QUANTIFYING THE FOLDING MECHANISM IN CHAOTIC DYNAMICS

Transcription

1 QUANTIFYING THE FOLDING MECHANISM IN CHAOTIC DYNAMICS V. BARAN, M. ZUS,2, A. BONASERA 3, A. PATURCA Faculty of Physics, University of Bucharest, 405 Atomiştilor, POB MG-, RO-07725, Bucharest-Măgurele, România 2 Maritime University of Constanţa, RO , Constanţa, România 3 Laboratori Nazionali del Sud, INFN, 9523 Catania, Italy baran@nipne.ro Received September 7, 205 In this work we discuss different measures aimed to characterize the folding mechanism which together with the stretching process determine the chaotic dynamics. We show that from a study of the evolution of the distance between two trajectories beyond the exponential stage until the asymptotic regime is possible to obtain a quantity which provide an insight about this mechanism and its dependence on the control parameter. The asymptotic mean distance d manifests a specific power law dependence at the transition to chaos and is quite complementary to Lyapunov exponent in characterizing the chaotic motion. Then based on the methods of inverse statistics applied to one-dimensional maps we advance an alternative measure able to reflect the folding mechanism on the strange attractors. In the final part we argue briefly that the inverse statistics can be a relevant tool to the study of earthquakes produced in the Vrancea region. Key words: Chaotic dynamics, strange attractors, inverse statistics, earthquakes. PACS: Ac, Pq, Tp.. INTRODUCTION One of the interesting features which emerge when dealing with nonlinear dynamics is the chaotic motion, i.e. an evolution which manifest a sensitive dependence on the initial conditions []. There has been a considerable effort to establish what are the conditions for a system (dissipative or conservative) to display chaos and what are the suitable quantities to describe this regime [2]. The Lyapunov exponent (LE) characterize the mean rate of separation between two adjacent trajectories in phase space. For a chaotic evolution these will diverge exponentially and the LE is greater than zero. The correlation function is related to the memory along one trajectory and decays quickly to zero in this regime. Another interesting feature associated to the dynamics, the power spectrum, changes from a discrete lines distribution specific to a quasi-periodic motion to a broad-band noise-like when chaos sets in. Is worth to mention that at the transition to chaos some of these quantities show an interesting power-law behavior as a function of the control parameter which resemble the behavior observed for various physical quantities (magnetization, densities, susceptibility) as a function of the temperature at the second order phase transition. RJP Rom. 60(Nos. Journ. Phys., 9-0), Vol , Nos. 9-0, (205) P , (c) 205 Bucharest, - v..3a*

2 264 V. Baran et al. 2 The chaotic dynamics is the result of the combination of two mechanisms: the stretching, the best characterized by the Lyapunov exponent and the folding, which keeps the trajectories inside the finite volume of phase-space. A specific quantity for the latter is not yet available however. The aim of this paper is to sought for the possible measures of this mechanism. We shall focus on the one-dimensional maps which in spite of their apparent simplicity are displaying most of the features observed in more complex systems. From the previous discussion is expected that a first insight can be provided by the separation distance between the two trajectories when this is considered beyond the exponential stage. In the asymptotic regime both trajectories are exploring the full available phase-space under the effects of back-folding process. In connection to this idea let us mention that the separation of two nearby fluid elements, or pair dispersion, was introduced by Richardson [4] when investigated the anomalous rise of turbulent diffusivity in the atmosphere dynamics. It proved to be very useful in characterizing the turbulent mixing and transport in the turbulent flows [5]. It is also of practical relevance when is surveyed the volcanic ashes spreading, air pollution, combustion or is investigated the clouds formation [6]. The pair dispersion manifests some peculiarities which are related to the dynamical sub-regime: an exponential time dependence in the dissipation sub-range, for separation distances much less than Kolmogorov length scale is followed by a slower, power-law time dependence over the inertial and diffusive scales [7]. In the second part of this work we employ the inverse statistics approach, previously considered to evidence new features of the turbulence by proposing the distance structure functions defined for a velocity field [8], to the study of one-dimensional maps. Inspired by the new results unveiled by the inverse statistics for the stock markets dynamics [9] we relate the observed asymmetry between the waiting iteration distributions corresponding to positive and negative values of the threshold parameter to another measure for the back-folding property. 2. CONNECTING THE FOLDING MECHANISM TO THE ASYMPTOTIC DISTANCE In the following we investigate the evolution determined by the one-dimensional maps: x n+ = f(x n,r) (2.) which exhibit a chaotic dynamics for certain values of the control parameter r. For an ensemble of N pairs of trajectories, separated initially by a very small distance d 0 we introduce the mean distance between them at the iteration n as: d n = N N i= x n (i) y n (i) (2.2) RJP 60(Nos. 9-0), (205) (c) v..3a*

3 3 Quantifying the folding mechanism in chaotic dynamics 265 where: x (i) n = f n (x (i) 0 ) ; y(i) n = f n (x (i) 0 + d 0) (2.3) The initial starting points x (i) 0 are chosen from an uniform distribution spanning the defining interval of the map. For a small enough initial separation the approximate dependence of the distance d n after n iterations is: d n = e nλ d 0 = e λ d n Λd n (2.4) Here Λ = e λ and λ is the Lyapunov exponent of the map: λ = lim n n ln f (x j ) (2.5) n j=0 The prime indicates the derivative of the function f(x i ) in respect to x i [2]. When n is sufficiently large the equation (2.4) is no longer valid. Actually the action of the map contains two components: the stretching, which leads to the exponential departure of neighboring trajectories, being characterized quantitatively by the LE and the folding process which keeps the trajectories bound. The relation (2.4) can be considered as a first order approximation in d n. Formally, starting from the eq. (2.) the following Taylor expansion is valid in terms of the difference x n : x n+ = k= k! f (k) (x n ) k x n = f (x n )x n + g(x n,x n ) 2 x n (2.6) where the function g(x n,x n ) includes the sum o higher order derivatives of f: g(x n,x n ) = 2! f (2) (x n ) + 3! f (3) (x n )x n +... (2.7) This is a bound function since both x n+ and x n are less than one. Now it is clear that when x n reaches values of the order of unit, the influence of the higher order terms has to be taken into account. For the one-dimensional maps we study in this work, which verify the relation d n < for any n, we consider a second order term correction to the relation (2.4): d n+ = Λd n Γd 2 n F (d n ) (2.8) The asymptotic value of the mean distance between two trajectories is defined as: d = lim The fixed points of (2.8) are d = 0 and: n n d i (2.9) n d 2 = d = Λ Γ RJP 60(Nos. 9-0), (205) (c) v..3a* i=0 (2.0)

4 266 V. Baran et al. 4 Thus the eq. (2.8) describes the irreversible approach of the system towards the equilibrium which corresponds to the fixed points solutions d or d 2. From the stability condition F (d i ) < results that d is a stable point for λ < 0, while d 2 = d is a stable point for 0 < λ < ln(3). An interesting case is associated Fig. (color online) The mean distance between trajectories vs. iteration for different maps. (a) Logistic map with the control parameter r = 4; (b) the same as (a) but for r = 3.77; (c) the triangular map at b = ; (d) the sine map at a = The various curves correspond to different starting values of d 0. The dotted lines are the predictions of the eq. (2.8). to the condition F (d ) = 0 which gives λ = ln(2) as being related to the superstable point of the map F (d n ). This value of the LE is obtained for the tent map f(x n ) = a( 2 2 x n ), 0 a and the logistic map f(x n ) = rx n ( x n ), 0 r 4 at a value of the control parameters corresponding to the fully developed chaos, i.e. b =.0 and r = 4.0 respectively. Therefore the ergodicity of the maps is equivalent to a superstable point of the generalized application (2.8) which describe the evolution of the mean distance between trajectories beyond the exponential separation. The value of Γ can be expressed in terms of λ and d : Γ = eλ d (2.) In Figure we plot d n versus n as obtained numerically for the logistic map, the triangular map and the sine map f(x n ) = x n + bsin(2πx n ), 0 a , for RJP 60(Nos. 9-0), (205) (c) v..3a*

5 5 Quantifying the folding mechanism in chaotic dynamics 267 three different initial values of d 0 and compare with the the predictions of the eq. (2.8) when Γ is given by (2.). The entire evolution of the distance between two trajectories is better approximated by the equation (2.8) which contain an additional parameter Γ. After a fast increase the distance between trajectories saturates at the value d as defined by (2.9), independent on the initial relative distance d 0. The agreement is quite good in all cases, supporting our hypothesis λ r d r Fig. 2 (color online) The Lyapunov exponent as a function of the control parameter r for the logistic map. Fig. 3 (color online) The asymptotic distances between trajectories vs. control parameter r for the logistic map. At the ergodic point, corresponding to fully developed chaos, we can calculate analytically λ and d as an average over the phase space by using the invariant distribution function ρ(x) [2]. For the logistic map ρ(x) = we obtain: π x( x) λ 0.4 d r r Fig. 4 (color online) The Lyapunov exponent as a function of the control parameter r for the tent map. Fig. 5 (color online) The asymptotic distances between trajectories vs. control parameter r for the tent map. RJP 60(Nos. 9-0), (205) (c) v..3a*

6 268 V. Baran et al. 6 λ e = d e = 0 0 ρ(x)ln 4 8x = ln2 (2.2) ρ(x)ρ(y) x y = 4 π 2 (2.3) For the triangular map with ρ(x) = the results are λ e = ln2 and d e = /3, the subscript e stands for ergodic. At other values of the control parameter it is not possible to obtain analytically λ and d. The numerical calculations of these quantities are reported in Figures 2 and 3 for the logistic map and in the Figures 4 and 5 for the tent map. Let us note that at some values of the control parameter d manifests discontinuities which indicate a change in the dynamics. Such jumps, as seen near a = 0.7 for the triangular map, are not present in the behavior of the LE. The sudden increase of d can be associated to the band splitting bifurcation and is evidenced also in the case of logistic map, see Fig. 3. Moreover d has a specific power-law behavior at the transition to chaos. It is well known that for the logistic map at this transition the LE has a power-law dependence on the control parameter [0 2], with the critical exponent β a function only on Feigenbaum universal constant = : λ (r r ) β ; β = ln2 (2.4) ln Here r is the accumulation value for the double period bifurcation cascade corresponding to the Feigenbaum attractor. In Fig. 2 the solid green line is derived from (2.4) and agree very well with the exact value of LE. The analogy with the usual second order phase transitions is more evident by recalling that at the critical point r the correlation function [2]: n C(m,f) = lim y i y i+m (2.5) n n where y i = f i (x 0 ) x av and x av = lim n n i= n x i, decays as a power law in m: C(m,f rc ) m η ; η = 2lnα (2.6) ln2 with a critical exponent η this time related only to the constant α = By employing the Renormalization Group (RG) approach and accounting for the properties of period-doubling operator T [3] is deduced that d (T f) = αd (f) which can be iterated to: d (f) = α n d (T n f). Following the same steps as for the LE is obtained that d has a power-law evolution too at the transition to chaos with a RJP 60(Nos. 9-0), (205) (c) v..3a* i=

7 7 Quantifying the folding mechanism in chaotic dynamics 269 critical exponent ν/2 which depends on both Feigenbaum constants: d (r r ) ν/2 ; ν/2 = lnα (2.7) ln In the Fig. 3 the green line obtained from (2.7) follows closely the behavior of d defining an envelop-like function as in the case of Lyapunov exponent. The three critical exponents satisfies the relation: ν = βη (2.8) which is expected to be valid for the transition to chaos through the double period bifurcation. To better grasp the meaning of the physical quantities discussed above let e+06 e Γ 000 Γ b Fig. 6 (color online) The folding parameter Γ as a function of the control parameter b for the triangular map r Fig. 7 (color online) The folding parameter Γ as a function of the control parameter r for the logistic map. us consider the unbound map: x n+ = 2x n. For this map λ = ln2, and Γ = 0. If instead is considered the bound application i.e. the Bernoulli shift, x n+ = 2x n mod, the LE remains the same while Γ = /d. Thus the LE is only sensitive to the stretching while Γ is sensitive to the folding and stretching mechanisms, eq.(2.). For a positive Lyapunov exponent Γ is greater than zero and consequently the corresponding term in the eq. (2.8) is limiting the exponential growth generated by the first term. Moreover, for a fixed λ, a larger d which indicates not only an enlarged phase space but also a less effective folding, determine a smaller value of Γ. For the tent map Γ exhibit a decreasing trend with the rise of the control parameter, see Fig. 6. It has discontinuities at the chaotic bands merging a feature observed also for the logistic map, Fig INVERSE STATISTICS FOR THE DETERMINISTIC CHAOS The inverse statistics was introduced firstly in studies of the turbulent regime by considering the averaged moments of the distance as a function of velocity difference RJP 60(Nos. 9-0), (205) (c) v..3a*

8 270 V. Baran et al. 8 rather than the moments of velocity differences in terms of the distance. In this way Jensen [8] constructed the distance structure functions which manifest a multiscaling spectrum quite different from that associated to the velocity structure function and which can be employed for the analysis of velocity measurements. This method was latter applied to characterize the stock markets dynamics. Instead of looking for the profit generated by an asset after a fixed time, one is asking about the time needed to appear a variation in price of a fixed value. The obtained waiting times define the investment horizon distribution whose maximum indicates the optimal investment horizon [9]. The comparison between equal positive and negative levels of return e+07 (a) e+07 (b) 6e+06 N 6e e+07 (c) N 6e e+07 (d) 6e Fig. 8 (color online) The dependence of the N on the absolute value of : blue lines refers to positive values while the red lines refers to negative values of. (a) r = 3.60, (b) r = 3.62, (c) r = 3.67, (d) r = prompted an asymmetry reflecting quantitatively the popular observation that the markets reacts faster to negative information and that it takes same time to drive up the prices [3]. Detailed studies within such approaches evidenced a kind of synchronization of the individual markets [4] as well as stronger correlations when the market is falling than in the case of rising markets [5]. In what follows our task is to extend the investigations based on the inverse statistics to the deterministic chaotic dynamics. For the one-dimensional maps this allows us to define another quantity for characterization of the folding mechanism. The waiting iteration (or the first passage iteration) n w () associated to the threshold value can be defined by analogy with the time series case [6]: n w () = inf{k > 0 ; x n+k x n } if > 0 (3.) n w () = inf{k > 0 ; x n+k x n } if < 0 (3.2) RJP 60(Nos. 9-0), (205) (c) v..3a*

9 9 Quantifying the folding mechanism in chaotic dynamics 27 e+07 (a) e+07 (b) 6e+06 N 6e e+07 (c) N 6e e+07 (d) 6e Fig. 9 (color online) The dependence of the N on the absolute value of. The blue lines refers to positive values while the red lines refers to negative values of. (a) r = 3.72; (b) r = 3.77; (c) r = 3.80, for r = 3.82 the black line corresponds to negative while the magenta line to positive ; (d) r = 3.90, the orange line is associated to positive and negative values of for r = where n is spanning the entire set of the N I generated iterations. Numerically, for a fixed value of the control parameter r, a sequence of N I = 0 7 iterations of the logistic map was obtained. The analysis was performed for several values of the threshold parameter = ±0.05,±0.,±0.2,±0.25,±0.3,±0.4,±0.5 which covers from a tenth up to half the size of the largest attractor. The total number of events N when a value n w () was found as one is spanning all the iterations is plotted as a function of abs() in Figures 8 and 9. At a fixed an asymmetry between the the number of returns corresponding to positive and negative respectively, is observed. The inequality N + < N takes place for all values of control parameter except r = 4.00 when they are very similar, see Fig. 9(d). With other words within a sequence of iterations, for a fixed absolute value of the threshold, is more likely to find negative returns rather than positive ones. Therefore we consider this ratio as appropriate to characterize the folding. In Fig. 0 the evolution with of the ratio N /N + is shown at different values of the control parameter r. We notice a rising trend towards larger values of but also a specific dependence on the control parameter r as can be seen in Fig.. Certainly a more detailed analysis on r dependence is required but we already remark a peculiar behavior with some local maxima for some values of which we interpret as a signature of an enhanced folding mechanism. Let us look at the waiting iterations distribution for a fixed threshold value. This is the analogous of the inverse statistics derived in the case of stock markets where now instead of prices the values assumed by the one-dimensional map for a RJP 60(Nos. 9-0), (205) (c) v..3a*

10 272 V. Baran et al N / Ν N / Ν r Fig. 0 (color online) The ratio N /N + as a function of return parameter for r = 3.60, 3.62, 3.67, 3.70, 3.72, 3.77, 3.80, 3.82, 3.90, Fig. (color online) The ratio N /N + as a function of the control parameter r for = 0.05,0.,0.2,0.25,0.3,0.4,0.5. given r are considered. For each ensemble specified by the pairs (r,) we define the P e-05 e P e-05 e n w (a) (b) Fig. 2 (color online) The probability distribution of waiting iterations for r = (a) = 0.05 (blue squares), = 0. (red squares), = 0.2 (maroon squares), = 0.25 (green squares); (b) = 0.3 (blue circles), = 0.4 (red circles) = 0.45 (maroon circles), = 0.5 (green circles), the black solid line corresponds to = distribution of probability for the waiting iterations as: P = n w() N (3.3) With this definition a comparison between different ensembles become possible. Figure 2 shows the distributions obtained in the case of fully developed chaos regime i.e. for r = 4.0. A clear power-law behavior is evidenced for between 0.05 and As is seen in Fig. 2(a) all plots are practically collapsing on the same straight RJP 60(Nos. 9-0), (205) (c) v..3a*

11 Quantifying the folding mechanism in chaotic dynamics 273 line in a log-log representation. The best fit with a parametrization P = An ɛ w () (3.4) is obtained for an exponent ɛ = 2.5. This value is quite close to that which was derived for the probability distribution of the normalized waiting time needed to reach a return level of for the DM against US Dollar in 998 exchange rate data [7]. We also note that a power-law distribution P (τ L ) = A L τ α L with α = 2.4 ± 0. of the waiting time between two successive maxima of the solar flares intensity was also firmly established [8] and interpreted as an analogous of Omori s law which states the time dependence of the frequency of the after-shocks following a major earthquake [9 2]. This behavior suggested the existence in the solar bursts dynamics of nontrivial correlations between successive events whose origin is related to the basic equations describing these processes. From these considerations we consider that an investigation based on inverse statistics can prove quite interesting also in the case of earthquakes produced in a given region. Therefore we shall present some preliminary results of such investigations for the seismic region Vrancea at the end of this section. The Figure 2(b) shows that for larger values of the return parameter deviations from the universal behavior manifests. These can be related to the finite size effects on the ergodic attractor. We also note that the waiting iterations distributions decreases monotonously with n w (), for all values of, at variance with the features observed in the case of financial markets where a well defined maximum was identified as an optimal investment time. The same type of analysis for r = 3.70 is presented in Fig. 3. Unlike the P e-05 e P e-05 e n w (a) (b) Fig. 3 (color online) The same as in Fig. 2 but for r = RJP 60(Nos. 9-0), (205) (c) v..3a*

12 274 V. Baran et al. 2 previous situation important changes for all values of threshold parameter are evidenced. The actual trends is likely to be related to the multi-spikes structure of the strange attractors of the system at values of control parameter below r = 4.00, see Fig. 4 where are represented the invariant distributions of probability for three different values of control parameter: r = 3.70, r = 3.90 and r = Some distributions present splittings for various s which points towards a coexistence of different dynamical regimes. Nevertheless even in these cases a memory of the universal behavior observed for r = 4.0 can be distinguished, at least in some limited domains of n w () values. We also performed a comparison between the waiting iterations 0.0 r=3.70 P r=3.90 P r=4.00 P x Fig. 4 (color online) The invariant density (the probability distribution) for the logistic map at the values of control parameter r = 3.70, 3.90 and distribution for positive and negative returns respectively. Figure 5 shows that the asymmetry is manifested even for r = 4.0 and small values of threshold parameter. In all presented cases a negative return is more likely than a positive for waiting iterations larger that around n w () = 0. Alternations are possible however at small n w () for some values of. Let us mention that accounting for the analogies (and differences) which may exists between qualitatively different physical phenomena such as the deterministic ones, which includes chaos, turbulence and solar flares and the stochastic ones, as stock markets and earthquakes, we were interested in the features of the inverse statistics related to the seismic events in the Vrancea region, Romania. Therefore here we present some preliminary results of these investigation, a more detailed discussion being the argument of a future publication. We selected for the analysis only the earthquakes with a magnitude larger that two, which were register in the period January July 205 at depths below 60 km. Their magnitude distribution is RJP 60(Nos. 9-0), (205) (c) v..3a*

13 3 Quantifying the folding mechanism in chaotic dynamics P e-05 e P e-05 e-06 (a) 0 00 (c) 0 00 n w e-05 e e-05 e-06 (b) 0 00 (d) 0 00 n w Fig. 5 (color online) The probability distribution of waiting iteration for the values of = 0.05,0.,0.2,0.25 at r = The blue (red) points are associated to the positive (negative) values of the return parameter. shown in Figure 6. By applying the same formula (3.3) a power-law distribution of the waiting times is obtained for positive return value of the earthquake magnitude ρ q = 0.25 as Figure 7 display. An analogous dependence is obtained for returns parameters until ρ q =.0 with an exponent γ =.67.7 in P q = A q τ γ q. 000 N(m) 00 0 P(τ) m τ(weeks) Fig. 6 (color online) Earthquakes magnitude distribution for events in the Vrancea region located at depths below 60 km and magnitude larger than two. Fig. 7 (color online) The probability distribution of waiting time needed to reach a return level in magnitude of ρ q = 0.25 for the earthquakes in the Vrancea region. 4. CONCLUSIONS In this work we explored the chaotic dynamics generated by the one-dimensional maps with the purpose to obtain a characterization of the folding mechanism. In the RJP 60(Nos. 9-0), (205) (c) v..3a*

14 276 V. Baran et al. 4 first part we introduced a generalized evolution of the mean distance between two trajectories, valid beyond the exponential stage which reproduces quite well, for various circumstances, the numerical results. We related the parameter which determine the saturation of the average distance to the folding and discussed its dependence on the control parameter. We concluded that an useful quantity to characterize the occurrence of chaos is the asymptotic distance d and is complementary to the Lyapunov exponent. Indeed, for example in the case of triangular map, the chaotic band merging is clearly signaled through jumps of this quantity when the control parameter is varying while the LE keeps a smooth evolution. Moreover at the transition to chaos d has a power-law dependence on control parameter, the exponent being a function of both Feigenbaum constants. In the second part of our work the inverse statistics was extended to the deterministic chaos generated by the logistic map. A power-law behavior for the waiting iteration distribution was identified in the case of fully developed chaos for a certain range of the return parameter. The asymmetry between the distributions corresponding to positive and negative values respectively, of the return parameter was confirmed also for this situation. This leads us to an additional measure of the folding mechanism. Finally we presented preliminary results concerning the inverse statistics applied to the earthquakes produced in the Vrancea region. As for other systems manifesting a stochastic evolution a more detailed investigation may provide additional insights concerning the underlying dynamical mechanisms which generate the observed features. Acknowledgements. The authors warmly thank A. Nicolin and Z. Neda for the useful discussions and suggestions. This work for V. Baran has been supported by the project from the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE REFERENCES. S.H. Strogatz, Nonlinear Dynamics and Chaos, (Westview Press, 994). 2. B. Mandelbrot, The Fractal Geometry of Nature, (Freeman, San Francisco, 983). J.L. McCauley, Chaos Dynamics and Fractals, (Cambridge Nonlinear Science Series 2, Cambridge University Press 993). E. Ott, Chaos In Dynamical Systems, (Cambridge University Press, England, 993). R.C. Hilborn, Chaos and Nonlinear Dynamics, (Oxford University Press, New York, 994). P.Cvitanovic, Universality in Chaos, (A.Hilger, Bristol,989). J.Froyland, Chaos and Coherence, (IOP, Bristol, 992). M.Tabor, Chaos and Integrability in Nonlinear Dynamics, (J.Wiley and Sons,New York,989). M.Schroeder, Fractals, Chaos, Power Laws, (W.H.Freeman et C.,99). 3. H.G.Schuster, Deterministic Chaos (VCH, New York, 995). 4. L. F. Richardson, Proc. R. Soc. Lond. Ser. A 0, (926) B. Sawford, Annu. Rev. Fluid Mech. 33 (200) 289. RJP 60(Nos. 9-0), (205) (c) v..3a*

15 5 Quantifying the folding mechanism in chaotic dynamics M. Bourgoin, N.I. Quellette, H. Xu, J. Berg, E. Bodenschatz, Science, 3 (2006) J. P. L. C. Salazar, L.R. Collins, Annu. Rev. Fluid Mech. 4 (2009) M.H. Jensen, Phys. Rev. Lett. 83, (999) I. Simonsen, M.H. Jensen and A. Johansen, Eur. Phys. J. B 27, (2002) B.A. Huberman, J. Rudnick, Phys. Rev. Lett. 45 (980) 54.. T. Geisel, J. Nierwetberg, J. Keller, Phys. Lett. 86A (98) J.P. Crutchfield, J.D. Farmer and B.A. Huberman, Phys. Rep. 92 (982) M.H. Jensen, A. Johansen and I. Simonsen, Int. J. Mod. Phys. B 22,23,24, (2003) A. Johansen, I. Simonsen, M.H. Jensen, Physica A 370, (2006) E. Balogh, I. Simonsen, B. Sz. Nagy, Z. Neda, Phys. Rev. E 82, (200) J. V. Siven and J. T. Lins, Phys. Rev. E 80, (2009) M.H. Jensen, A. Johansen, F. Petroni, I. Simonsen, Physica A 340, (2004) P. Giuliani, V. Carbone, P. Veltri, Physica A 280, (2000) F. Omori, J. coll. Sci., Tokyo Imp. Univ., 7, 895 (); Rep. Earth Inv. Commmun. 2, (984) T. Utsu, Y. Ogata, S. Matsu uara, J. Phys. Earth, 43, (995). 2. D.L. Turcotte, Fractals and Chaos in Geology and Geophysics, (Cambridge University Press, 997). RJP 60(Nos. 9-0), (205) (c) v..3a*