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

Characterization of R¨ossler and Duffing maps with R´enyi entropy and generalized complexity measures B. God´o and ´ A. Nagy Department of Theoretical Physics, University of Debrecen, Debrecen, Hungary E-mail: [email protected] Abstract. R´enyi entropy and generalized complexity measures are used to describe the chaotic behaviour of dynamical systems. These measures are found to be sensitive to the fine details of the R¨ossler and the Duffing maps. They are good descriptors of chaotic behaviour. Periodic windows and the fractal character of the chaotic dynamics are nicely detected. 1. Introduction There exist several quantities to study the chaotic behaviour of dynamical systems. Complexity measures have proved to be especially efficient in this respect. One of these measures is the LMC (Lopez-Ruiz - Mancini - Calbet) statistical complexity [1]. A couple of years ago, a oneand a two-parameter extension [2] of this measure were put forward. These generalizations are based on the R´enyi entropy. First, some simple quantum systems (H-atom, harmonic oscillator and square well) were studied with these measures. Recently, it has been demonstrated [3] that these generalized complexity measures are suitable to describe chaotic behavior. The logistic and Tinkerbell maps were analyzed. In this work the R¨ossler and the Duffing maps are studied with the R´enyi entropy and the generalized complexity measures. 2. R´enyi Entropy and Generalized Statistical Complexity Measures Consider a set of discrete probabilites p1, ..., pNwith PN i=1 pi= 1. The R´enyi entropy of order αhas the form R(α)=1 1−αln Xpα i,0< α < ∞, α 6= 1.(1) The limit α→1 gives the Shannon entropy: S=−Xpiln pi.(2) The LMC complexity was defined as the product of two important information-theoretical quantities: C=HQ, where H=eSis the Shannon entropy power, while Q=e−D=e−R(2) is the logarithm of the R´enyi entropy or order 2. The disequilibrium Dquantifies the deviation of the probability distribution from uniformity. The Shannon entropy S, on the other hand, is a measure of uncertainty. A one-parameter extension of the generalized statistical measure of complexity [2] is C(α)=eR(α)−R(β=2) . If α→1 we obtain the LMC complexity. x b Figure 1. Bifurcation diagram, R´enyi entropy(α= 6) and generalized complexity(α= 3, β= 6) for the Duffing map. In the two-parameter extension, on the other hand, the generalized statistical measure of complexity [2] has the form ˜ C(α,β)=eR(α)−R(β),0< α, β < ∞.(3) Certainly, the special case α→1 and β= 2 gives back the LMC complexity. Important properties of the generalized complexity are detailed in [2]. It has been shown that the generalized complexity extends the complexity measure to any kind of well behaved distribution. -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 Figure 2. Enlarged bifurcation diagram of the Duffing map for −0.2981 < b < −0.2979 and the R´enyi entropy in the vicinity of a bifurcation point. 3. Application: Duffing and R¨ossler maps Now, we apply the generalized complexity measure to characterize the Duffing and R¨ossler maps. The Duffing map has the form: xn+1 =yn, yn+1 =−bxn+ayn−y3 n.(4) The parameter ais taken as a= 2.75 and the parameter bis selected as a control parameter. The initial coordinates were: x= 0.1 and y= 0.1. Fig.2 shows the xcoordinate. (ybehaves similarly.) The probabilities piwere determined [4] by subdividing the interval [−2,2] into 10000 equal bins. The number of iterates falling within a bin divided by the total number of iterations (104) gives the probability. For an n-periodic dynamics there are only nprobabilities that are not zero. As these probabilities are all equal, the R´enyi entropy is ln n, independent from the parameter α, therefore the complexity is 1. From the definition (3) follows that ˜ C(α,β)≥1 if α < β and ˜ C(α,β)≤1 if α > β. As one expects that complexity is larger for a more complex behaviour, the case α < β is selected. The upper panel of Fig. 1 presents the bifurcation diagram. (The values of xare plotted against the parameter b.) Fig. 1 also shows the R´enyi entropy for α= 6 (middle panel) and the Figure 3. R¨ossler bifurcation diagram and generalized complexity(α= 3, β= 6). generalized complexity for α= 3 and β= 6 (lower panel) for the interval 0 < b < 1. Periodic and chaotic behaviour can be seen in the bifurcation diagram, and can also be detected by the Figure 4. Enlarged R¨ossler bifurcation diagram and R´enyi entropy. R´enyi entropy and the generalized complexity. In the bifurcation points both the R´enyi entropy and the generalized complexity icreases abruptly. Fig. 2b enlarges the R´enyi entropy in the vicinity of a bifurcation point. Fig. 2a shows an enlargement of the bifurcation diagram: a very interesting behaviour in the intervals −0.86 < x < −0.94 and −0.2981 < b < −0.2979. At b=−0.298075 the diagram is shifted, at b=−0.29801 it goes back to the original position. There is another shift in the interval −0.29799 < b < −0.297985. A similar behaviour can be observed for other values of x. These shifts can not be detected in the R´enyi entropy and the generalized complexity, because the values of the probabilities do not change. The R¨ossler model is given by dx dt =−y−z, dy dt =x+ay, dz dt =b+z(x−c).(5) The parameters aand bwere taken as a= 0.2, b= 0.2 and cis the control parameter. The initial coordinates were: x= 0, y=−5 and z= 0. The differential equations were solved numerically by the Runge-Kutta (second order). Poincar´e sections were taken at x= 0 and the figures show the coordinate y. Fig. 3 presents the bifurcation diagram and the generalized complexity(α= 3, β= 6) for 1 < c < 15. Fig. 4 shows the enlarged bifurcation diagram and the R´enyi entropy for 6.75 < c < 7.1. It is a very rich structure, the bifurcation diagram and the R´enyi entropy reflects different aspects. The regular and chaotic parts can be clearly distinguished. When periodic windows appear, the R´enyi entropy decreases. Further enlargements (not presented here) would reveal additional fine details and the fractal character of the chaotic dynamics. In summary, we used the R´enyi entropy and the generalized complexity measures to describe R¨ossler and the Duffing maps. These measures nicely show the regular and the chaotic behaviour of dynamical systems. Periodic windows and the fractal character of the chaotic dynamics are clearly detected. Acknowledgments The work is also supported by the TAMOP 4.2.1/B-09/1/KONV-2010-0007 and the TAMOP 4.2.2/B-10/1-2010-0024 projects. The project is co-financed by the European Union and the European Social Fund. Grant OTKA No. K 100590 is also gratefully acknowledged. References [1] R. Lopez-Ruiz, H. L. Mancini, and X. Calbet, Phys. Lett. A 209, 321 (1995); R.G. Catalan, J. Garay, and R. L´opez-Ruiz, Phys. Rev. E 66, 011102 (2002). [2] E. Romera, R. Lopez-Ruiz, J. Sanudo and ´ A. Nagy, Int. Rev. Phys. 3, 207 (2009); R. Lopez-Ruiz, ´ A. Nagy, E. Romera and J. Sanudo, J. Math. Phys. 50, 123528 (2009). [3] B. God´o and ´ A. Nagy, Chaos 85, 023118 (2012). [4] G. L. Ferri, I. Pennini and A. Plastino, Phys. Lett. A 373, 2210 (2009).