Generalized Complexity Measures and Chaotic Maps B

Generalized complexity measures and chaotic maps B. Godó and Á. Nagy

Citation: Chaos 22, 023118 (2012); doi: 10.1063/1.4705088 View online: http://dx.doi.org/10.1063/1.4705088 View Table of Contents: http://chaos.aip.org/resource/1/CHAOEH/v22/i2 Published by the American Institute of Physics.

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Generalized complexity measures and chaotic maps B. Godo´ and A´ . Nagy Department of Theoretical Physics, University of Debrecen, H–4010 Debrecen, Hungary (Received 24 November 2011; accepted 5 April 2012; published online 24 April 2012) The logistic and Tinkerbell maps are studied with the recently introduced generalized complexity measure. The generalized complexity detects periodic windows. Moreover, it recognizes the intersection of periodic branches of the bifurcation diagram. It also reﬂects the fractal character of the chaotic dynamics. There are cases where the complexity plot shows changes that cannot be seen in the bifurcation diagram. VC 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4705088]

ð Complexity is a key concept in modern science. A lot of ðaÞ 1 a R ¼ ln ½ f ðrÞ dr for 0< a< 1; a 6¼ 1; papers addressed the problem of quantification of com- f 1 a plexity. The LMC (Lo´pez-Ruiz-Mancini-Calbet) statisti- (1) cal complexity has proved to be useful in studying several systems. Recently, this measure has been generalized where r stands for r1;Ð :::; rD. The limit a ! 1 gives the Shan- using Reńyi entropy. In this paper, it is presented that non entropy Sf ¼ f ðrÞln f ðrÞdr. The limit a !1,on this generalized complexity measure is suitable to ðaÞ the other hand Rf !lnjj f jj1, where jj f jj1 ¼ supr f ðrÞ describe chaotic behavior. Analysing the logistic and Tin- represents the maximum reached by f over its whole support. kerbell maps, it is shown that there are cases where this One of the factors in LMC-like statistical complexities measure is more sensitive to the fine details of the maps measures the broadening of the distribution. Originally it than the bifurcation diagram. was taken to be the Shannon entropy power H,

Sf Hf ¼ e : (2) I. INTRODUCTION The other factor quantifies the narrowness of the distribution There is a growing interest in studying complex behav- f: iour in several fields of science. Statistical measures have Rð2Þ proved to be very useful in describing properties of physical Qf ¼ e f : (3) systems. Information entropies and statistical complexities1 have been studied on different quantum and classical sys- The logarithm of this disequilibrium gives the second order tems (e.g., atomic properties in position and momentum Reńyi entropy. The LMC measure was defined as the product 2–4 5–7 6 spaces). Especially, the so called LMC complexity was of the Shannon entropy power Hf and the distribution Qf , used from H-atom8,9 to classical chaotic maps.10 The LMC statistical complexity is defined as a product of the power Cf ¼ Hf Qf : (4) Shannon entropy and the disequilibrium. (The logarithm of the LMC complexity coincides with the structural entropy When the Shannon entropy of the statistical complexity Cf is that characterizes the shape of a distribution).11,12 Recently, replaced with the Reńyi entropy of order a, we obtain a one- the Reńyi entropy13 has been a role of growing importance. parameter extension of the generalized statistical measure of 14 In two recent papers,14,15 the LMC statistical complexity complexity, has been generalized using Reńyi entropy. The one- and a ðaÞ ðb¼2Þ ð Þ Rf Rf two-parameter extensions have been applied on different Cf ¼ e : (5) quantum systems: H-atom, harmonic oscillator and square well. In this paper, usefulness of these measures in describ- In the limit a ! 1 it tends to Cf . ing chaotic behaviour is explored. In the two-parameter extension of the generalized statis- In Sec. II, the generalized LMC statistical complexity tical measure of complexity,15 the Reńyi entropy of order 2 measures are summarized. Section III presents the general- is replaced by the Reńyi entropy of order b, ized complexity measures for logistic and Tinkerbell maps.

ða;bÞ RðaÞRðbÞ C~ ¼ e f f ; 0 < a; b < 1: (6) II. GENERALIZED STATISTICAL COMPLEXITY f MEASURES ~ð1;2Þ ~ða;2Þ ðaÞ Certainly, we can recover Cf ¼ Cf and Cf ¼ Cf as Consider a D-dimensionalÐ density function f ðrÞ, (with special cases. The generalized complexity has several impor- 15 ~ða;bÞ f ðrÞ nonnegative and f ðrÞdr ¼ 1). The Re´nyi entropy of tant properties. We emphasize here only that Cf 1if ~ða;bÞ order a of the density function f is given by a < b and Cf 1ifa > b.

1054-1500/2012/22(2)/023118/7/$30.00 22, 023118-1 VC 2012 American Institute of Physics 023118-2 B. Godo´ and A´ . Nagy Chaos 22, 023118 (2012)

Note that we might find a value d 6¼ a such that jjf jjd Then we counted how many iterates fall within each bin. Di- ðd;bÞ 3 is convergent and C~ takes a finite value even if jjf jj is vision of this number by the total number of iterations (10 ) f a gives the probability. Calculations have been done for other divergent. The same can be said for the b parameter. In this values for the number of bins. We found (in agreement sense, the generalized complexity extends the complexity with10) that at least 1000 bins are needed for an adequate measure to any kind of well behaved distribution. description. (If the number of bins are larger than 1000, there In this paper we will apply the generalized statistical is almost no change in the results.) complexity measures for discrete maps, therefore these For an n-periodic dynamics, there are only n probabil- measures are also given for discrete distributions. Consider a P ities that are not zero. As these probabilities are all equal the set of discrete probabilities, p ; :::; p with N p ¼ 1. 1 N i¼1 i Reńyi entropy is ln n independent from the parameter a, Then the Reńyi entropy of order a has the form therefore the complexity is 1. X 1 a Fig. 1 presents the bifurcation diagram, the Reńyi en- RðaÞ ¼ ln ½p ; for 0< a< 1; a 6¼ 1; (7) 1 a i tropy for a ¼ 5 and the generalized complexity for the a ¼ 2 and b ¼ 5 for the interval 3 < al < 3:9. In accordance with If a ! 0 the Shannon entropy is recovered, ða;bÞ X the definition (6), we see that C~ 1ifa < b (lower S ¼ p ln p : (8) ða;bÞ i i panel of Fig. 1) and C~ 1ifa > b. We will present results only for the case a < b. It corresponds to the intuitive Using expressions (7) and (8) in Eqs. (5) and (6), the discrete expectation that complexity is larger for a more complex versions of the generalized statistical complexity measures behaviour. We can observe the period doubling as a step are obtained. structure in the Reńyi entropy. There is an increase in the III. APPLICATION TO THE LOGISTIC AND complexity at the bifurcation points. We see how the com- TINKERBELL MAPS plexity shows the chaotic nature of the map. The fractal character of the chaotic parts can also be clearly seen in the The generalized complexity measure is used now to figures. It also detects the occurrence of periodic windows. study chaotic maps. Consider first the logistic map, Moreover, we can observe periodic windows more clearly in

xnþ1 ¼ alxnð1 xnÞ; 0 < xn < 1 and 0 < al < 4 the complexity plot than in the bifurcation diagram. (9) As a second example consider the Tinkerbell map,

2 2 as an example of a one-dimensional complex system. We xnþ1 ¼ xn yn þ axn þ byn 1 can ﬁnd periodic behaviour for al < a ¼ 3:59699:::. For l ynþ1 ¼ 2xnyn þ cxn þ dy : (10) 1 n al > al , the dynamics is much more complicated. It is mainly chaotic but there are several periodic windows. We The parameters a, b,andd are generally taken as a ¼ 0.9, considered the solutions for the values 3 < al < 4. b ¼0.6013, and d ¼ 0.5 and c is selected as a control param- First, the probabilities pi were calculated following Ferri eter. Initial values are x0 ¼0:72 and y0 ¼0:64. The upper et al.10 The interval [0, 1] is subdivided into 1000 equal bins. panel of Fig. 2 presents the bifurcation diagram. (The values

FIG. 1. Bifurcation diagram (upper panel), Renyi entropy for a ¼ 5 (middle panel), and the generalized complexity for the a ¼ 2 and b ¼ 5 (lower panel) as a function of the control parameter al for the logistic map. 023118-3 B. Godo´ and A´ . Nagy Chaos 22, 023118 (2012)

FIG. 2. Bifurcation diagram (upper panel), Renyi entropy for a ¼ 5 (middle panel), and the generalized complexity for the a ¼ 2 and b ¼ 5 (lower panel) as a function of the control parameter c for the Tinkerbell map. of x are plotted against the parameter c.) Fig. 2 also shows the intersection means that the periodicity of the orbital changes, Reńyi entropy for a ¼ 5 (middle panel) and the generalized for example, the cycle period 8 around c ¼ 1.377 (Fig. 2). complexity for a ¼ 2andb ¼ 5 (lower panel) for the interval Before the intersection the probabilities are 1/8. At the inter- 0:9 < c < 2. As we see the Reńyi entropy itself reflects the section two lines merges and the probability increases to 1/4, complex structure of the map. The complex nature of the map while the other probabilities remain 1/8. The Reńyi entropies is even more apparent from the complexity plot. The com- are Rða¼2Þ ¼ 1:856 and Rðb¼5Þ ¼ 1:690. Therefore, the gener- ða¼2;b¼5Þ plexity measure detects several periodic windows for alized complexity takes the value C~ ¼ 1:181. 1 < c < 2. We can observe a very interesting behaviour of Fig. 3 enlarges the map for the interval 1:00 < c < 1:03, the periodic structure in the interval 1:35 < c < 1:55. There where a very special structure can be found. The Reńyi en- are intersections in the bifurcation diagram (upper panel of tropy and especially the generalized complexity reflect this Fig. 2). At the intersection point, there is a change in interesting behaviour. It has certain regularity, though it is the Reńyi entropy and in the generalized complexity. The not periodic. There is an abrupt decrease in the complexity at

FIG. 3. Bifurcation diagram (upper panel), Renyi entropy for a ¼ 5 (middle panel), and the generalized complexity for the a ¼ 2 and b ¼ 5 (lower panel) as a function of the control parameter c for the Tinkerbell map. Enlargement of Fig. 2 for the interval 1 < c < 1:03. 023118-4 B. Godo´ and A´ . Nagy Chaos 22, 023118 (2012)

FIG. 4. x-y diagram of the Tinkerbell map for 6 different values of c: regular behaviour (c ¼ 0.847 and c ¼ 0.918) almost regular behaviour (c ¼ 1.017 and c ¼ 1.029) and chaotic behaviour (c ¼ 1.676 and 1.947). c ¼ 1.017. We can observe it in the Re´nyi entropy also, but it for c ¼ 1.029. Chaotic behaviour can be found, e.g., for cannot be seen in the bifurcation diagram. To gain a more c ¼ 1.676 and 1.947. clear insight into this interesting behaviour, both x and y are Fig. 5 presents another interesting behaviour at c ¼ 1.526. shown in Fig. 4 for that particular value of c (c ¼ 1.017). To There is an abrupt change in the bifurcation diagram. The compare this behaviour with regular one, plots are also branches suddenly end and unexpected new branches appear. shown for values c ¼ 0.847 and c ¼ 0.918 on Fig. 4. A simi- This sudden change causes increasing complexity. A similar lar, somewhat more irregular picture has been obtained behaviour can be found around c ¼ 1.62 in Fig. 6.Foralittle

FIG. 5. Bifurcation diagram (upper panel), Renyi entropy for a ¼ 5 (middle panel), and the generalized complexity for the a ¼ 2 and b ¼ 5 (lower panel) as a function of the control parameter c for the Tinkerbell map. Enlargement of Fig. 2 for the interval 1:34 < c < 1:58. 023118-5 B. Godo´ and A´ . Nagy Chaos 22, 023118 (2012)

FIG. 6. Bifurcation diagram (upper panel), Renyi entropy for a ¼ 5 (middle panel), and the generalized complexity for the a ¼ 2 and b ¼ 5 (lower panel) as a function of the control parameter c for the Tinkerbell map. Enlargement of Fig. 2 for the interval 1:5 < c < 1:7. larger c the branches disappeared around c ¼ 1.62, come into parts have a very similar form to the original one. It reflects sign again (just as if they had not disappeared). This structure a self-similarity. A similar plot can be obtained for the can also be observed in the Reńyi entropy and especially in complexity. the generalized complexity. Fig. 7 presents further interesting All presented examples show that both the Reńyi en- behaviour in the periodic window 1:72 < c < 1:735. We can tropy and the complexity measure are good descriptors of observe intersection of branches of the bifurcation diagram. chaotic behaviour. The advantage of generalized complexity The Reńyi entropy shows a step structure, while we can see measure in comparison to the Reńyi entropy is that the com- an increased complexity in Fig. 7. plex nature is even more apparent from the complexity plot, Fig. 8 presents the fractal nature of the Reńyi entropy the complexity measure is more sensitive to the fine details plot for the interval 1:48 < c < 1:59. A rectangle around of the map (see especially Figs. 3–5). c ¼ 1.533 in enlarged. Then another rectangle is selected and In summary, we studied chaotic maps with the general- enlarged. The process is repeated three times. The enlarged ized complexity measure introduced recently. The new

FIG. 7. Bifurcation diagram (upper panel), Renyi entropy for a ¼ 5 (middle panel), and the generalized complexity for the a ¼ 2 and b ¼ 5 (lower panel) as a function of the control parameter c for the Tinkerbell map. Enlargement of Fig. 2 for the interval 1:7 < c < 1:8. 023118-6 B. Godo´ and A´ . Nagy Chaos 22, 023118 (2012)

FIG. 8. Re´nyi entropy for the interval 1:48 < c < 1:59. A rectangle around c ¼ 1:533 in enlarged. Then another rectangle is selected and enlarged. The process is repeated three times.

measure detects periodic windows. Moreover, it recognizes Union and the European Social Fund. Grant OTKA No. K the intersection of periodic branches of the bifurcation dia- 100590 is also gratefully acknowledged. gram. It also reﬂects the fractal character of the chaotic dynamics. 1Statistical Complexity, edited by K. D. Sen (Springer, Berlin, 2011). ACKNOWLEDGMENTS 2S. R. Gadre, S. B. Sears, S. J. Chakravorty, and R. D. Bendale, Phys. Rev. A 32, 2602 (1985). The work is supported by the TAMOP-4.2.2/B-10/1- 3K. Ch. Chatzisavvas, Ch. C. Moustakidis, and C. P. Panos, J. Chem. Phys. 2010-0024 project. The project is co-ﬁnanced by the European 123, 174111 (2005). 023118-7 B. Godo´ and A´ . Nagy Chaos 22, 023118 (2012)

4A´ . Nagy and S. B. Liu, Phys. Lett. A 372, 1654 (2008). 11J. Pipek and I. Varga, Phys. Rev. A 46, 3148 (1992). 5R. Lo´pez-Ruiz, H. L. Mancini, and X. Calbet, Phys. Lett. A 209, 321 12I. Varga and J. Pipek, Phys. Rev. E 68, 26202 (2003). (1995). 13A. Re´nyi, Proceedings of the 4th Berkeley Symposium on Mathematical 6R. G. Catalan, J. Garay, and R. Lo´pez-Ruiz, Phys. Rev. E 66,011102 Statistics and Probability, Volume 1: Contributions to the Theory of Statis- (2002). tics (University of California Press, Berkeley, 1961), p. 547. 7X. Calbet and R. Lo´pez-Ruiz, Phys. Rev. E 63, 066116 (2001). 14E. Romera, R. Lo´pez-Ruiz, J. Sanudo, and A´ . Nagy, Int. Rev. Phys. 3, 207 8J. San˜udo and R. Lo´pez-Ruiz, Phys. Lett. A 372, 5283 (2008). (2009). 9J. San˜udo and R. Lo´pez-Ruiz, J. Phys. A 41, 265303 (2008). 15R. Lo´pez-Ruiz, A´ . Nagy, E. Romera, and J. Sanudo, J. Math. Phys. 50, 10G. L. Ferri, I. Pennini, and A. Plastino, Phys. Lett. A 373, 2210 (2009). 123528 (2009).