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
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[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).