Characterization of Rössler and Duffing maps with Rényi entropy and generalized complexity measures

Cha ac e iza ion o R¨ossle and Du ing maps wi h R´enyi en opy and gene alized complexi y measu es B. God´o and ´ A. Nagy Depa men o Theo e ical Physics, Uni e si y o Deb ecen, Deb ecen, Hunga y E-mail: [email protected] Abs ac . R´enyi en opy and gene alized complexi y measu es a e used o desc ibe he chao ic beha iou o dynamical sys ems. These measu es a e ound o be sensi i e o he ine de ails o he R¨ossle and he Du ing maps. They a e good desc ip o s o chao ic beha iou . Pe iodic windows and he ac al cha ac e o he chao ic dynamics a e nicely de ec ed. 1. In oduc ion The e exis se e al quan i ies o s udy he chao ic beha iou o dynamical sys ems. Complexi y measu es ha e p o ed o be especially e icien in his espec . One o hese measu es is he LMC (Lopez-Ruiz - Mancini - Calbe ) s a is ical complexi y [1]. A couple o yea s ago, a one- and a wo-pa ame e ex ension [2] o his measu e we e pu o wa d. These gene aliza ions a e based on he R´enyi en opy. Fi s , some simple quan um sys ems (H-a om, ha monic oscilla o and squa e well) we e s udied wi h hese measu es. Recen ly, i has been demons a ed [3] ha hese gene alized complexi y measu es a e sui able o desc ibe chao ic beha io . The logis ic and Tinke bell maps we e analyzed. In his wo k he R¨ossle and he Du ing maps a e s udied wi h he R´enyi en opy and he gene alized complexi y measu es. 2. R´enyi En opy and Gene alized S a is ical Complexi y Measu es Conside a se o disc e e p obabili es p1, ..., pNwi h PN i=1 pi= 1. The R´enyi en opy o o de αhas he o m R(α)=1 1−αln Xpα i,0< α < ∞, α 6= 1.(1) The limi α→1 gi es he Shannon en opy: S=−Xpiln pi.(2) The LMC complexi y was de ined as he p oduc o wo impo an in o ma ion- heo e ical quan i ies: C=HQ, whe e H=eSis he Shannon en opy powe , while Q=e−D=e−R(2) is he loga i hm o he R´enyi en opy o o de 2. The disequilib ium Dquan i ies he de ia ion o he p obabili y dis ibu ion om uni o mi y. The Shannon en opy S, on he o he hand, is a measu e o unce ain y. A one-pa ame e ex ension o he gene alized s a is ical measu e o complexi y [2] is C(α)=eR(α)−R(β=2) . I α→1 we ob ain he LMC complexi y. x b Figu e 1. Bi u ca ion diag am, R´enyi en opy(α= 6) and gene alized complexi y(α= 3, β= 6) o he Du ing map. In he wo-pa ame e ex ension, on he o he hand, he gene alized s a is ical measu e o complexi y [2] has he o m ˜ C(α,β)=eR(α)−R(β),0< α, β < ∞.(3) Ce ainly, he special case α→1 and β= 2 gi es back he LMC complexi y. Impo an p ope ies o he gene alized complexi y a e de ailed in [2]. I has been shown ha he gene alized complexi y ex ends he complexi y measu e o any kind o well beha ed dis ibu ion. -0.94 -0.93 -0.92 -0.91 -0.9 -0.89 -0.88 -0.87 -0.86 -0.29806 -0.29804 -0.29802 -0.298 -0.29798 -0.29796 -0.29794 -0.29792 x b 0 0.5 1 1.5 2 2.5 0.3746 0.3748 0.375 0.3752 0.3754 0.3756 0.3758 0.376 R(α) b Figu e 2. Enla ged bi u ca ion diag am o he Du ing map o −0.2981 < b < −0.2979 and he R´enyi en opy in he icini y o a bi u ca ion poin . 3. Applica ion: Du ing and R¨ossle maps Now, we apply he gene alized complexi y measu e o cha ac e ize he Du ing and R¨ossle maps. The Du ing map has he o m: xn+1 =yn, yn+1 =−bxn+ayn−y3 n.(4) The pa ame e ais aken as a= 2.75 and he pa ame e bis selec ed as a con ol pa ame e . The ini ial coo dina es we e: x= 0.1 and y= 0.1. Fig.2 shows he xcoo dina e. (ybeha es simila ly.) The p obabili ies piwe e de e mined [4] by subdi iding he in e al [−2,2] in o 10000 equal bins. The numbe o i e a es alling wi hin a bin di ided by he o al numbe o i e a ions (104) gi es he p obabili y. Fo an n-pe iodic dynamics he e a e only np obabili ies ha a e no ze o. As hese p obabili ies a e all equal, he R´enyi en opy is ln n, independen om he pa ame e α, he e o e he complexi y is 1. F om he de ini ion (3) ollows ha ˜ C(α,β)≥1 i α < β and ˜ C(α,β)≤1 i α > β. As one expec s ha complexi y is la ge o a mo e complex beha iou , he case α < β is selec ed. The uppe panel o Fig. 1 p esen s he bi u ca ion diag am. (The alues o xa e plo ed agains he pa ame e b.) Fig. 1 also shows he R´enyi en opy o α= 6 (middle panel) and he Figu e 3. R¨ossle bi u ca ion diag am and gene alized complexi y(α= 3, β= 6). gene alized complexi y o α= 3 and β= 6 (lowe panel) o he in e al 0 < b < 1. Pe iodic and chao ic beha iou can be seen in he bi u ca ion diag am, and can also be de ec ed by he Figu e 4. Enla ged R¨ossle bi u ca ion diag am and R´enyi en opy. R´enyi en opy and he gene alized complexi y. In he bi u ca ion poin s bo h he R´enyi en opy and he gene alized complexi y ic eases ab up ly. Fig. 2b enla ges he R´enyi en opy in he icini y o a bi u ca ion poin . Fig. 2a shows an enla gemen o he bi u ca ion diag am: a e y in e es ing beha iou in he in e als −0.86 < x < −0.94 and −0.2981 < b < −0.2979. A b=−0.298075 he diag am is shi ed, a b=−0.29801 i goes back o he o iginal posi ion. The e is ano he shi in he in e al −0.29799 < b < −0.297985. A simila beha iou can be obse ed o o he alues o x. These shi s can no be de ec ed in he R´enyi en opy and he gene alized complexi y, because he alues o he p obabili ies do no change. The R¨ossle model is gi en by dx d =−y−z, dy d =x+ay, dz d =b+z(x−c).(5) The pa ame e s aand bwe e aken as a= 0.2, b= 0.2 and cis he con ol pa ame e . The ini ial coo dina es we e: x= 0, y=−5 and z= 0. The di e en ial equa ions we e sol ed nume ically by he Runge-Ku a (second o de ). Poinca ´e sec ions we e aken a x= 0 and he igu es show he coo dina e y. Fig. 3 p esen s he bi u ca ion diag am and he gene alized complexi y(α= 3, β= 6) o 1 < c < 15. Fig. 4 shows he enla ged bi u ca ion diag am and he R´enyi en opy o 6.75 < c < 7.1. I is a e y ich s uc u e, he bi u ca ion diag am and he R´enyi en opy e lec s di e en aspec s. The egula and chao ic pa s can be clea ly dis inguished. When pe iodic windows appea , he R´enyi en opy dec eases. Fu he enla gemen s (no p esen ed he e) would e eal addi ional ine de ails and he ac al cha ac e o he chao ic dynamics. In summa y, we used he R´enyi en opy and he gene alized complexi y measu es o desc ibe R¨ossle and he Du ing maps. These measu es nicely show he egula and he chao ic beha iou o dynamical sys ems. Pe iodic windows and he ac al cha ac e o he chao ic dynamics a e clea ly de ec ed. Acknowledgmen s The wo k is also suppo ed by he TAMOP 4.2.1/B-09/1/KONV-2010-0007 and he TAMOP 4.2.2/B-10/1-2010-0024 p ojec s. The p ojec is co- inanced by he Eu opean Union and he Eu opean Social Fund. G an OTKA No. K 100590 is also g a e ully acknowledged. Re e ences [1] R. Lopez-Ruiz, H. L. Mancini, and X. Calbe , Phys. Le . A 209, 321 (1995); R.G. Ca alan, J. Ga ay, and R. L´opez-Ruiz, Phys. Re . E 66, 011102 (2002). [2] E. Rome a, R. Lopez-Ruiz, J. Sanudo and ´ A. Nagy, In . Re . Phys. 3, 207 (2009); R. Lopez-Ruiz, ´ A. Nagy, E. Rome a and J. Sanudo, J. Ma h. Phys. 50, 123528 (2009). [3] B. God´o and ´ A. Nagy, Chaos 85, 023118 (2012). [4] G. L. Fe i, I. Pennini and A. Plas ino, Phys. Le . A 373, 2210 (2009).