-dimensional the 3.2 , Figure in diagram bifurcation two-dimensional the to illustration an As h onayo h eoac oecnit fsdl–oebfrain o h period-1, the for bifurcations saddle–node of consists zone resonance the of boundary the 3.4, Figure In 2 L SN h roi ou nege e eidduln transition. period-doubling new a undergoes torus ergodic the , 4 R r ont eiet h ag feitnefrtepro- eoac yais(o iil in visible (not dynamics resonance period-4 the for existence of range the delineate to born are PD ETD n-iesoa cnaogtedirection the along scan One-dimensional 1 U .Fromherewecanfollowthetwosolutions(nowasasaddlecycleandadoublyunstable whereitsdifferentlayersbegintomix. PD PD 2 U TD h adend iucto curves bifurcation saddle–node The . 2 S o h tbepro- yl n h bifurcation the and cycle period-2 stable the for L and –9.0 –3.0 y TD 1.05 PD R httk lc nteqaieidcrgm usd h resonance the outside regime quasiperiodic the in place take that SN SN 1 S while the corresponding 1:1 saddle solution suffers its first period first its suffers solution saddle 1:1 corresponding the while c 1 L 1 L h aro adeadsal : ylsmrein merge cycles 2:2 stable and saddle of pair The . otne oices,tesm rcs eet vradover and over repeats process same the increase, to continues SN otelf ntefgr.Ti adend iucto defines bifurcation saddle–node This figure. the in left the to TD 1 L 2 L n tbend : eoac ylsmreat merge cycles resonance 2:2 node stable and e SN A 2 L ω h aro adeaddul ntbe1:1 unstable doubly and saddle of pair The 3.2. Figure in adoko plctoso ho Theory Chaos of Applications of Handbook c =–51ω+58 PD U 1 SN PD 2 L S and 1 PD 2 S SN ,andthesaddleperiod-2solution 1.10 TD PD 2 R 2 L . 2 U aretangenttotheseperiod- o h orsodn sad- corresponding the for B After 3.1. Figure in PD SN 1 S and 2 L ,leadingtoa PD SN 1 U ,the SN 2 L to 4 L Downloaded By: 10.3.98.104 At: 17:44 01 Oct 2021; For: 9781466590441, chapter3, 10.1201/b20232-5 obigpoesol cusi etitdrgo ntesdso h eoac oge ttepoint the At tongue. resonance the of torus- sides The the torus. on ergodic region period-4 restricted a torusisdestroyed,andtotheleftof a cycles. to in birth saddle gives occurs and that only bifurcation node process saddle–node period-1 doubling a synchronized in ends the torus both period-4 of resonant bifurcations The period-doubling subsequent two follow can eo t on fitreto with intersection of point its below peet h iuto ertergthn ieo h ogewhere tongue the of side right-hand the near situation the presents 3.5c Figure it o nrnvHp)bifurcation Andronov–Hopf) (or birth ogeeg.Hence, edge. tongue osdr h iuto ttelf-adsd ftersnnetnu hr h e adend bifurca- saddle–node new the 3.5b where Figure tongue resonance cycles. the period-1 the of curve for side tion left-hand points the bifurcation at fold-flip situation two the considers the mag- (or around shows of quasiperiodic regions c and the 3.5b the region Figure of gray. nifications dark and is shades zone different resonance in the outside displayed dynamics are chaotic) nonsynchronized cycles resonance etc., 4:4, 2:2, the 1:1, from ble period-doubling saddle–node and the new of zone the branch resonance unstable of the emergence the and the of stable side, the side given from curves. one a place takes in for Moreover, emerge alternately curve side. curve curves bifurcation other bifurcation period-doubling the saddle–node ampli- the in new in forcing of branch the to unstable branch that increase close is stable with curve situation decreases the generic period-doubling points from The the bifurcation vanishes. fold-flip on never the points but to from tude, not distance but do corresponding The points, curves the to points. bifurcation bifurcation these saddle-node fold-flip is tangent new the the from bifurcation above, emerge explained saddle–node As the curve. where bifurcation points period-doubling bifurcation fold-flip the to Bifurcation Fold-Flip Tounderstandtheabovetransitionsinmoredetailweneedtofocusonthebifurcationstructureclose a Near Structure Bifurcation 3.3 3.4 FIGURE Chaos Synchronized Phase to Transition the On utb uciia.Ti lossdl–oebfrain otk lc ln htpr of part that along place take to bifurcations saddle–node allows This subcritical. be must in generated torus two-branched unstable the which through vr h eto of section the ever, ntesal branch stable the on hssdl–oebfraincurve. bifurcation saddle–node this yl rdcdat produced cycle obe roi ou 2] h lentv aeweetesdl–oebfraincreeegsfrom emerges curve bifurcation saddle–node the of branch where stable case the alternative The [24]. torus ergodic doubled © 2016 Group, by Taylor LLC & Francis ocoetegpbtentesdl–oebfraincurves bifurcation saddle–node the between gap the close To i hc h ein feitnefrtesta- the for existence of regions the which in 3.2 Figure of version redrawn slightly a shows 3.5a Figure SN n-iesoa cnaogtedirection the along scan One-dimensional 2 L mre rmtepoint the from emerges PD PD PD PD SN 1 S 1 S a saefo h eoac oe h ytmmksueo uciia torus- subcritical a of use makes system the zone) resonance the from escape can 1 htcnet h on twhich at point the connects that 2 R 1 S osntlaeagptruhwihte22rsnnecce a sae How- escape. can cycles resonance 2:2 the which through gap a leave not does ftepro-obigcurve. period-doubling the of utintersect must –8.0 –3.0 ETD y 1.06 SN h ytmdsly osnhooschaos. nonsynchronous displays system the 1 R Q SN .Wealsonotethat T SN 2 ETD 2 1 L nteusal branch unstable the on 1 ncnucinwt ope e fgoa bifurcations global of set complex a with conjunction in R PD n hratr tlati einn,poedt h ih of right the to proceed beginning, in least at thereafter, and U 2 PD S 2 B ω stefrigfrequency forcing the As 3.2. Figure in SN c =–51ω+58.42 2 R SN PD sbr ihtepito agnywith tangency of point the with born is 2 R SN U 1 T sbr otelf ftearayexisting already the of left the to born is 2 PD 1 L stasomdit tbeperiod- stable a into transformed is and 1 U PD 1.10 ftepro-obigcre and curve, period-doubling the of SN S 1 2 L adaodta h period-2 the that avoid (and SN 2 R mre rmapoint a from emerges ETD ω SN srdcd we reduced, is ,theergodic 2 R htfalls that SN G 45 1 R 2 Downloaded By: 10.3.98.104 At: 17:44 01 Oct 2021; For: 9781466590441, chapter3, 10.1201/b20232-5 ytmtclastenwsdl–oebfraincre opoedatraeyisd n usd the outside and inside above alternately the that proceed conclude to can we curves time bifurcation same saddle–node resonancezonedelineatedbytheimmediatelyp the period-doubling At new the [16–18]. the of tongue character leads resonance cyclic the systematic a characteristic along of the phase edge for in opposite the basis along the alternate is transition scenarios This different zone. bifurca- two resonance period-doubling the the of indicated, subsequent sides already the have of we points as bifurcation and, fold-flip tions the near arise structures Similar adend bifurcations saddle–node vriwo h is orpro-obigbfrain.Ol h is opeo adend iuctosaogthe along around region the bifurcations of saddle–node of Enlargement (b) scale. couple applied the first the in resolved Only be can bifurcations. edge period-doubling zone four first the of Overview 3.5 FIGURE 46 h it n rnfraino h ag eid2egdctrsta rs ntetrsbrhbifurcation torus-birth the in point arise the that from torus emanates ergodic period-2 large the T of transformation and birth the Torus Ergodic Letusconsidertheabovetransitionsoncemore,butnowwithadirectfocusontheprocessesthatleadto the of Doubling Period 3.4 oebfraincreeegsfo h tbebac ftepro-obigcre ato hsbac mre by (marked branch this of Part curve. period-doubling the of branch stable the smallblacksquares)thenhastobesubcritical. from emerges curve bifurcation node lc near place © 2016 Group, by Taylor LLC & Francis 2 ncneto ihFgr .bw aearaydsusdhwtesdl–oebfraincurve bifurcation saddle–node the how discussed already have we 3.5b Figure with connection In . hscmltsordsuso ftedtie iucto tutr ertefl-lpbfrainpoint. bifurcation fold-flip the near structure bifurcation detailed the of discussion our completes This (b) c 2.935 3.015 G 2 c nagmn ftecrepnigrgo otergti h eoac oe eetenwsaddle– new the Here zone. resonance the in right the to region corresponding the of Enlargement (c) . 1.06846 iucto tutr o h : eoac ogei h eidclyfre öse ytm (a) system. Rössler forced periodically the in tongue resonance 1:1 the for structure Bifurcation T 2 SN SN Q G 1 L Q 2 2 and L 1 2 nteusal rnho h eidduln curve period-doubling the of branch unstable the on ω SN c 2 L 2.0 5.0(a) scoe ytetrsbrhbifurcation torus-birth the by closed is 1:1 SN 1.055 TD TD L 2 L 2:2 2 PD L 1 1.06874 PD 1 S U 1 SN SN eeigsdl–oebfraincurves. bifurcation saddle–node receding L L 1 2 (c) c PD adoko plctoso ho Theory Chaos of Applications of Handbook 2.69 2.75 PD U PD 2 ω PD 2 S PD 4 S 1.1108 U 1 S 1 SN R PD 2 U 1 T 2:2 2 n h lblbfrain httake that bifurcations global the and SN PD 1.12 TD TD R 1 1 R 2 R 1 and ω PD SN PD S,sub 1 1 PD L h a ewe the between gap The . 1 U 1 S .Thispointdoes 1:1 1.1118 SN SN R 1 R 2 SN 2 L Downloaded By: 10.3.98.104 At: 17:44 01 Oct 2021; For: 9781466590441, chapter3, 10.1201/b20232-5 h litccredenoted curve elliptic the it iucto.A rvosyntd h curve the noted, previously As bifurcation. birth o h ycrnzto oe ial,tetrsduln curve torus-doubling the boundary Finally, required zone. the establish synchronization thereby and the torus for period-2 ergodic the stabilize to serve that bifurcations eid2egdctrstasom noteegdcpro- ou hteit usd h eoac tongue parameter resonance nonlinearity the outside the exists of that torus values period-1 for ergodic the into transforms torus ergodic period-2 w first we 3.7 Figure of range intermediate the through bifurcation right saddle–node to left the from meet transitions the follow we As zone. observedintheregionaroundthebirthofthesaddle–nodebifurcationcurveSN imdaeyalw st oaetesdl–oebifurcation saddle–node the locate to the us escape allows not immediately do 3.5b Figure cycles of (which curves 2:2 3.6 version the Figure redrawn of that Inspection slightly curves. ensure a bifurcation to saddle–node is the place by left in gap the are through bifurcations area resonance global and local additional 3.3, dynamics,thusleavingagapintheboundaryoftheresonancezone.However,asdescribedinSection hs iucto igasaetogtt edanaeidctdi iue36b h letters the by 3.6 Figure in indicated are drawn be to thought are diagrams bifurcation these iga eosreteegdcpro- ou hteit below exists that torus period-1 ergodic the observe we diagram o oniewt h on twhich at point the with coincide not Chaos Synchronized Phase to Transition the On r h tbeadusal rnhso h eidduln iucto uv and curve bifurcation period-doubling the of branches unstable and stable the are 3.6 FIGURE and , 3.7 Figure in seen torus ergodic noa11sdl yl hl rdcn tbe22ccetases otetrssrae eefe follows Hereafter surface. torus the to transverse cycle 2:2 stable a sequence producing the while cycle saddle 1:1 a into riaypro- roi ou.Nt htsm ftesbtutr iapaswe h ocn mltd becomes amplitude forcing the when disappears substructure the of some large. that sufficiently Note torus. ergodic period-1 ordinary uvs tti ou iucto,tesal eid2ccepoue ttelwrbac fteperiod- the of branch lower the at produced cycle period-2 curve stable doubling the bifurcation, torus this At curves. on h ou iucto curve bifurcation torus The born. on At born. ees ou-obigbfraina h point the at bifurcation torus-doubling reverse ou htdmntsms ftergthn ieo h iga n oa ntbetobac torus torus-birth two-branch subcritical unstable the an in disappears to again and torus diagram two-branch the unstable bifurcation of The cycle. side focus right-hand 2:2 the the of around most dominates that torus 2 adend iucto curve bifurcation saddle–node © 2016 Group, by Taylor LLC & Francis ntepeec fteeadtoa iuctos h rt oc iucto iga acltdalong calculated diagram bifurcation force brute the bifurcations, additional these of presence the In gv lae con ftesrcueo oa n lblbifurcations global and local of structure the of account clearer a give 3.8 Figure in sketches The SN PD 1 L and T 1 S 2 hsi olwdb h eidduln iucto nwihte11nd stransformed is node 1:1 the which in bifurcation period-doubling the by followed is this , G anfcto fpr ftebfraindarmna h odfi iucto point. bifurcation fold-flip the near diagram bifurcation the of part of Magnification hl h ag eid2trscniust xs ni tudrosteaforementioned the undergoes it until exist to continues torus period-2 large the while 2 SN PD fcoeystae oa n lblbfrain htgv it obt h ag period- large the both to birth give that bifurcations global and local situated closely of 2 L 1 S n o rnfre noasal ou,lssissaiiyi uciia torus- subcritical a in stability its loses focus, stable a into transformed now and , h w rnhso h eidduln curve period-doubling the of branches two the , c C SN a 2.94 2.96 2.98 3.02 3.0 nFgr . ae h omilsrtdin illustrated form the takes 3.6 Figure in 2 L starts. .641.0685 1.0684 TD T SN 2 1 L TD PD G stetrsduln iucto nwihti ou stasomdit an into transformed is torus this which in bifurcation torus-doubling the is sse obig h a ewe h w adend bifurcation saddle–node two the between gap the bridge to seen is 1 L 2 1 L 1 attheedgeoftheresonancetonguewherethe1:1nodecycleis c representsthesequenceofbifurcationsthatgiverisetothelargeperiod-2 stnett h deo h ycrnzto einfrperiod-1 for region synchronization the of edge the to tangent is below D TD PD TD T G 1 L U 2 1 . 2 1 L . ersnsasto lsl iutdlcladglobal and local situated closely of set a represents ω SN G 2 Q 1 L 1.0686 2 TD C a 1 L TD 1:1 ersnstepoessb hc the which by processes the represents C PD b 1 L PD 2:2 1 n otelf fteresonance the of left the to and tbt nso this of ends both At 3.7. Figure ,andthepoint S 1.0687 1 PD SN Q – C 2 2 U L 1 c eoe h on nwihthe which in point the denotes 2 L .Thecurvesalongwhich Q 2 C where a PD , C 1 S b and ,and SN PD 2 L C 47 is 1 U c . Downloaded By: 10.3.98.104 At: 17:44 01 Oct 2021; For: 9781466590441, chapter3, 10.1201/b20232-5 rt oc iga nFgr .,teusal w-rnhtrspoue nti iucto disappears point bifurcation the this to in the close produced with bifurcation torus accordance two-branch global unstable In a the focus. in 3.7, unstable Figure an in into diagram turns force and brute bifurcation torus-birth subcritical a undergoes adecce r ona h ytmetr h eoac oan eefe olw h period-doubling the follows Hereafter domain. resonance the enters system bifurcation the as born are cycles saddle n ausof values ing trigfo h otm panel bottom, the from Starting 3.7 FIGURE 48 IUE3.8 FIGURE nst h adend iucto curve bifurcation saddle–node the to ends rnfrsit nodnr eid1egdctorus. ergodic period-1 ordinary an into transforms htgv it otelreegdcpro- ou and torus period-2 ergodic large the to birth give that h oiinalong position the etblzsit obyusal yl.Ti atrccefnlydsper nasdl–oebifurcation asaddle–node in disappears finally cycle latter This cycle. unstable doubly a into destabilizes obyusal ade nege ees eidduln iucto in bifurcation period-doubling reverse a undergoes saddle) unstable doubly a ie obyusal oeo ntbefcssltos oedtie ecito ftetasomtosta take that transformations the of dotted description and detailed solutions, more saddle A at lines solutions. dashed place focus solutions, unstable focus or or node node unstable doubly stable represent lines lines Full zone. resonance the from uciia ou-it bifurcation torus-birth subcritical eoac oe eefe olw is h period-doubling the first follows Hereafter zone. resonance © 2016 Group, by Taylor LLC & Francis G 2 a efudi u eetwr [28]. work recent our in found be may PD θ iucto igasdanaogprso h curves the of parts along drawn diagrams Bifurcation rt oc iucto iga acltdaogteelpi curve elliptic the along calculated diagram bifurcation force Brute h iga is hw h adend bifurcation saddle–node the shows first diagram the 1 S C nte11nd.At node. 1:1 the on a ih0(n 2 (and 0 with G 2 T PD C y 2 a –5.4 –5.2 –4.8 –4.6 1 S –5 T SN C ersnigteototrgtpitand point right outmost the representing π) 2 a SN L 1 nwihte22ccelssisstability. its loses cycle 2:2 the which in 2–1 –2 first shows the saddle–node bifurcation through which the 1:1 nodeand the which through bifurcation saddle–node the shows first 1 L T PD SN SN 2 h : yl eeae nti iucto nwasal focus) stable a (now bifurcation this in generated cycle 2:2 the , U 1 1 L G 1 L .Notehowthe2:2cyclesinallcasesarecapturedbeforetheycanescape 2 G .Aswecontinuethescan,theunstable2:2focuscycle(now 2 C T PD PD b 2 TD 1 S 01234 S 1 1 L SN PD G eoe h ou-obigbfraini hc hstorus this which in bifurcation torus-doubling the denotes SN θ 2 L 1 PD U 1 1 L adoko plctoso ho Theory Chaos of Applications of Handbook 1 S T htpoue h tbe22rsnn oeadthe and node resonant 2:2 stable the produces that 2 SN C SN 2 L a , PD C 1 L C c b htocr hntesse neste1:1 the enters system the when occurs that ,and 1 S G SN 2 π C SN ersnsasqec fbifurcations of sequence a represents TD L 1 c h ums etpit o increas- For point. left outmost the C in 1 L L 1 a PD iue3.6 Figure in SN U PD 1 iue3.6. Figure 2 L 1 U hl h : saddle 1:1 the while n xeddi both in extended and θ samaueof measure a is Downloaded By: 10.3.98.104 At: 17:44 01 Oct 2021; For: 9781466590441, chapter3, 10.1201/b20232-5 obigcre h lentv iuto hr h adend uv mre rmtesal rnhof branch stable the from emerges curve saddle–node PD the where situation alternative The curve. doubling apart. move torus thecasewherethenewsaddle–nodebifurcationcurveemergesfromtheunstablebranchoftheperiod- period-2 the of windings two the how observe can one eaainbtentetownig sse ocniu ogo,adfrlre auso h parameter the of values larger for and grow, to As continue curve. to invariant seen closed is the windings two of the periphery between transition the separation period-doubling to a transverse undergone has direction originates torus the the the in of which indication from another mode is resonance This the other. that the fact over route one chooses alternately oscillator quasiperiodic the iin oedtie ecito ftepoessascae ihti rniinhsbe rsne in presented been has transition this with associated processes [28]. the work previous of description detailed more A sition. panel otepito agnywith tangency of point the to . 3.3 Section in discussed scenario the recover we and cycles, 2:2 the iucto srqie.Hwvr h eidduln uv htcnet h on where point the connects that curve period-doubling the However, required. is bifurcation fteegdctrsfrdfeetpositions different for torus ergodic the of cla h adend iucto curve bifurcation saddle–node the as ical h ytmna h o ftescin hsi h elkonidcto httesse prahsa approaches system the that indication well-known the resonance. 1:1 is the of This is "hesitation" it section. case a the this indicating in periphery, of zone, resonance the top along the points near of system distribution uneven the an is there that however, it of birth rigt idapt rmte22rsnn eint roiiywtotcrossing without to ergodicity region 2:2 resonant the from a path find to trying ttetnu edge tongue the at Chaos Synchronized Phase to Transition the On incurve tion 3.9 FIGURE SN rsnsasre fPicr etoso h roi ou bevdi h n-iesoa rt force brute one-dimensional the in parameter the observed 3.7, torus Figure to ergodic the caption C the in of defined sections As diagram. Poincaré bifurcation of series a presents h tblt fte22cces hti a niiaewt h : adeccea h pe rnho the of branch upper the at cycle saddle 2:2 the with annihilate panel can in it Finally, that curve. so period-doubling cycle 2:2 the of stability the adeccewt igeusal ieto.I hswytetrsbrhbifurcation torus-birth the way this In direction. unstable single a with cycle saddle below adend bifurcation saddle–node © 2016 Group, by Taylor LLC & Francis a hscmltsordsuso ftemi iucto tutr ertefl-lpbfrainpitfor point bifurcation fold-flip the near structure bifurcation main the of discussion our completes This fw eoetepro-obigbifurcation period-doubling the denote we If nodrt rvd la mrsintetrsduln iucto htocr along occurs that bifurcation torus-doubling the impression clear a provide to order In As 1 R 1 S in .. eo h on ftnec,i on ttergthn deo h eoac oe nti aethe case this In zone. resonance the of edge right-hand the at found is tangency, of point the below i.e., , . θ For 3.6. Figure C PD a srdcd n a bev o h nain uv trst pi notodfeetwnig,as windings, different two into split to starts curve invariant the how observe can one reduced, is SN ssbrtcl h orsodn eidduln iucto npanel in bifurcation period-doubling corresponding the subcritical, as TD 1 U 2 R .Moreover, 1 L osntlaeagpbtenrsnn n roi yais hscnesl ecekdby checked be easily can This dynamics. ergodic and resonant between gap a leave not does lutaino h rniinta cusa h ocdRslrsse rse h ou obigbifurca- doubling torus the crosses system Rössler forced the as occurs that transition the of Illustration ntedrcinidctdb h arrow the by indicated direction the in SN θ 1 L esalrfrt h euneo iuctosta cu at occur that bifurcations of sequence the to refer shall We . = SN SN .,tePicr eto hw nodnr ie,pro-)egdctrs Note, torus. ergodic period-1) (i.e., ordinary an shows section Poincaré the 3.8, 2 R 0.0504 0.0508 0.0512 2 L utcross must SN hasovertakentheroleofdelineatingtheedgeoftheresonancetonguefor 3.4 1 z R utb uciia nodrt con o htpr of part that for account to order in subcritical be must 3.5 θ ln h curve the along SN C SN 3.6 1 R c h ou iucto nte22ccen ogrocr,the occurs, longer no cycle 2:2 the on bifurcation torus the andatleastatthebeginningproceedalongtherightsideof 2 L θ a rnfre h obyusal : adeit 2:2 a into saddle 2:2 unstable doubly the transformed has 3.7 PD D h iuepeet eiso onaésections Poincaré of series a presents figure The 3.6. Figure in 1 U 3.8 hted h ieo h : yl nteupredof end upper the in cycle 2:2 the of life the ends that C a opr ihtebfraindarmin diagram bifurcation the with Compare . –5.0 –4.9 –4.8 y –4.7 θ steageaogtecurve the along angle the is SN C θ b 2 sfrhrrdcd the reduced, further is R G a eoesupercrit- become has ec,n additional no Hence, . 2 T satrsfl tran- torus-fold a as 2 evst degrade to serves TD SN 1 L SN 2 R iue3.9 Figure , htexists that iue3.7 Figure 2 R sborn is 49 c . , Downloaded By: 10.3.98.104 At: 17:44 01 Oct 2021; For: 9781466590441, chapter3, 10.1201/b20232-5 . 3.2 Figure the in outside diagram exist bifurcation that 2D the tori to ergodic refer resonance the again shall multilayered with we communicate so-called discussion structures the this these In of zone. how reconstruction resonance examine and to formation and the [51] study tori to us leads This cascade. eoac oecycle. node resonance e ftrsadgoa iuctosa h ih-adedge. right-hand the at bifurcations global and torus of set cation,exceptthatherethesubcriticalcaseisfoundattheleftedgeandthesupercriticalwiththeadditional eid1cycle period-1 h etpolmi odsrb h nenlognzto ftemn ifrn eoac ylsta arise that cycles parameter resonance different nonlinearity many the the of of organization values internal increasing the as describe oscillator, to period-doubling forced is periodically problem a next for the structure bifurcation the of discussion above the After Multi-Layered of Reconstruction and Formation 3.5 50 cnline scan 3.10 FIGURE saddle fe h is eidduln ftend yl.()Paeprri fe h rnvrepro obigo h origi- the of doubling period transverse the after portrait Phase (c) cycle. cycle node saddle the nal of period-doubling first the after uefrafre ii-yl siltr ln h curve the Along oscillator. struc- resonance limit-cycle normal the forced to a corresponds this for 3.10a, ture Figure in shown As torus. resonance the represents eidduln iucto.Ti ie iet h omto fasal eid2rsnnecycle resonance period-2 stable a of formation the to rise gives This bifurcation. period-doubling © 2016 Group, by Taylor LLC & Francis eo h is eidduln curve period-doubling first the Below iia iucto tutrsaeosre ttepit ftnec ttenx eidduln bifur- period-doubling next the at tangency of points the at observed are structures bifurcation Similar S eoac Tori Resonance 2 C n h ntbemnfl ftesdl.()Paeprri fe h eodpro-obigo h original the of period-doubling second the after portrait Phase (d) saddle. the of manifold unstable the and , a rgnl11rsnnetrswt t tbenode stable its with torus resonance 1:1 Original (a) 3.2. Figure in (c) –8.0 hs otat ftersnnesrcuea ifrn auso h olnaiyprmtraogthe along parameter nonlinearity the of values different at structure resonance the of portraits Phase S –8.0 N 1 6.0(a) 6.0 .Thesystemnowdisplaysastabledouble-lay 1 y y n orsodn adecycle saddle corresponding a and ω –6.5 –6.5 S = 0 1.08. S S S 2 1 2 F F x x c =3.098 c =2.66 PD N 2 S 1 1 S * h ocdRslrsse ipasasal synchronized stable a displays system Rössler forced the , N 1 N 7.5 7.5 2 c ae h siltru hog h period-doubling the through up oscillator the takes S (b) (d) 1 ohstae ntecoe nain uv that curve invariant closed the on situated both , PD adoko plctoso ho Theory Chaos of Applications of Handbook –8.0 –9.0 6.0 7.0 rdrsnnetrscnitn ftenode the of consisting torus resonance ered 1 S y y h tbepro- yl nege t first its undergoes cycle period-1 stable the , –6.5 –9.0 S N S 1 1 2 n adecycle saddle and S 2 F x x F N S 4 N c =3.75 S 1 * 2 1 * c =2.8 S S 1 2 b hs portrait Phase (b) . * N N 7.5 9.0 S 2 4 2 * N 2 N while 2 ,the Downloaded By: 10.3.98.104 At: 17:44 01 Oct 2021; For: 9781466590441, chapter3, 10.1201/b20232-5 rse noargm hr h rgnlrsnnenode resonance original the where regime a into crosses nadrcintases otersnnetrs oe oee,hwteusal aiodo h new the of manifold unstable the how however, Note, torus. cycle resonance saddle the period-2 to transverse direction a in acd hra h rgnlsdl cycle saddle original the whereas cascade saddle ievle ftesdl cycle saddle the of eigenvalues frhge eeso h eidduln cascade. period-doubling the of levels higher for 3.6 Figure in observed adecycle saddle ntbemnfl fti yl.Wt ute nraeo h olnaiyprmtr h eid2cycle period-2 the parameter, nonlinearity the of increase further With cycle. this of manifold unstable eid2sdl cycle saddle period-2 iue31 hw anfcto ftergo around region the of magnification a shows 3.11 Figure hsbfraingvsbrht h tbepro- eoac cycle resonance period-4 stable the to torus. birth resonance gives original bifurcation this the 3.10d , to Figure transverse in direction shown the As in bifurcation period-doubling new a undergoes cycle saddle period-1 a into transformed is node resonance original the Chaos Synchronized Phase to Transition the On IUE3.11 FIGURE eal steproiiyo h tbend yl nrae.Wt vnfrhrices in increase further even With increases. cycle node stable the of periodicity the as details sstae wyfo h eoac ou,adteusal aiodof manifold unstable the cycle and period-2 torus, stable resonance The torus. the resonance from the away to transverse situated direction is a in place takes doubling period this rnvrepro-obigcsaewt h omto fsdl ylso period-4 its continues of cycle 3.13b cycles saddle saddle Figure resonance of formation the the while with chaotic cascade now period-doubling is transverse cycle node original the around dynamics the o otistases tblt n e,tes-alddul-aee eoac ou 5,4,hsbeen node has [51,54], period-2 torus the resonance of double-layered consists so-called torus the This new, a born. and stability transverse its lost now h oddapoc rmteusal eid1rsnnecycle resonance period-1 is unstable This the nodes. from resonance approach period-16 folded and for the c period-8 and of 3.12a Figure formation in the illustrated with process period-doubling verse unstable the of direction in unity than larger numerically manifold. and negative and manifold stable the of direction l eid4node period-4 ble S l solution dle © 2016 Group, by Taylor LLC & Francis 1 ∗ rvasa neetn tutr fteusal aiodfo h period-1 the from manifold unstable the of structure interesting an reveals 3.10d Figure of Inspection ihfrhrices ftenniert parameter nonlinearity the of increase further With ihfrhrices ftenniert parameter nonlinearity the of increase further With prahstesal eid4cycle period-4 stable the approaches S 1 ∗ si prahstesal eid4cycle period-4 stable the approaches it as S S n period-16 and 1 0 aon h adecycle saddle the around 3.10d Figure in portrait phase the of part of Magnification nege eidduln iucto hsgvn it otedul ntbeperiod-1 unstable doubly the to birth giving thus bifurcation period-doubling a undergoes n oanwpro- saddle period-2 new a to and N 4 oe o h ntbemnfl from manifold unstable the how Note, . S S 2 ∗ 2 . oncst h eid2node period-2 the to connects iue3.13c. Figure y S –5.0 2 ∗ 0.2 hs ievle r oiieadnmrclyls hnuiyi the in unity than less numerically and positive are eigenvalues These . c .55.75 4.15 = .3 and 3.936 N 4 S 1 na siltr anrta per ob otoldb the by controlled be to appears that manner oscillatory an in N S a eidduldol ne silsrtdin illustrated As once. only doubled period has 4 2 N ssonin shown As . 2 oehrwt h orsodn adecycle saddle corresponding the with together c N = 4 dmntae how demonstrates 3.12b Figure respectively. 3.947, nodrt iulz hssrcuei oedetail, more in structure this visualize to order In . c x h ytmcosstecurve the crosses system the , S S c N S 1 ∗ 1 ∗ h tbersnnend otne t trans- its continues node resonance stable the , N 2 * 1 siltsa tapproaches it as oscillates and 2 hascompleteditstransverseperiod-doubling h rgnlpro- eoac ou has torus resonance period-1 original The . hstasto gi ae place takes again transition this 3.10c , Figure S 1 * N 4 S oe o h ntbemnfl from manifold unstable the how Note, . N 1 ∗ 4 otne odvlpfnradfiner and finer develop to continues S S 1 ∗ 1 ∗ silsrtdin illustrated As . stases oti torus. this to transverse is N period-8 3.13a, Figure 4 .Asimilarstructureis PD 1 U .Here,thesad- S N c iue3.10b Figure 1 ∗ ,thesystem iue3.13, Figure 4 n h sta- the and S n othe to and 2 n the and N 51 2 , Downloaded By: 10.3.98.104 At: 17:44 01 Oct 2021; For: 9781466590441, chapter3, 10.1201/b20232-5 h eidduln curves period-doubling the ono h eid1trsivle pitn fteusal aiodof manifold unstable the of break- splitting 3.14b, a Figure in involves shown torus As unstable. period-1 now reso- the double-layered is of torus a resonance down of period-1 formation original the the to and birth torus, given nance have transitions These torus. resonance period-1 e snwcnie h hnmn htaiei scan a in arise that phenomena the consider now us direction Let the 3.2. Figure along in diagram 2D-bifurcation the of scan from is eidduln rniin ftepro- oeadsdl yls( cycles saddle and node period-1 the of transitions period-doubling the the first is, there- into where zone. transforms this region of scan zone the a left of synchronization the into the to system in exists idea The that torus the torus resonance leads ergodic [5,43]. double-layered period-doubled scan bifurcation the the how period-doubling that examine first to shows a fore, 3.2 undergone Figure has in torus zone ergodic resonance the of left curves btentesdl cycle saddle the between 3.6a Figure in portrait phase the of part of Magnification (b) node. resonance 1 3.12 FIGURE 52 noapro-obe roi ou fsmlrsaeadsz.Finally, size. and shape similar torus of resonance double-layered torus ergodic a period-doubled of transformation a a into to of dynamics form period-2 resonant the from takes boundary quasiperiodicity zone period-doubled the across communication way, this In 3.14c ). (Figure crosses hereafter system the As manner. alternating curve an bifurcation in saddle–node torus the double-layered stable the approach may eidduldegdctrstgte ihapi fsnl n obyusal adecycles saddle unstable doubly and singly of pair a with together torus ergodic period-doubled ohatatdb h obelyrdrsnnecce hsipista rjcoysatn ls to close starting trajectory a that implies This cycle. resonance double-layered the by attracted both tbepro- node period-8 stable © 2016 Group, by Taylor LLC & Francis Theabovedescriptionhaspresentedsomeofthemainstructuresthatonecanobserveinavertical adsosteognzto ftersnnecce fe the after cycles resonance the of organization the shows and 3.10c Figure to similar is 3.14a Figures S 1 ∗ SN otesal oeccecniust cu stepro-obigpoesproceeds. process period-doubling the as occur to continues cycle node stable the to 2 L and y hs otat fe h hr a n h orh()tases eidduln fteoiia period- original the of period-doubling transverse (c) fourth the and (a) third the after portraits Phase –9.0 SN 7.0(a) N 1 L A –9.0 8 n notergo fqaieidct.Ave ntebfrainsrcuet the to structure bifurcation the on view A quasiperiodicity. of region the into and hl eeoigfnradfnrsrcue,tefle praho h ntbemanifold the unstable of approach folded the structures, finer and finer developing While . → A S 2 .. ln h line the along i.e., 3.2, Figure in PD S 1 U 2 and (c) x y F SN –9.0 PD 7.0 2 L ,theresonancecycles N S c =3.936 2 S –9.0 1 * 8 ,thisscantakesthesystemacrossthesaddle–nodebifurcation N S 9.0 2 8 S 2 (b) y x –5.0 adoko plctoso ho Theory Chaos of Applications of Handbook F 0.2 c c =3.947 N S 4.15 =− 1 * 16 N N 2 8 and 51ω N 16 S 9.0 S 4 * 1 ∗ S + N 2 and 8 8 triga on between point a at Starting 58. eg n iapa evn a leaving disappear and merge N x S S 1 0 S 2 * and 0 notobace htare that branches two into lomreaddisappear and merge also N 8 S ω 1 )transverselytothe S = 4 * 1.08. N 6.0 8 S S 1 ∗ 1 ∗ n the and and S S 0 0 Downloaded By: 10.3.98.104 At: 17:44 01 Oct 2021; For: 9781466590441, chapter3, 10.1201/b20232-5 uv,adte33sdl yl obe t eidaogteuprbac.Smlry h lsdcurve closed the Similarly, branch. upper this the of along branch period lower its doubles the cycle along saddle doubles 3:3 period the cycle and curve, node 3:3 resonant The bifurcation. period-doubling hscrecoe oei h deo h eoac ogebetween tongue resonance the of edge the in hole a closes curve This 3.5c. Figure iucto uv htdlnae h ie ftersnnezn o h : cycles. 6:6 the for zone resonance the of sides the delineates that curve bifurcation ntesdl–oebifurcation saddle–node the in dynamics the node, this around region the In (a) 3.13 FIGURE Chaos Synchronized Phase to Transition the On satrsbrhbfraincreta evsaproesmlrt hto h ou iucto curve bifurcation torus the of that to similar period- purpose the a serves in that generated curve cycle bifurcation torus-birth saddle a 3:3 is the curve bifurcation which the in structure, bifurcations bifurcation cusp doubling saddle–node pronounced its branch of With lower couple the branch. a at upper period represents the its at doubling cycle period saddle cycle the node and the with doubling period second the represents forced the in bifurcations period-doubling of region bifur the similar that for demonstrating analysis by system bifurcation Rössler our complete Window us Period-3 Let the in Structure Bifurcation 3.6 anbfrainsrcueo h eid3wno.Here, parameter window. nonlinearity period-3 the the of values of higher structure for bifurcation regime main chaotic the in exist that hc h : eoatnd n adecce r on hscrecniusu ln h w ie of sides two the along up continues curve This born. denoted are now cycles tongue, saddle resonance and the node resonant 3:3 the which ftesse a undergodic. turned has system the of obigpoestruhtefraino b eid8ad()pro-6sdl cycles. saddle period-16 (c) period- and its period-8 continues (b) cycle of saddle resonance formation the the increased, through further process is doubling parameter nonlinearity the As region. chaotic the © 2016 Group, by Taylor LLC & Francis hs otat fe opeino h rnvrepro-obigfrteoiia eoac node. resonance original the for period-doubling transverse the of completion after portraits Phase y –9.0 7.0(a) –9.0 PD 3 S is oe n usqetyrgissaiiyi eodr ieto.Finally, direction. in a secondary stability regains subsequently and loses first 4 S SN 4 1 L (c) y x .Hwvr hshpesol fe h tbedynamics stable the after only happens this However, 3.14d). (Figure SN –9.0 F 7.0 –9.0 3 L S 1 and * c =4.2 isnowchaotic,andtheunstablemanifoldofS S SN 16 9.0 3 R ,respectively.TheclosedcurvePD ainpeoeatk lc nteproi windows periodic the in place take phenomena cation S 16 (b) y x –9.0 F 7.0 –9.0 SN ω =1.08 c =4.46 3 B eoe h adend iucto in bifurcation saddle–node the denotes S 8 9.0 S 8 x F ω SN c = sosthe shows 3.15 Figure . c =4.42 3 L 1.08. n h saddle–node the and 3 4 ersnstefirst the represents evsa nistto inset an as serves 9.0 T SN PD 2 53 T in 3 C 3 6 Downloaded By: 10.3.98.104 At: 17:44 01 Oct 2021; For: 9781466590441, chapter3, 10.1201/b20232-5 ftebfrainsrcuei h : eincnim that confirms region 3:3 the in structure bifurcation the of bifurcation ln h direction the along 3.14 FIGURE 54 ln h de ftersnnezn n otecniuto fteecre noterneo phase of dependent, range model the are into that curves bifurcations these three of to continuation two first the curves the to bifurcation for and saddle–node the perhaps, zone of Except, resonance accumulation chaos. the the to synchronized of relate edges that the problems the on along focus finally us Bifurcations Let Saddle–Node of Set Accumulating 3.7 cycles Destruction resonance (b) period-2 unstable. the is of torus Disappearance resonance (c) period-1 torus. corresponding resonance The period-1 cycles. the saddle of and node period-1 original ntbe and unstable) eieotietersnnezone. resonance the outside regime © 2016 Group, by Taylor LLC & Francis ncoeacrac ihtebfrainsrcueosre o h : eoac eie examination regime, resonance 1:1 the for observed structure bifurcation the with accordance close In .Adtoa ou n lblbfrain ev ocoetegpbtentenwadteprevious the and new the between gap the close to serve bifurcations that global bifurcations and saddle–node torus of Additional pair 3. new a generates bifurcations period-doubling of pair Each 2. of edge the At interconnected. are solutions saddle and node the for cascades period-doubling The 1. odr ftersnnezone. resonance the cascade. the of in borders level next the at zone resonance the of edges the define edge solution. tongue saddle the the before from bifurcates away type) forcing but applied the simultaneous, (with are solution node bifurcations the two the tongue, synchronization the SN (c) S 2 L 1 ∗ y y .Thismarksthetransitiontoergodicdynamics.(d)Disappearanceofthesaddlecycles tgsi h rniinfo obelyrdrsnnetrst eidduldegdctrsi scan a in torus ergodic period-doubled to torus resonance double-layered from transition the in Stages ntesdl–oebfraincurve bifurcation saddle–node the in –9.0 –9.0 A 7.0 7.0(a) → 909.0 –9.0 –9.0 A a obelyrdrsnnetrspoue hog eidduln fthe of doubling period through produced torus resonance Double-layered (a) . 3.2 Figure in ω =1.0668 S 0 F F S S x x 0 2 ω =1.076 S N * 1 2 S * 1 N 9.0 2 SN 1 L (b) (d) hsfnlbfraintkspaei h quasiperiodic the in place takes bifurcation final This . y y adoko plctoso ho Theory Chaos of Applications of Handbook –9.0 –9.0 7.0 7.0 –9.0 –9.0 ω =1.065 S 0 S S F F 2 2 x x N 2 ω =1.0742 S and N * 1 2 S 2 ntesaddle–node the in N 9.0 9.0 2 S 0 (doubly Downloaded By: 10.3.98.104 At: 17:44 01 Oct 2021; For: 9781466590441, chapter3, 10.1201/b20232-5 oeslto iuctn ttelwrbranc lower the at bifurcating solution node oecreetnsu ln ohtnu edges. tongue both along up extends curve node to ehius[01]hsalwdu oflo h adend iuctosta eiet h left-hand the delineate that bifurcations saddle–node the follow to us allowed has [10,19] techniques ation 3.16. Figure in outlined form the on bifurcation saddle–node cascade of period-doubling unstable generation the the the in with from curves associated emerge behavior cyclic that characteristic curves the represent bifurcation can curve saddle–node bifurcation hand, saddle–node other preceding the the branchproceedinsidetheresonancezonedefinedbytheformersaddle–nodebifurcationcurve.Hence,we On by defined edge. period- as same a boundary the of zone branch along stable the the out from cross emerging curve curves doubling bifurcation saddle–node formation, their after ately generations formulation [16–18]. many their criticality for for C-type basis for observed the theory represents be scaling This can special period-256. zone a to of synchronization up least the at of bifurcations, bifurca- sides period-doubling the of two of alternation the the that between demonstrated have processes [16] tion al. et Kuznetsov by performed analyses scaling parameter nonlinearity the of values increasing For solutions. resonance period-3 the of onset the 3.15 FIGURE Chaos Synchronized Phase to Transition the On incre ln h deo h eoac oe emnsdenoted Segments zone. resonance the of edge the along curves tion 3.16 FIGURE oebfraincre ln ie ieatraeymv otergtadtelf ftepeeigsaddle–node preceding the of left the and right the to move alternately side curve. given bifurcation a along curves bifurcation node n emnsdenoted segments and © 2016 Group, by Taylor LLC & Francis adesstesm rbe rmamr lblpito iw s fcontinu- of Use view. of point global more a from problem same the addresses 3.17 Figure in sketch The immedi- least at that, argued have we 3.2 , Section in presented analyses bifurcation the on Based ktht lutaeteatraino w ifrn cnro o h it fnwsdl–oebifurca- saddle–node new of birth the for scenarios different two of alternation the illustrate to Sketch anbfrainsrcuei h eid3wno.Telwrsdl–oebfraincremarks curve bifurcation saddle–node lower The window. period-3 the in structure bifurcation Main PD 1 , PD c 2 ,andPD 4.8 5.0 5.2 5.4 5.6 5.8 6.0 SN T T 2 8 4 SN 6 1.06 L ersn uciia eidduln iuctos oe o e saddle– new how Note, bifurcations. period-doubling subcritical represent PD handthesaddlesolutionattheupperbranch. SN 3 L 2 sub 3 C PD T 3 3 1.07 and SN PD ω 6 C 3 representthefirstandsecondperioddoublingwiththe 1.08 Period-3 PD Chaos PD SN T 6 2 PD 1 sub 6 , R T 4 sub 4 PD 1.09 ,and SN 3 SN T 3 R 4 T 3 B 8 ersn ou-it bifurcations torus-birth represent c h aesaddle– same the , 55 Downloaded By: 10.3.98.104 At: 17:44 01 Oct 2021; For: 9781466590441, chapter3, 10.1201/b20232-5 eino osnhooschaos. nonsynchronous of region h tbend yl.A etoe bv,ti mle httepro-obe roi ou htexists that torus of ergodic period-doubled value the given that implies a this for above, consis- mentioned periodicity As curves, highest cycle. the node bifurcation of stable the curve saddle–node bifurcation the of of saddle–node value the given finds the several by for it until structure unaffected the penetrates dynamics, tently torus ergodic of region o h ul eeoe hs ycrnzdchaos. synchronized curves bifurcation phase saddle–node chaotic developed the the of fully organization into the the deeper alter- for of progresses of description system a structure provide well-organized the cannot as the we dissolve Hence, that regime. to note tends to curves important bifurcation is saddle–node it nating description this complete To size. and ae and cade, uln h tutr fteacmltn e fsdl–oebfrain.Tecre eoe ,2 ,8, 4, 2, 1, denoted curves The bifurcations. curves bifurcation saddle–node saddle–node of the to set represent thus accumulating etc., and bifurcations the period-doubling of of structure generations eight the to outline seven through zone resonance the of side observed the preserves sketch the scales, the distorting While oscillator. Rössler forced systematicofthestructure.Thecurvesdenoted1,2,4,8,etc.,representthesaddle–nodebifurcationcurves periodically the for zone nance 3.17 FIGURE SN 56 to many different problems in physics, chemistry, biology, and other fields of science and technology and science of fields applications and other obvious with biology, chemistry, research in physics, of area problems interesting different extremely many an to is synchronization phase Chaotic Conclusions 3.8 oebfraincurves, bifurcation node SN eidduln curves. period-doubling o hi w ore,asseai tutr onmtraie nwihtesdl–oebifurcation saddle–node the which in curve materializes accumulation final soon a structure around systematic alternate a curves courses, own their low osnhooscasi ace ihaga akrud sbfr,tebfrainprmtr r the are parameters bifurcation the before, of As region the background. and gray gray, a frequency is with forcing windows) hatched periodic is of chaos occurrence the nonsynchronous (neglecting chaos synchronized phase of whitebackground,theregioninwhichergodictori(of region the is hatched, exist periodicity) different © 2016 Group, by Taylor LLC & Francis 4 lsriseto fFgr .7rvasta lhuhtefrtfwsdl–oebfraincre fol- curves bifurcation saddle–node few first the although that reveals 3.17 Figure of inspection Closer L , SN 8 L t.Tega oerpeet h eino hs ycrnzdcas n h ace ryzn sthe is zone gray hatched the and chaos, synchronized phase of region the represents zone gray The etc. , ETD kthilsrtn h acd fsdl–oebfrain ln h ethn ieo h : reso- 1:1 the of side left-hand the along bifurcations saddle–node of cascade the illustrating Sketch ersnstecreo roi ou etuto.Tersnnezn speetdwt a with presented is zone resonance The destruction. torus ergodic of curve the represents ω c n h olnaiyparameter nonlinearity the and c omnctswt ut-aee eoac ou ftesm eidct,form, periodicity, same the of torus resonance multi-layered a with communicates hsi lotesdl–oebfrainta rvdstebudr feitnefor existence of boundary the provides that bifurcation saddle–node the also is This . SN PD ∞ ∞ SN eoe h ueial eemndacmlto uv o h saddle– the for curve accumulation determined numerically the denotes iial ersnsteacmlto uv o h eidduln cas- period-doubling the for curve accumulation the represents similarly ∞ c Nonsynchronous ETD chaos ω SN Ergodic c SN 1 L torus fteRslroscillator. Rössler the of , SN ∞ adoko plctoso ho Theory Chaos of Applications of Handbook 8 ti loitrsigt oeta h hatched the that note to interesting also is It . 2 L , 1 SN synchronized 2 4 L Phase chaos Resonant 4 , SN torus 8 L PD PD ,etc.,andPD PD 1 2 ∞ 1 , PD 2 oaetefirst the locate 1 L , SN 2 L , Downloaded By: 10.3.98.104 At: 17:44 01 Oct 2021; For: 9781466590441, chapter3, 10.1201/b20232-5 ftetasto ocatcpaesnhoiaini eidclyfre öse siltr ihthis With within oscillator. occur that Rössler bifurcations forced the follow periodically to [10] a method continuation in standard a synchronization applied have phase we purpose, chaotic to transition the of [14]. kidney the p of nephrons the neighboring between such synchronization of phase frequency of the of occurrence matter the a becomes then not or 2π occurs synchronization phase whether question The small margin a relatively within constant remains difference phase the that synchronization finds margin often phase small Theoretically, one the value. margin if constant occur small some to sufficiently near said a be fall may with to if, continues phases occur two to the said between be two inter- then between may synchronization oscillators 1:1phase chaotic ratios, acting, locking other for definitions pha similar instantaneous trans- With series. theHilbert the time define long, sufficiently to is one this series If allows the formation Provided synchroniza- oscillators. phase chaotic role. a of interacting degree two the determine between to tion approach play transformation Hilbert a also of use may the suggested have oscillator chaotic unforced intosev- up the distribution break may in the points groups. connection loops stroboscopic of eral this thedistribution different In inhomogeneous, the sufficiently distribution. is of distribution basic times same the return return of again to reorganization major thatthedistributionofdotsacrosstheforcedstatefromtimetotimeturnsunstableandundergoesa presence the with noise. associated be of may phenomenon revealed sync rounding be the similar a then of studies, may edges experimental In sharp synchronization thresholds. otherwise frequency the follows precise of approximately of rounding system lack the The chaotic by frequency. forced forcing the of the of frequency restricted variation mean a the the delineate can where one frequencies that forcing means similar variation of This a 3.1a . range intro- displays Figure typically actually in instance, without shown for chapter characteristics frequency, synchronization this mean the The of to [33]. beginning phase the a in of concept described the tools ducing diagnostic the of some use often nerthatinmanywaysresemblesthereactionobservedforthesimpleRösslersystem.Moreover,onecan [8]. we exponent Lyapunov vanishing values of stable of the neutrally direction the determination the i.e., parameter also involve direction, may practice, In the studies theoretical variable. phase More for [33]. the of sufficient often is arise point definition in the not consistent a direction for do fixed any allow between to type angle simple this sufficiently the of is onto attractor dynamics Rössler the the problems of regime projection the the However, , 3.1b in Figure chosen in xy illustrated are As attractor. study. parameters present the the in if funnel used system have Rössler the so-called for the even arises of situation This [33]. cycle a longertransitiontimes,anditmaynotbepossible,inaconsistentmanner,todecidewhichloopsconstitute of other the or [49]. one attractor points around the Lorenz equilibrium rotation for symmetric between manner, two for instance, random the apparently case, the an not is in may shifts, it This dynamics situations [8,33]. the many solution where in satisfactory a and problem, find complicated to a possible present be questions can even many oscillator particular, chaotic given In a complete. for from phase far still is area this relatedtothesynchronizationofmulti-modeoscillationsremainunsolved.Alreadythedefinitionofthe of understanding useful many our contributed [6,27,33], already has results synchronization phase chaotic on work although However, [1,23,40]. Chaos Synchronized Phase to Transition the On eido iet hnjm n rmr 2 more or one jump then to time of period © 2016 Group, by Taylor LLC & Francis paefrteeprmtrvle rdcsarglrrtto ftepaepitaon rg.Hence, origo. around point phase the of rotation regular a produces values parameter these for -plane h ups fti hpe a ent rsn h eut farltvl ealdbfrainanalysis bifurcation detailed relatively a of results the present to been has chapter this of purpose The [35] al. et Pikovsky series, time experimental for relationships phase the analyzing of primarily, view, In find might one applied, is c and 3.1b Figure in illustrated technique projection stroboscopic the When man- a in forcing external an to react systems chaotic many that clear is it complications these of spite In much with loops and loops small between shifts random less or more display systems chaotic Other tp.W aescesul sdteHlettasomto praht eosrt,frinstance, for demonstrate, to approach transformation Hilbert the used successfully have We steps. xy μ paeadtedrcinfo rg oteisatnosphase- instantaneous the to origo from direction the and -plane  π tp steoclaosframmn os synchronization. loose moment a for oscillators the as steps neprmna tde fitrcigcatcoscillators chaotic interacting of studies experimental In 2π. e(n mltd)o neprmnal observed experimentally an of amplitude) (and se oia uua rsueoclain o arof pair a for oscillations pressure tubular roximal rnzto hrceitca h synchronization the at characteristic hronization μ,thedifference μ o certain a for 57 Downloaded By: 10.3.98.104 At: 17:44 01 Oct 2021; For: 9781466590441, chapter3, 10.1201/b20232-5 eutapis rvddta h : r14snhoiainfrteitra oe fteidvda oscil- individual the of modes internal coupling. the the for by synchronization unaffected 1:4 remained or lator 1:5 This the states. that anti-symmetric provided respectively applies, symmetric, result the into essentially synchronization phase synchronization demonstrated for has structures to chaotic reso- [23] bifurcation (nephrons) the same the systems transition two-oscillator of of identical part nearly study coupled, that arecent two is for that phase-modulated modes of note occurrence finally anti-symmetric the us includes the Let This quasiperiodicity. modes. for quasiperiodic different diagram by invaded bifurcation is zone the nance of reconstruction this torus the of where point the to down way the all born. was curve curve bifurcation the bifurcation on saddle–node points corresponding the by and chaos, curve to transition the to close supported, curves. bifurcation are the saddle–node and They corresponding curve curve. period-doubling corresponding bifurcation the torus between intersection former bifurcation of torus points The the modes. from resonance emerge unstable stable curves triply where formed and is doubly region with a way, coexist this oscillations In quasiperiodic zone. resonance the inside but curves, bifurcation saddle–node from quasiperiodicity. transition to the dynamics mark saddle– longer periodic former no the stable curves of bifurcation right zone. saddle–node the resonance to the However, and the curve. left of bifurcation the edge to node manner the alternating along an up in occurs continue from still diagram and process emerge bifurcation this Moreover, curve curves the bifurcations period-doubling in saddle–node corresponding Like the the system, cycles. bifurca- on Rössler resonance points saddle–node forced the of periodically of of the cascade A existence the edge for of 3.2) type. range (Figure Here, the different [23]. delineate of to structure bifurcations bifurcation serves of tions different cascades very involves a zone resonance observe the can one states, anti-symmetric to attractor. Rössler symmetricstatesisessentiallythesameasthestructurewehavedescribedfortheperiodicallyforced bifurcations subsequent the and bifurcations. space, homoclinic phase and of processes torus-birth part involve larger space typically much with a phase move oscillators to of two made are the part excursions state, anti-symmetric phases, explored an opposite into the occurs synchronization phase, the hand, in bifurc other the move subsequent ofa on oscillators the two and presence narrow, the the relatively state, that is remains symmetric a notice into to place thing takes symmetri first into modes the synchronized the systems, splits coupling period-doubling identical nearly coupled, of [17,18]. tongue resonance a period-doubling of the edge characterizes the that along criticality) bifurcation bifur- (cyclic different structure of scaling alternation special this the precisely produce is that It from cations space. modes parameter torus-birth of resonance regions of stable designated involvement the their alternating from restricting the escaping bifurca- in (v) bifurcations and doubling zone, period-doubling torus resonance subcritical the the and (iv) of tori bifurcations zone, edges the resonance elimination the resonance along and the occur between outside transformation, that communication exist tions formation, that the the oscillations (iii) quasiperiodic (ii) tori, the zone, resonance and multi-layered resonance node of the the forms for in different bifurcations occur of period-doubling that interconnected cycles the saddle between and relation the (i) including nomena, n ln h de fte11snhoiainzn sicesn auso h ocle nonlinearity so-called the of values increasing as zone synchronization 1:1 the of parameter edges the along and 58 © 2016 Group, by Taylor LLC & Francis h ou iucto uvspoueasto eidduldqaieidcsae,adteoealresult overall the and states, quasiperiodic doubled period of set a produce bifurcation curves bifurcation torus torus the The between stretches bifurcations homoclinic of curve a points, these of each From the along up stretch that curves bifurcation torus of cascade a by over taken been has function This leads typically locking phase the where zone, synchronization the to of leads edge that other tongue the synchronization along However, a of edge the along observe can one structure bifurcation The pair a for synchronization phase chaotic to transition the consider to investigation the extends one If phe- interesting of number significant a between interplay the examine to us allowed has approach This c aetesse ptruhtecsaeo eidduln bifurcations. period-doubling of cascade the through up system the take tosi eea ananteoiia ymty If, symmetry. original the maintain general in ations n nismercsae 3] fsynchronization If [38]. states anti-symmetric and c adoko plctoso ho Theory Chaos of Applications of Handbook Downloaded By: 10.3.98.104 At: 17:44 01 Oct 2021; For: 9781466590441, chapter3, 10.1201/b20232-5 References C-type with associated structure bifurcation the about criticality. discussions valuable for Kuznetsov S.P. thank We Acknowledgments Chaos Synchronized Phase to Transition the On 2 .S ad n .G oebu.Snhoiainadcatzto foclain nculdself- coupled in oscillations of chaotization and Synchronization Rosenblum. G. M. and dynamical Landa chaotic in S. P. bifurcation Riddling 22. Venkataramani. C. S. and Yorke, A. J. Grebogi, C. Lai, Y.-C. 21. 20. Yu.A.Kuznetsov,H.G.E.Meijer,andL.VanVeen.Thefold-flipbifurcation. Kuznetsov. A. Yu. 19. 3 .F eg,T .Crol n .M eoa eycrnzto yproi orbits. periodic by Desynchronization Pecora. M. L. and Carroll, L. T. Heagy, F. J. 13. 8 .P unto n .R aav nvraiyadsaigfrtebekpo hs ycrnzto at synchronization phase of breakup the for scaling and Universality Sataev. R. I. and Kuznetsov and P. universality S. situations, critical 18. Multiparameter Sataev. R. I. and Kuznetsov, in P. classes A. Kuznetsov, universality P. period-doubling S. of 17. variety A Sataev. R. I. and Kuznetsov, P. S. Kuznetsov, P. A. 16. Schreiber. T. and Kantz H. 15. phenomena Synchronization Mosekilde. E. and Sosnovtseva, V. O. Yip, K.-P. Holstein-Rathlou, N.-H. 14. arteries small pressurized isolated, of contractions Rhythmic Nilsson. H. and Bülow, A. Gustafsson, H. 12. 1 .I ymn .S ad,adY .Nyak ycrnzn h hoi siltosb xenlforce. external by oscillations chaotic the Synchronizing Neymark. I. Y. and Landa, S. P. Dykman, I. G. 11. 10.E.J.Doedel,R.C.Paffenroth,A.R.Champneys,T.F.Fairgrieve,Yu.A.Kuznetsov,B.E.Oldeman, © 2016 Group, by Taylor LLC & Francis .A.AhlbornandU.Parlitz.Experimentalobservationofchaoticphasesynchronizationofaperiodically1. .A ryi,D yrds n .Lgre l ho-ae omncto sn omrilfbeoptic fibre commercial using communication Chaos-based al. et Lager L. and Syvridis, D. basins. Argyris, A. Riddled chaos. 4. of Synchronization Kan. Safonova. A. M. I. and Postnov, E. and D. Vadivasova, You, E. T. Anishchenko, Z. S. V. Yorke, 3. A. J. Alexander, C. J. 2. .C .S ait,A .Btsa .A .d ots .L in,adS .Lps hoi hs synchro- phase Chaotic Lopes. R. S. and Viana, L. R. Pontes, de 8.S.Boccaletti,J.Kurths,G.Osipov,D.L.Valladares,andC.S.Zhu.Thesynchronizationofchaotic C. A. J. Batista, M. A. Batista, S. A. C. Sosnovtseva. 7. O. and Postnov, D. Janson, N. Balanov, A. 6. .D .DSae,R rbn .Ot n .Ry eetn hs ycrnzto nacatclsrarray. laser chaotic a in synchronization phase Detecting Roy. R. and Ott, E. Breban, R. DeShazer, J. D. 9. A.Arnéodo,P.H.Coullet,andE.A.Spiegel.Cascadeofperioddoublingsoftori.5. siltn systems. oscillating systems. 2004. 14:2253–2282, ouae rqec-obe DYGlaser. ND:YAG frequency-doubled modulated links. Chaos Bifurcat. J. 1992. h ne fcasi eidclydie öse oscillator. Rössler driven periodically a in chaos of onset the maps. period-doubling two-dimensional in scaling chaos. to transition of analysis multi-parameter interaction. nephron–nephron in 1995. 52:R1253–R1256, rat. from systems. neurons. bursting of networks scale-free in nization 2009. Berlin, Springer-Verlag, 1983. ho oi Fract. Soli. Chaos 2002. Technology, of Institute California report, differential Technical ordinary homcont). for (with software equations bifurcation and Continuation 2000: Auto Wang. X. and Sandstede, B. Phys.Rev.Lett. Nature hs e.Lett. Rev. Phys. hs Rep. Phys. caPyil Scand. Physiol. Acta 3:4–4,2005. 438:343–346, , 7040,2001. 87:044101, , :3–4,1992. 2:633–644, , :3–5,1991. 1:339–353, , lmnso ple iucto Theory Bifurcation Applied of Elements 6:–0,2002. 366:1–101, , pl eh Rev. Mech. Appl. 75–8 1996. 77:55–58, , olna ieSre Analysis Series Time Nonlinear 5:4–5,1994. 152:145–152, , Chaos 64446 1993. 46:414–426, , 14746 2001. 11:417–426, , pisLett. Optics hsc D Physica .Sa.Phys. Stat. J. hs e.E Rev. Phys. pigrVra,Bri,2004. Berlin, Springer-Verlag, . abig nvriyPes K 1997. UK, Press, University Cambridge . 0:112 1997. 109:91–112, , 425–76 2009. 34:2754–2756, , ycrnzto:Fo ipet Complex to Simple From Synchronization: hs e.E Rev. Phys. 2:9–4,2005. 121:697–748, , 6061,2007. 76:016218, , n.J i.Chaos Bif. J. Int. 4061,2001. 64:046214, , n.J iuct Chaos Bifurcat. J. Int. hs et A Lett. Phys. Phys.Rev.E 2:795–813, , 94:1–6, , Int. 59 , . , Downloaded By: 10.3.98.104 At: 17:44 01 Oct 2021; For: 9781466590441, chapter3, 10.1201/b20232-5 2 .F ukv mgso ycrnzdcas xeiet ihcircuits. with Experiments chaos: oscillators. synchronized chaotic of Images of Rulkov. F. synchronization N. Phase Kurths. 42. J. and Pikovsky, S. A. Rosenblum, G. M. 41. 40. E.Rosa,W.B.Pardo,C.M.Ticos,J.A.Walkenstein,andM.Monti.Phasesynchronizationofchaosin and Anishchenko, S. V. Balanov, G. A. Sosnovtseva, V. O. Vadivasova, E. T. for bursting. Postnov, A mechanism E. D. intermittency: On–off 37. Tresser. C. and Spiegel, A. E. Platt, N. 36. 43. M.Sekikawa,N.Inaba,T.Yoshinaga,andT.Tsubouchi.Bifurcationstructureofsuccessivetorusdou- chaos. of synchronization phase to Transition Hess. H. M. and Ott, E. Rosa, E. 39. period- identical coupled, symmetrically in quasiperiodicity of Emergence Mosekilde. E. and Reick C. 38. attractors. strange of interaction 35. the A.S.Pikovsky,M.G.Rosenblum,G.V.Osipov,andJ.Kurths.Phasesynchronizationofchaoticoscil- On Pikovsky. S. A. 34. Kurths. J. and Rosenblum, M. Pikovsky, eyelet A. and collision 33. Attractor–repeller Kurths. J. and Zaks, M. Rosenblum, M. Osipov, G. Pikovsky, A. 32. 8 .Ynhc,Y.Mitek,adE oeid.Ls fsnhoiaini ope öse systems. Rössler coupled in synchronization of Loss Mosekilde. E. and Maistrenko, Yu. Yanchuck, S. 48. 7 .W n .L.Gnrlzdpoetv ycrnzto ftefatoa re hnhyperchaotic Chen order fractional the of synchronization projective Generalized Lu. systems. Y. dynamical of and Wu control X. and synchronization 47. for framework unified A Chua. O. L. and Wu W. C. 46. semicon- chaotic with generation bit random physical 45. Fast T.E.Vadivasova,A.G.Balanov,O.V.Sosnovtseva,D.E.Postnov,andE.Mosekilde.Synchronization al. et Inoue M. and Amano, K. Uchida, A. 44. systems. chaotic in Synchronization Carroll. inter- L. T. on–off and and Pecora basins M. L. riddled of 30. occurrence The bifurcations: Blowout Sommerer. C. J. syn- phase and of Ott structure E. the On 29. Yanochkina. O. and Laugesen, L. J. Zhusubaliyev, T. Zh. Mosekilde, E. 28. Postnov. D. and Maistrenko, Yu. Mosekilde, E. 27. 1 .Pn,Y in,adF hn eeaie rjciesnhoiaino rcinlodrcatcsys- chaotic order fractional of synchronization projective Generalized Chen. F. and Jiang, Y. Peng, G. 31. neuronal a in synchronization phase chaotic determined Experimentally Llinás. R. and Makarenko V. 26. 5 .L asrno .L asrno n .Mosekil E. and Maistrenko, L. Y. Maistrenko, L. V. 25. oscil- period-doubling 24.J.L.Laugesen,E.Mosekilde,andZh.T.Zhusubaliyev.BifurcationstructureoftheC-typeperiod- of Synchronization Holstein-Rathlou. N.-H. and Mosekilde, E. Laugesen, L. J. 23. 60 © 2016 Group, by Taylor LLC & Francis e.Lett. Rev. tube. discharge plasma a 1998. 1645, synchronization. phase chaotic to transition the in multistability of Role Mosekilde. E. Lett. bling. systems. doubling 1999. 232, driving. external by lators 2001. UK, Press, University Cambridge synchronization. phase to transition the at intermittency hsc D Physica system. Chaos Bif. J. Int. bifurcations. and Diagnostics systems: chaotic driven in lasers. ductor mittency. chaos. chronized 2002. Singapore, Scientific, World tems. system. ini ope iemaps. sine coupled in tion transition. doubling nephrons. coupled vascular in lations 02922 1993. 70:279–282, , hsc A Physica hs et A Lett. Phys. ol Dyn. Nonl. 5177172 1998. 95:15747–15752, , Sci. Acad. Natl. Proc. 610–87 1996. 76:1804–1807, , 5:64,2001. 154:26–42, , hs et A Lett. Phys. auePhoton. Nature 8:7834,2008. 387:3738–3746, , :7–9,1994. 4:979–998, , ho oio.Fract. Soliton. Chaos hs e.E Rev. Phys. 4:8–9,2006. 348:187–194, , 72–5 2009. 57:25–35, , hsc D Physica 8:94,1994. 188:39–47, , n.J iuct Chaos Bifurcat. J. Int. hsc D Physica n.J i.Chaos Bif. J. Int. :2–3,2008. 2:728–732, , 4:8–9,2012. 241:488–496, , 211–45 1995. 52:1418–1435, , 0:1–3,1997. 104:219–238, , 62–7 2013. 46:28–37, , Chaos ycrnzto:AUieslCneti olna Science Nonlinear in Concept Universal A Synchronization: 526–17 2007. 15:2161–2177, , 1032,2011. 21:033128, , 025–53 2000. 10:2551–2563, , hoi ycrnzto:App Synchronization: Chaotic e hoi ycrnzto n antisynchroniza- and synchronization Chaotic de. adoko plctoso ho Theory Chaos of Applications of Handbook .Py.B Phys. Z. hs et A Lett. Phys. hs e.Lett. Rev. Phys. 51912 1984. 55:149–152, , Phys.Rev.Lett. 5:67,1999. 253:66–74, , Chaos 94–0 1997. 79:47–50, , iain oLvn Systems. Living to lications Phys.Rev.Lett. :6–7,1996. 6:262–279, , 48184 1991. 64:821–824, , Chaos 80:1642– , hs Rev. Phys. 9:227– , Phys. . Downloaded By: 10.3.98.104 At: 17:44 01 Oct 2021; For: 9781466590441, chapter3, 10.1201/b20232-5 4 h .Zuuaie n .Mskle oe otst ho hog ou radw nnon- in breakdown torus through chaos to routes Novel Mosekilde. E. and Zhusubaliyev T. Zh. 54. 3 h .Zuuaie n .Mskle utlyrdtr nasse ftoculdlgsi maps. logistic coupled two of system a in tori Multilayered Mosekilde. E. and Zhusubaliyev T. Zh. map 53. coupled in tori multilayered of destruction and Formation Mosekilde. period- E. to and Zhusubaliyev tori T. resonance Zh. multi-layered 52. From chaotic Mosekilde. of E. synchronization and phase Laugesen, Noise-enhanced L. J. Hudson. Zhusubaliyev, L. T. Zh. J. and 51. Kiss, Z. I. Kurths, J. Zhou, C. 50. 9 .Za,Y-.Li .Wn,adJ-.Go iist hoi hs synchronization. phase chaotic to Limits Gao. J.-Y. and Wang, R. Lai, Y.-C. Zhao, L. 49. Chaos Synchronized Phase to Transition the On © 2016 Group, by Taylor LLC & Francis netbemaps. invertible A Lett. systems. tori. ergodic doubled oscillators. 2004. 66:324, 7:4–5,2009. 373:946–951, , Chaos hs e.Lett. Rev. Phys. 8072,2008. 18:037124, , hsc D Physica hs et A Lett. Phys. 9040,2002. 89:014101, , 3:8–0,2009. 238:589–602, , 7:5423,2010. 374:2534–2538, , u.Py.Lett. Phys. Eur. Phys. 61 ,