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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. Related Articles On finite-size Lyapunov exponents in multiscale systems Chaos 22, 023115 (2012) Exact folded-band chaotic oscillator Chaos 22, 023113 (2012) Components in time-varying graphs Chaos 22, 023101 (2012) Impulsive synchronization of coupled dynamical networks with nonidentical Duffing oscillators and coupling delays Chaos 22, 013140 (2012) Dynamics and transport in mean-field coupled, many degrees-of-freedom, area-preserving nontwist maps Chaos 22, 013137 (2012) Additional information on Chaos Journal Homepage: http://chaos.aip.org/ Journal Information: http://chaos.aip.org/about/about_the_journal Top downloads: http://chaos.aip.org/features/most_downloaded Information for Authors: http://chaos.aip.org/authors CHAOS 22, 023118 (2012) 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 À 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 Re´nyi 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 ÀRð2Þ 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 Re´nyi 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 Re´nyi entropy of order a, we obtain a one- the Re´nyi 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 Re´nyi entropy. The one- and a ðaÞ ðb¼2Þ ð Þ Rf ÀRf 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 Re´nyi entropy of order 2 measures are summarized. Section III presents the general- is replaced by the Re´nyi entropy of order b, ized complexity measures for logistic and Tinkerbell maps. ða;bÞ RðaÞÀRðbÞ 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 Re´nyi entropy is ln n independent from the parameter a, Then the Re´nyi entropy of order a has the form therefore the complexity is 1. X 1 a Fig. 1 presents the bifurcation diagram, the Re´nyi en- RðaÞ ¼ ln ½p ; for 0< a< 1; a 6¼ 1; (7) 1 À 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 Re´nyi 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 char- acter 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 À 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 À 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.
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