Deterministic coherence resonance in off intermittency and delayed feedback

Transcription

1 Wasaw Uivesit of Techolog Facult of Phsics Koszkowa 75 PL--66 Wasaw Polad Tel: (48) 66767; fax: (48) 687; Jaosław aw Buk Adzej Kawiecki ad Teodo Buche Detemiistic coheece esoace i sstems with o-off off itemittec ad delaed feedback 4th Chaotic odelig ad Simulatio Iteatioal Cofeece Agios Nikolaos Cete Geece

2 J. Buk A. Kawiecki ad T. Buche Detemiistic coheece esoace i sstems with o-off itemittec ad delaed feedback O-off itemittec is a sot of exteme bustig which occus i sstems posessig a chaotic attacto withi a ivaiat maifold whose dimesio is less tha that of the phase space [e.g N. Platt E.A. Spiegel ad C. Tesse Phs. Rev. Lett (993); J.F. Heag N. Platt ad S.. Hammel Phs. Rev. E49 4 (994)] x p<p p>p c c ivaiat maifold tasvesel stable chaotic attacto tasvesel ustable As a cotol paamete p cosses a cetai theshold p c this attacto udegoes a supecitical blowout bifucatio ad loses tasvese stabilit ad a ew attacto is fomed which ecompasses that cotaied withi the ivaiat maifold [e.g. E. Ott ad J. Sommee Phs. Lett. A88 39 (994)] x chaotic bust lamia phase t Attacto bubblig occus if the ivaiat maifold is destabilized below the theshold fo the blowout bifucatio e.g. due to the actio of oise i the diectio tasvese to the ivaiat maifold. Chaotic busts occu alead fo p<p c. 4th Chaotic odelig ad Simulatio Iteatioal Cofeece Agios Nikolaos Cete Geece

3 J. Buk A. Kawiecki ad T. Buche Detemiistic coheece esoace i sstems with o-off itemittec ad delaed feedback Example Logistic map with time-depedet cotol paamete [J.F. Heag N. Platt ad S.. Hammel Phs. Rev. E49 4 (994)] ζ ( ) + aζ chaotic pocess ( ) ivaiat maifold Chaacteistics of o-off itemittec ae ot sesitive to the details of the damics withi the ivaiat maifold (ζ ) povided that its coelatio time is small; thus ζ ca be appoximated b adom pocessξ with uifom distibutio o the iteval (). ( ).7... ζ ξ a c e Chaacteistics of o-off itemittec Distibutio of legths of lamia phases P ( τ) τ 3 ea duatio of lamia phases τ p p c Example Diffusivel coupled chaotic oscillatos at the edge of idetical schoizatio H( z ) x& & f f ( x) + H( x) ( ) + H( x ) liea couplig fuctio O-off itemittec occus as the cotol paamete (e.g. the stegth of the liea diffusive couplig) deceases below the theshold fo idetical schoizatio. The tajectoies of the two oscillatos ae schoized duig log time itevals (lamia phases) but occassioall deviate fom each othe (chaotic busts). I this wok we ae iteested i the effect of delaed feedback (memo) o sstems with o-off itemittec x ( tτ) x( t) 4th Chaotic odelig ad Simulatio Iteatioal Cofeece Agios Nikolaos Cete Geece

4 J. Buk A. Kawiecki ad T. Buche Detemiistic coheece esoace i sstems with o-off itemittec ad delaed feedback 4th Chaotic odelig ad Simulatio Iteatioal Cofeece Agios Nikolaos Cete Geece The logistic map with time-depedet cotol paamete ad delaed feedback ( ) ( ) k K a K + + ζ Fo ζ cost this map is a model fo chaos cotol [T. Buche ad J.J. śebowski Phs. Rev. E63 6 ()] Fo the damics is well apoximated b the lieaizatio ( ) k K a K + + ζ Itoducig ew vaiables ( ) ( ) ( ) ( ) + + k j k j k K K the lieaized map ca be witte i the fom of a k+ dimesioal map ( ) ( ) ( ) ( ) + + ˆ ˆ k a K L O ζ with the ivaiat maifold Tasvese stabilit of the attacto withi the ivaiat maifold is detemied b the tasvese Lapuov expoet ( ) { } : ˆ ˆ ˆ l lim N N N T N K ζ λ

5 J. Buk A. Kawiecki ad T. Buche Detemiistic coheece esoace i sstems with o-off itemittec ad delaed feedback Itemittec theshold a c is defied as the value of the cotol paamete a fo which the attacto withi the ivaiat maifold loses tasvese stabilit i.e. the tasvese Lapuov expoetλ T cosses zeo. Itemittec theshold a c vs. K fo vaious k (see leged) fo the map withζ ξ (white oise o the uit iteval) Fo a > a c the output sigal exhibits itemittet busts sepaated b quiescet phases duig which. Time seies fo k K. a. (just above a c ) 4th Chaotic odelig ad Simulatio Iteatioal Cofeece Agios Nikolaos Cete Geece

6 J. Buk A. Kawiecki ad T. Buche Detemiistic coheece esoace i sstems with o-off itemittec ad delaed feedback Output sigal Z < > th th. Fo K> the distibutio of lamia phase legths N(τ) fo a just above a c exhibits a seies of maxima at the values of τ equal to k ad its multiples supeimposed o a powe-law ted tpical of o-off itemittec N(τ) τ -3/. Histogam of the umbe of lamia phases N(τ) of duatio τ fo k K.3 a. vetical lies ae daw at multiples of k A boad peak ceteed at the fequec /k appeas i the powe spectum desit (PSD) of Z ; simila peaks occu at the multiples of /k Powe spectum desit fom the time seies Z fo k64 K.3 a. f/k 4th Chaotic odelig ad Simulatio Iteatioal Cofeece Agios Nikolaos Cete Geece

7 J. Buk A. Kawiecki ad T. Buche Detemiistic coheece esoace i sstems with o-off itemittec ad delaed feedback Detemiistic coheece esoace i the logistic map with delaed feedback Coheece esoace maifests itself as a peak of egulait of the output sigal of cetai oliea stochastic sstems fo optimum itesit of the iput oise ad without a exteal peiodic stimulatio [e.g. H. Gag et al. Phs. Rev. Lett (993); W.-J. Rappel ad S.H. Stogatz Phs. Rev. E5 349 (994); A. Logti Phs. Rev. E (997); A.S. Pikovsk ad J. Kuths Phs. Rev. Lett (997)]. Coheece esoace was obseved i sstems with delaed fedback icludig bistable sstems [e.g. L.S. Tsimig ad A. Pikovsk Phs. Rev. Lett (); K. Paajotov et al. Phs. Rev. A69 8 (4);. Aizaleta Ateaga et al. Phs. Rev. Lett (7)] excitable sstems [e.g. G.C. Sethia J. Kuths ad A. Se Phs. Lett. A364 7 (7)] theshold cossig detectos [e.g. R. ose ad A. Logti Phs. Lett. A (6)]. SPA ad SNR show maxima as fuctios of the cotol paamete a SPA ad SNR vs. a fom the PSD of Z fo k64 K.3. SPA is defied as the absolute height of the maximum of the PSD at f/k; SNR is defied as the elative height of the maximum of the PSD at f/k with espect to the mea value of the PSD o the iteval (-/k 3/k) SPA 5 x SNR Hee maximum egulait of the output sigal is obseved as the damics withi the ivaiat maifold cotolled b the paamete a is chaged without exteal oise; so this pheomeo is a example of detemiistic coheece esoace [e.g. J.F. atiez Avila H.L.D. de S. Cavalcate ad J.R. Rios Leite Phs. Rev. Lett (4)] 4th Chaotic odelig ad Simulatio Iteatioal Cofeece Agios Nikolaos Cete Geece

8 J. Buk A. Kawiecki ad T. Buche Detemiistic coheece esoace i sstems with o-off itemittec ad delaed feedback Diffusivel coupled Rössle oscillatos with delaed feedback x& & z& x& & z& b ( + z) x+ a+ k( ) + Ks( τ) + z( x c) diffusive delaed ( + z ) couplig feedback x+ a+ k( ) Ks( τ) ( b+ δb) + z ( x c) a. b s ( τ) ( tτ) ( tτ) ( tτ) δb. c models small paamete mismatch i expeimetal sstems which leads to the occuece of attacto bubblig Fo K adδb the oscillatos ae ideticall schoized if k > k c. ad fo k < k c itemittet busts of deschoizatio (o-off itemittec) occu Fo K adδb> chaotic bustig occus alead fo k > k c (due to attacto bubblig) Fo K adδb ad k > k c. chaotic attacto exists withi the ivaiat schoizatio maifold x ; z z x ; which loses tasvese stabilit fo k < k c 4th Chaotic odelig ad Simulatio Iteatioal Cofeece Agios Nikolaos Cete Geece

9 J. Buk A. Kawiecki ad T. Buche Detemiistic coheece esoace i sstems with o-off itemittec ad delaed feedback Output sigal Z( t) ( t) ( t) ( t) ( t) ( t) ( t) <. >. Fo k < k c (below the itemittec theshold whe the schoizatio is lost) i the sstem of diffusivel coupled oscillatos ad delaed feedback similia pheomea ae obseved as i the logistic map with time-depedet cotol paamete. Fo k < k c the output sigal (t) exhibits itemittet busts sepaated b quiescet phases duig which (t). Time seies (t) fo τ5 k. δb -4 A boad peak ceteed at the fequec /τ appeas i the powe spectum desit (PSD) of Z(t); simila peaks occu at the multiples of /τ Powe spectum desit fom the time seies Z(t) fo τ5 k. δb -4 f/τ 4th Chaotic odelig ad Simulatio Iteatioal Cofeece Agios Nikolaos Cete Geece

10 J. Buk A. Kawiecki ad T. Buche Detemiistic coheece esoace i sstems with o-off itemittec ad delaed feedback Detemiistic coheece esoace i the sstem of diffusivel coupled Rössle oscillatos with delaed feedback the case of attacto bubblig the case of o-off itemittec SPA vs. k fo δb (cicles the case of o-off itemittec) adδb -4 (dots the case of attacto bubblig) Detemiistic coheece esoace occus i the sstem of diffusivel coupled Rössle oscillatos ad the output sigal exhibits maximum egulait fo optimum value of the paamete k which cotols the iteal damics withi the ivaiat schoizatio maifold. Hece the esults of umeical simulatios suggest that detemiistic coheece esoace ca be obseved expeimatall i sstems of coupled chaotic oscillatos at the edge of idetical schoizatio. 4th Chaotic odelig ad Simulatio Iteatioal Cofeece Agios Nikolaos Cete Geece

11 J. Buk A. Kawiecki ad T. Buche Detemiistic coheece esoace i sstems with o-off itemittec ad delaed feedback Coclusios The ifluece of delaed feedback o sstems with o-off itemittec was studied usig geeic maps with the time-depedet cotol paamete ad schoized chaotic oscillatos. Delaed feedback ca decease the theshold fo the blowout bifucatio ad its effect esembles that of oise applied tasvesel to the ivaiat maifold. Detemiistic coheece esoace was obseved i sstems ude cosideatio chaacteized b the appeaace of a seies of maxima at the multiples of the dela time i the pobabilit distibutio of the legths of lamia phases supeimposed o the powe-law ted tpical of o-off itemittec ad b the pesece of a stog peiodic compoet i the itemittet time seies with peiod equal to the dela time. The stegth of the latte compoet exhibits maximum as the cotol paamete is vaied due to the chages of the iteal damics of the sstem withi the ivaiat maifold i aalog with the case of coheece esoace i stochastic sstems. 4th Chaotic odelig ad Simulatio Iteatioal Cofeece Agios Nikolaos Cete Geece

12 Thak ou fo ou attetio coutes of D. P. Jóźwiak Svalbad