- Email: [email protected]
LMC-complexity and various chaotic regimes
Physics Letters A 373 (2009) 2210–2214
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
LMC-complexity and various chaotic regimes G.L. Ferri a,∗ , F. Pennini b,c , A. Plastino b a b c
Facultad de Ciencias Exactas y Naturales, Universidad Nacional de La Pampa. Uruguay 151, Santa Rosa, La Pampa, Argentina Dep. de Física, Universidad Católica del Norte, Antofagasta, Chile Exact Sciences Fac., National University and National Research Council (CONICET), C.C. 727, (1900) La Plata, Argentina
a r t i c l e
i n f o
Article history: Received 2 September 2008 Received in revised form 15 April 2009 Accepted 23 April 2009 Available online 8 May 2009 Communicated by A.R. Bishop
a b s t r a c t A variant of the statistical complexity originally advanced by López-Ruiz et al. [R. López-Ruiz, H.L. Mancini, X. Calbet, Phys. Lett. A 209 (1995) 321], is here used in conjunction with Fisher’s information measure so as to explore fine details of chaotic dynamics. As a main result we can easily distinguish between (i) periodicity and chaos or (ii) between distinct chaotic dynamics belonging to different attractors. © 2009 Elsevier B.V. All rights reserved.
PACS: 05.45.-a Keywords: Chaotic dynamics Fisher information Complexity
1. Introduction Fisher’s information measure (FIM) is much in vogue in the physics world (as examples see [1–4] and references therein). Here we are concerned with the possibility of joining forces between FIM’s F and the statistical-complexity C concept (among many others see Refs. [5,6]), in order to be in a position to grasp fine details of the rich physics of chaos. We choose to this end the LMC C -version [5] for concocting a new planar [7,8]) F –C -representation devised so as to be able to detect significant changes in the behavior of nonlinear dynamical systems (NLDS) [9–12]. In order to quantify differences in NLDS-behavior useful tools include the Liapunov exponents’ technique, dynamical entropies, topological entropies and information measures, etc. (see, for example, [9–12] and references therein). We are going to show that the above referred to conjunction between FIM and statistical complexity measure allows for a practical procedure, easier to compute and yielding better visualization of the intricacies of chaotic dynamics than previous approaches. It provides detailed insights that any physicist will readily appreciate.
2. Theoretical tools Rapid review of Fisher concepts. FIM arises as a measure of the expected error in a measurement [1]. If f (x) denotes a continuous probability distribution function, the Fisher information measure (FIM) F associated to f writes [1] F [ f ] =
Corresponding author. E-mail addresses: [email protected] (G.L. Ferri), [email protected] (F. Pennini), plastino@fisica.unlp.edu.ar (A. Plastino). 0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.04.062
dx
|∇ f |2 f
. Let
{ p j } Nj=1
now P = stand for a normalized set of discrete probabilities. This will be the case of interest here since, although the signal generated by our nonlinear dynamical system has a continuous distribution, it will be evaluated numerically. The associated problem of loss of information due to the discretization has been studied for various measures of information, the Fisher and Renyi measures in particular (see, for instance, [13,14] and references therein). This entails the loss of FIM invariance under displacement, which is of no importance for our present purposes. The corresponding discrete approximation to the evaluation of Fisher’s measure used in this communication is a slight improvement (in the denominator) on the form used in [11] (see also Ref. [14]), i.e., we place in that denominator the average value ( p j +1 + p j )/2 instead of old plain p j so as to considerably diminish the chance of a zero-denominator. We have F [ P ] = (1/4)
*
N −1
2( p j +1 − p j )2 /( p j +1 + p j ) .
(1)
j =1
The factor 1/4 is included for normalization purposes. The maximum value attainable for F is the unity and occurs when almost
G.L. Ferri et al. / Physics Letters A 373 (2009) 2210–2214
all the p j -s are zeros save for a few isolated p j -s (that are finite). Whenever p j = p j +1 = 0 for some j, the jth term in Eq. (1) is taken to be zero. Defining p j = p j +1 − p j , one notes that lim( p j →0) [( p j )2 /( p j + p j /2)] = 0. It is clear that some properties of Fisher’s measure may be lost in this discrete approximation, which is immaterial for our present purposes. More to the point, it was shown in [11] that F distinguishes between chaos and periodicity in the Logistic Map. Here we wish to dig deeper into this F -sensitivity and ascertain that new vistas become then accessible. Statistical complexity measures. López-Ruiz, Mancini, and Calbet (LMC) invented today’s canonical form of a statistical complexity measure (SCM) [5]. It is a product of a measure of “disorder”, or entropy S [ P ], times a “disequilibrium”. Their idea was that a SCM-definition should not be made in terms of just “disorder” or “information” S, but needs adopting some kind of distance D of the given probability distribution P to the equilibrium distribution P e of the accessible states of the system, which motivates defining the “disequilibrium” as
Q[ P ] = Q0 · D[ P , P e ],
(2)
where [11] Q 0 = −2[( NN+1 ) ln( N + 1) − 2 ln(2N ) + ln N ]−1 , so that 0 Q 1. The disequilibrium Q would reflect on the systems’ “architecture”, being different from zero if there exist “privileged”, or “more likely” states among the accessible ones. Thus, the SCM is the product C = H [ P ] Q [ P ] of Q[ P ] with a normalized (∈ [0, 1]) Shannon’s entropy H constructed as [15] S[P ] = −
N
H [ P ] = S [ P ]/ ln N .
p j ln p j ,
(3)
j =1
As a measure of disequilibrium we prefer here the Jensen–Shannon divergence (JSD) D JS
D JS [ P , P e ] = S
P + Pe
2
−
1 2
S[P ] −
1 2
ln N ,
(4)
but we will compare the concomitant results below with those obtained using different disequilibrium forms. The JSD choice is made on account that, using it, one is in a position to obtain a generalized SCM that is: (i) able to grasp essential details of the dynamics and (ii) an intensive quantity, which does not happen in the case of other disequilibrium forms [16,17]. 3. Application to maps The logistic map. Because of its paradigmatic nature it pays well concentrating efforts on it in researching new NLDS tools. It reads xn+1 = α xn (1 − xn ), with 0 < xn < 1, and 0 < α 4 has an attractor set that reduces to a fixed point for α < 3. This fixed point is x = 0 if 0 < α < 1, and x = 1 − 1/α , if 1 < α < 3. At α = 3 things become hectic. There ensues a cascade of perioddoubling that ends at α∞ = 3.5699456 . . . , where the dynamics becomes chaotic. However, we do not encounter a chaotic solution for all control-values α∞ < α < 4. For example, there is a period-six attractor within a window around α = 3.64, a periodfive attractor at a window about α = 3.72 and also a period-three attractor at a window about α = 3.84, that is the most noticeable of them. We choose to scrutinize the dynamics of the logistic map around this period-three window so as to begin testing the performance of the conjunction of the Fisher F (α )-pluscomplexity C (α ) approach. Let us now have a look at Fig. 1(a). With a fine-enough resolution one detects the presence of additional periodic windows. The period-three attractor arises through a saddle-node bifurcation at α1 3.828445 . . . and “exists” within the interval α1 < α < 3.842 . . . . The (chaotic) dynamics present before reaching α1 is called “chaos 1” in Fig. 1(a). As α is increased
2211
beyond αend1 3.842 . . . , the period-three solutions experience a new sequence of period-doubling bifurcations that ends in a totally chaotic dynamics at αcha = 3.845 . . . . The chaotic attractor consists of three narrow disjoint segments and is referred to as “chaos 2”. At α2 = 3.856807 . . . this chaotic attractor is replaced by another one (designed as “chaos 3”) which “lives” within a wider segment that includes the three parts of the previous attractor. For 3.845 . . . < α < α2 the dynamics alternates among chaos (“chaos 2”) and small periodic windows. Fig. 1(a) depicts, within the interval 3.82 < α < 3.87, the logistic map’ bifurcation diagram for 1000 equally spaced α -values, with δ α = 5 × 10−5 . For each α -value, (1) the map is iterated 105 after discarding a transient region of 104 iterations, (2) the interval [0, 1] is subdivided into 103 equal bins of size 10−3 , and a histogram is built on the basis of how many iterates “fall” within each bin. The normalized histogram yields the equilibrium distribution P = { p j } Nj=1 , where N is initially equal to 103 . Of course, many p j -s vanish. For an n-periodic dynamics, we have only n nonnull p j -s (out of N). In the so-called chaotic zones very many p-values are finite for the equilibrium distribution. Before evaluating the statistical complexity of this PDF one eliminates all van˜ elements, with ishing p j -s and we are left with a PDF containing N ˜ ≡ N˜ (α ) < N. If the dynamics is of period m, our PDF consists of N ˜ = m identical p j -s, which makes the complexity to vanish. InN spection of Fig. 1 allows one to notice some interesting features. Fisher’s measure F (α ) clearly detects the occurrence of periodic windows. For an α -value that yields periodic behavior the equilibrium PDF consists of a few disjoint peaks. F should be large then, since it is associated to derivatives of the PDF. Additionally, for periodic windows we expect, and do find, that the complexity will vanish, since by definition a periodic signal is not a complex one. These points had been already made in [11]. What follows are novel features.
• There exist in the graph peaks for F (α ) associated to low complexity values. The interesting point here is that these peaks are associated to α -values for which periodic windows cannot be distinguished in the bifurcation diagram! This is due to the fact that, in order to observe them one needs a much finer resolution in the drawings. Thus, at a given resolution level F provides more information than the bifurcation diagram itself. • Both F and C detect period-doubling, the later in a more noticeable fashion. • Focus attention in the borders of the transition delimiters between different dynamical regimes, i.e., periodic dynamicschaotic one and chaotic–chaotic of different type. C experiments large jumps there. In particular, this shows that the complexity is able to distinguish between different chaotic regimes. • F is insensitive to attractor changes in the chaos’ type. See details of the chaos2 –chaos3 transition in Fig. 3. The following concepts are crucial to the message we are conveying here. At α1 = 3.828445 . . . there is a transition between chaos and a period 3 regime. For α slightly smaller than α1 typical orbits exhibit intermittencies that “separate” time-intervals of quasi-periodic motion. The intermittencies “explore” in sporadic fashion the remainder of the attractor. Precisely in this zone of intermittencies between chaotic and periodic regimes we have large complexity values. C diminishes as the distance between α and α1 grows. When α traverses through α2 = 3.856807 . . . a sudden attractor-change ensues. If α is slightly smaller than α2 , the attractor is chaotic. It consists of three narrow intervals among which the orbit moves. If α is slightly larger than α2 , the attractor is still chaotic but consists now of a rather large interval that contains within it the above mentioned three small ones. We speak of an
2212
G.L. Ferri et al. / Physics Letters A 373 (2009) 2210–2214
Fig. 1. (a) Bifurcation diagram for a logistic map in the vicinity of the period-three window; (b) Plot of Fisher’s information measure F for de equilibrium probability distribution vs. α ; (c) Graph of the statistical complexity C (α ) vs. α . Important α -values referred to in the text are α1 3.828, αend1 3.842, αcha 3.845, and α2 3.8568.
Fig. 2. The Fisher-Complexity plane for the logistic map. Different dynamic regimes (see inset at the north-east corner) are plotted in a complexity (C ) vs. Fisher measure (F ) graph. It is noticed that the three distinct kinds of chaos are located at different places, chaos 3 being the most complex one.
“interior crisis” [9]. A typical orbit for α slightly larger than α2 cycles between the three small pre-crisis subintervals during long time-intervals, followed by intermittent “bursts” during which the orbit explores a wide region, returning afterwards to the previous behavior. In the small region of the logistic diagram that we are analyzing here, the left side border of the window exhibits an intermittent dynamics of the kind: chaos1 → approximately periodic → chaos1 , while in the right side border it can be described as: chaos2 → chaos3 → chaos2 . The statistical complexity is noticeably larger when the orbit switches intermittently between two attractors (one or both of them chaotic) that when it is purely chaotic, and distinguishes clearly between different kinds of chaos.
A complementary type of insights are provided by Fig. 2, in which we introduce as a new tool, the “Fisher-Complexity” plane, to characterize different dynamical regimes. We can readily appreciate the fact that each distinct regime is located at different places in this plane. For instance, we appreciate that, in the case of the chaos-periodicity transition near α1 , F grows in rapid fashion, while C maintains rather high values before such control-value. In the transition “chaos 2” → “chaos 3”, it is C the quantity rapidly growing at α2 , and keeping the large values there attained. Note also how the 1st. period doubling is as complex as that the onset of periodicity. Chaotic dynamic is a low Fisher phenomenon except during intermittencies near α1 . “Chaos 3” is more complex than
G.L. Ferri et al. / Physics Letters A 373 (2009) 2210–2214
2213
Fig. 3. (a) Bifurcation diagram for the logistic map near a transition among two kinds of chaotic dynamics (α2 = 3.856807); (b) Complexity calculated using three different kinds of disequilibrium, namely, Jensen–Shannon, Wooters, and Euclidean, respectively. As we see, the JS-Complexity exhibits a larger jump than the other two at the transition between chaotic attractors. Moreover, the Euclidean one is not able to “see” such type of change; (c) Neither FIM nor the variance of the probability distribution can detect these transitions. Only the complexity does the job.
Fig. 4. Ikeda map. The JS-complexity detects changes in the dynamic before we can appreciate them in the bifurcation diagram. FIM only detects periodic windows.
part of z. We choose the parameter values a = 0.85, b = 0.9, and
Table 1 Chaos 1
Chaos 2
Chaos 3
Period doubling
Periodic
F low C = C 1 0.25
F medium 0 C < C1
F minimum C rather high
F high C medium-high
F maximum C =0
“chaos 2”, etc. The following Table 1 summarizes the properties of the various dynamical regimes of Fig. 2. In Fig. 3(a) we plot the bifurcation diagram for the logistic map in the vicinity of a transition among two kinds of chaotic dynamics (α2 = 3.856807 . . .). The complexities are now evaluated by recourse to three disequilibrium-forms. It is clear that the JS one yields the best results (Fig. 3(b)). We also compare what the Fisher measure is able to discern in relation to what one can “see” using as the discriminating-tool the variance of the pertinent probability distribution. The later is seen to fail in miserable fashion (Fig. 3(c)). iη Ikeda’s map. It reads [9] zn+1 = a − bzn exp (i κ − 1+|z |2 ); z = n
x + iy ∈ C. The ensuing time series is constructed with the real
κ = 0.4. The control parameter is η, that varies here between 7.22 and 7.5, traversing the critical point ηc = 7.26884894 . . . [9]. The equilibrium PDF is reached after evaluation of 103 values of η . For each of them we iterate the map 105 times, after discarding a transient of 104 iterations. In this way we build an histogram of 103 bins of equal size in the interval [−1.4, 1.1] so as to find our PDF. The ensuing dynamics is chaotic throughout, but at ηc = 7.26884894 . . . one detects a sudden growth of the attractor’s size, an interior crisis. For η < ηc , the attractor occupies the range −0.2 till +0.6, approximately. As η crosses ηc one encounters orbit-“bursts” that take it outside this range. The burst-frequency augments as η continues growing. Intermittencies are also detected in this map for the appreciable η -range that one needs to traverse while the new attractor begins to establish itself. The associated bifurcation diagram is depicted in Fig. 4, together with Fisher’s measure F and the statistical complexity C corresponding to the equilibrium PDF. F does not distinguishes between differ-
2214
G.L. Ferri et al. / Physics Letters A 373 (2009) 2210–2214
ent chaotic regimes, but C , instead, clearly “detects” the presence of ηc . C is high in the transient zone between the two attractors, and diminish as the second one stabilizes itself. For η 7.32 and η 7.36 one encounters F -peaks, indicative of periodic windows, that are not accompanied by a diminution of C , that even peaks there. This is due to the fact that periodicity ensues after a very long transient. For some η -values in the interior of the window a periodic orbit is not reached after then thousand iterations. Thus, whenever F and C frow simultaneously we are detecting a narrow periodic window within a large transient zone. 4. Conclusions We studied dynamical systems here so as to estimate the power of the Fisher-measure-Complexity conjunction as a tool for their analysis, with results showing that a rather powerful weapon for such research is thereby being added. Although FIM by itself does distinguishes chaos from periodicity [11], it is unable to detect intermittency or discriminate between distinct chaotic regimes. Dramatic improvements are seen when the statistical complexity C enters the scene. It vanishes for periodic dynamics, is small for chaotic ones, and high for intermittency (or interior crisis). As far as we know, ours is the only approach that allows for distinguishing between different kinds of chaos without explicit knowledge of the map’s structure. The plane Fisher-Complexity permits a clear visu-
alization of the nonlinear phenomena, locating at different places different kinds of chaos and n-period augmentations. Acknowledgement F. Pennini thanks FONDECYT-support, grant 1080487. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
R. Frieden, Science from Fisher Information, Cambridge Univ. Press, 2004. F. Pennini, A. Plastino, Phys. Rev. E 69 (2004) 057101. F. Pennini, A.R. Plastino, A. Plastino, Physica A 258 (1998) 446. M.J.W. Hall, Phys. Rev. A 62 (2000) 012107. R. López-Ruiz, H.L. Mancini, X. Calbet, Phys. Lett. A 209 (1995) 321. J.P. Crutchfield, K. Young, Phys. Rev. Lett. 63 (1989) 105. C. Vignat, J.-F. Bercher, Phys. Lett. A 312 (2003) 27. M. Fuentes, H. Larrondo, M.T. Martin, A. Plastino, O.A. Rosso, Phys. Rev. Lett. 99 (2007) 154102. E. Ott, Chaos in Dynamical Systems, Cambridge Univ. Press, N.Y., 1993; J.C. Sprott, Chaos and Time Series Analyis, Oxford Univ. Press, Oxford, 2004. A.H. Nayfeh, B. Balachandran, Applied Nonlinear Dynamics, Wiley, NY, 1998. M.T. Martin, J. Perez, A. Plastino, Physica A 291 (2001) 523. M.T. Martin, A. Plastino, O.A. Rosso, Physica A 369 (2006) 439. K. Sografos, K. Ferentinos, T. Papaiannous, Canad. J. Stat. 14 (2008) 355. M. Madiman, O. Johnson, I. Kontoyiannis, IEEE Int. Symp. Inform. Th., Nice, June 2007. C.E. Shannon, The Mathematical Theory of Communication, University of Illinois Press, IL, USA, 1949. P.W. Lamberti, M.T. Martin, A. Plastino, O.A. Rosso, Physica A 334 (2004) 119. A. Majtey, P.W. Lamberti, M.T. Martin, A. Plastino, Eur. Phys. J. D 32 (2005) 413.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
LMC-complexity and various chaotic regimes G.L. Ferri a,∗ , F. Pennini b,c , A. Plastino b a b c
Facultad de Ciencias Exactas y Naturales, Universidad Nacional de La Pampa. Uruguay 151, Santa Rosa, La Pampa, Argentina Dep. de Física, Universidad Católica del Norte, Antofagasta, Chile Exact Sciences Fac., National University and National Research Council (CONICET), C.C. 727, (1900) La Plata, Argentina
a r t i c l e
i n f o
Article history: Received 2 September 2008 Received in revised form 15 April 2009 Accepted 23 April 2009 Available online 8 May 2009 Communicated by A.R. Bishop
a b s t r a c t A variant of the statistical complexity originally advanced by López-Ruiz et al. [R. López-Ruiz, H.L. Mancini, X. Calbet, Phys. Lett. A 209 (1995) 321], is here used in conjunction with Fisher’s information measure so as to explore fine details of chaotic dynamics. As a main result we can easily distinguish between (i) periodicity and chaos or (ii) between distinct chaotic dynamics belonging to different attractors. © 2009 Elsevier B.V. All rights reserved.
PACS: 05.45.-a Keywords: Chaotic dynamics Fisher information Complexity
1. Introduction Fisher’s information measure (FIM) is much in vogue in the physics world (as examples see [1–4] and references therein). Here we are concerned with the possibility of joining forces between FIM’s F and the statistical-complexity C concept (among many others see Refs. [5,6]), in order to be in a position to grasp fine details of the rich physics of chaos. We choose to this end the LMC C -version [5] for concocting a new planar [7,8]) F –C -representation devised so as to be able to detect significant changes in the behavior of nonlinear dynamical systems (NLDS) [9–12]. In order to quantify differences in NLDS-behavior useful tools include the Liapunov exponents’ technique, dynamical entropies, topological entropies and information measures, etc. (see, for example, [9–12] and references therein). We are going to show that the above referred to conjunction between FIM and statistical complexity measure allows for a practical procedure, easier to compute and yielding better visualization of the intricacies of chaotic dynamics than previous approaches. It provides detailed insights that any physicist will readily appreciate.
2. Theoretical tools Rapid review of Fisher concepts. FIM arises as a measure of the expected error in a measurement [1]. If f (x) denotes a continuous probability distribution function, the Fisher information measure (FIM) F associated to f writes [1] F [ f ] =
Corresponding author. E-mail addresses: [email protected] (G.L. Ferri), [email protected] (F. Pennini), plastino@fisica.unlp.edu.ar (A. Plastino). 0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.04.062
dx
|∇ f |2 f
. Let
{ p j } Nj=1
now P = stand for a normalized set of discrete probabilities. This will be the case of interest here since, although the signal generated by our nonlinear dynamical system has a continuous distribution, it will be evaluated numerically. The associated problem of loss of information due to the discretization has been studied for various measures of information, the Fisher and Renyi measures in particular (see, for instance, [13,14] and references therein). This entails the loss of FIM invariance under displacement, which is of no importance for our present purposes. The corresponding discrete approximation to the evaluation of Fisher’s measure used in this communication is a slight improvement (in the denominator) on the form used in [11] (see also Ref. [14]), i.e., we place in that denominator the average value ( p j +1 + p j )/2 instead of old plain p j so as to considerably diminish the chance of a zero-denominator. We have F [ P ] = (1/4)
*
N −1
2( p j +1 − p j )2 /( p j +1 + p j ) .
(1)
j =1
The factor 1/4 is included for normalization purposes. The maximum value attainable for F is the unity and occurs when almost
G.L. Ferri et al. / Physics Letters A 373 (2009) 2210–2214
all the p j -s are zeros save for a few isolated p j -s (that are finite). Whenever p j = p j +1 = 0 for some j, the jth term in Eq. (1) is taken to be zero. Defining p j = p j +1 − p j , one notes that lim( p j →0) [( p j )2 /( p j + p j /2)] = 0. It is clear that some properties of Fisher’s measure may be lost in this discrete approximation, which is immaterial for our present purposes. More to the point, it was shown in [11] that F distinguishes between chaos and periodicity in the Logistic Map. Here we wish to dig deeper into this F -sensitivity and ascertain that new vistas become then accessible. Statistical complexity measures. López-Ruiz, Mancini, and Calbet (LMC) invented today’s canonical form of a statistical complexity measure (SCM) [5]. It is a product of a measure of “disorder”, or entropy S [ P ], times a “disequilibrium”. Their idea was that a SCM-definition should not be made in terms of just “disorder” or “information” S, but needs adopting some kind of distance D of the given probability distribution P to the equilibrium distribution P e of the accessible states of the system, which motivates defining the “disequilibrium” as
Q[ P ] = Q0 · D[ P , P e ],
(2)
where [11] Q 0 = −2[( NN+1 ) ln( N + 1) − 2 ln(2N ) + ln N ]−1 , so that 0 Q 1. The disequilibrium Q would reflect on the systems’ “architecture”, being different from zero if there exist “privileged”, or “more likely” states among the accessible ones. Thus, the SCM is the product C = H [ P ] Q [ P ] of Q[ P ] with a normalized (∈ [0, 1]) Shannon’s entropy H constructed as [15] S[P ] = −
N
H [ P ] = S [ P ]/ ln N .
p j ln p j ,
(3)
j =1
As a measure of disequilibrium we prefer here the Jensen–Shannon divergence (JSD) D JS
D JS [ P , P e ] = S
P + Pe
2
−
1 2
S[P ] −
1 2
ln N ,
(4)
but we will compare the concomitant results below with those obtained using different disequilibrium forms. The JSD choice is made on account that, using it, one is in a position to obtain a generalized SCM that is: (i) able to grasp essential details of the dynamics and (ii) an intensive quantity, which does not happen in the case of other disequilibrium forms [16,17]. 3. Application to maps The logistic map. Because of its paradigmatic nature it pays well concentrating efforts on it in researching new NLDS tools. It reads xn+1 = α xn (1 − xn ), with 0 < xn < 1, and 0 < α 4 has an attractor set that reduces to a fixed point for α < 3. This fixed point is x = 0 if 0 < α < 1, and x = 1 − 1/α , if 1 < α < 3. At α = 3 things become hectic. There ensues a cascade of perioddoubling that ends at α∞ = 3.5699456 . . . , where the dynamics becomes chaotic. However, we do not encounter a chaotic solution for all control-values α∞ < α < 4. For example, there is a period-six attractor within a window around α = 3.64, a periodfive attractor at a window about α = 3.72 and also a period-three attractor at a window about α = 3.84, that is the most noticeable of them. We choose to scrutinize the dynamics of the logistic map around this period-three window so as to begin testing the performance of the conjunction of the Fisher F (α )-pluscomplexity C (α ) approach. Let us now have a look at Fig. 1(a). With a fine-enough resolution one detects the presence of additional periodic windows. The period-three attractor arises through a saddle-node bifurcation at α1 3.828445 . . . and “exists” within the interval α1 < α < 3.842 . . . . The (chaotic) dynamics present before reaching α1 is called “chaos 1” in Fig. 1(a). As α is increased
2211
beyond αend1 3.842 . . . , the period-three solutions experience a new sequence of period-doubling bifurcations that ends in a totally chaotic dynamics at αcha = 3.845 . . . . The chaotic attractor consists of three narrow disjoint segments and is referred to as “chaos 2”. At α2 = 3.856807 . . . this chaotic attractor is replaced by another one (designed as “chaos 3”) which “lives” within a wider segment that includes the three parts of the previous attractor. For 3.845 . . . < α < α2 the dynamics alternates among chaos (“chaos 2”) and small periodic windows. Fig. 1(a) depicts, within the interval 3.82 < α < 3.87, the logistic map’ bifurcation diagram for 1000 equally spaced α -values, with δ α = 5 × 10−5 . For each α -value, (1) the map is iterated 105 after discarding a transient region of 104 iterations, (2) the interval [0, 1] is subdivided into 103 equal bins of size 10−3 , and a histogram is built on the basis of how many iterates “fall” within each bin. The normalized histogram yields the equilibrium distribution P = { p j } Nj=1 , where N is initially equal to 103 . Of course, many p j -s vanish. For an n-periodic dynamics, we have only n nonnull p j -s (out of N). In the so-called chaotic zones very many p-values are finite for the equilibrium distribution. Before evaluating the statistical complexity of this PDF one eliminates all van˜ elements, with ishing p j -s and we are left with a PDF containing N ˜ ≡ N˜ (α ) < N. If the dynamics is of period m, our PDF consists of N ˜ = m identical p j -s, which makes the complexity to vanish. InN spection of Fig. 1 allows one to notice some interesting features. Fisher’s measure F (α ) clearly detects the occurrence of periodic windows. For an α -value that yields periodic behavior the equilibrium PDF consists of a few disjoint peaks. F should be large then, since it is associated to derivatives of the PDF. Additionally, for periodic windows we expect, and do find, that the complexity will vanish, since by definition a periodic signal is not a complex one. These points had been already made in [11]. What follows are novel features.
• There exist in the graph peaks for F (α ) associated to low complexity values. The interesting point here is that these peaks are associated to α -values for which periodic windows cannot be distinguished in the bifurcation diagram! This is due to the fact that, in order to observe them one needs a much finer resolution in the drawings. Thus, at a given resolution level F provides more information than the bifurcation diagram itself. • Both F and C detect period-doubling, the later in a more noticeable fashion. • Focus attention in the borders of the transition delimiters between different dynamical regimes, i.e., periodic dynamicschaotic one and chaotic–chaotic of different type. C experiments large jumps there. In particular, this shows that the complexity is able to distinguish between different chaotic regimes. • F is insensitive to attractor changes in the chaos’ type. See details of the chaos2 –chaos3 transition in Fig. 3. The following concepts are crucial to the message we are conveying here. At α1 = 3.828445 . . . there is a transition between chaos and a period 3 regime. For α slightly smaller than α1 typical orbits exhibit intermittencies that “separate” time-intervals of quasi-periodic motion. The intermittencies “explore” in sporadic fashion the remainder of the attractor. Precisely in this zone of intermittencies between chaotic and periodic regimes we have large complexity values. C diminishes as the distance between α and α1 grows. When α traverses through α2 = 3.856807 . . . a sudden attractor-change ensues. If α is slightly smaller than α2 , the attractor is chaotic. It consists of three narrow intervals among which the orbit moves. If α is slightly larger than α2 , the attractor is still chaotic but consists now of a rather large interval that contains within it the above mentioned three small ones. We speak of an
2212
G.L. Ferri et al. / Physics Letters A 373 (2009) 2210–2214
Fig. 1. (a) Bifurcation diagram for a logistic map in the vicinity of the period-three window; (b) Plot of Fisher’s information measure F for de equilibrium probability distribution vs. α ; (c) Graph of the statistical complexity C (α ) vs. α . Important α -values referred to in the text are α1 3.828, αend1 3.842, αcha 3.845, and α2 3.8568.
Fig. 2. The Fisher-Complexity plane for the logistic map. Different dynamic regimes (see inset at the north-east corner) are plotted in a complexity (C ) vs. Fisher measure (F ) graph. It is noticed that the three distinct kinds of chaos are located at different places, chaos 3 being the most complex one.
“interior crisis” [9]. A typical orbit for α slightly larger than α2 cycles between the three small pre-crisis subintervals during long time-intervals, followed by intermittent “bursts” during which the orbit explores a wide region, returning afterwards to the previous behavior. In the small region of the logistic diagram that we are analyzing here, the left side border of the window exhibits an intermittent dynamics of the kind: chaos1 → approximately periodic → chaos1 , while in the right side border it can be described as: chaos2 → chaos3 → chaos2 . The statistical complexity is noticeably larger when the orbit switches intermittently between two attractors (one or both of them chaotic) that when it is purely chaotic, and distinguishes clearly between different kinds of chaos.
A complementary type of insights are provided by Fig. 2, in which we introduce as a new tool, the “Fisher-Complexity” plane, to characterize different dynamical regimes. We can readily appreciate the fact that each distinct regime is located at different places in this plane. For instance, we appreciate that, in the case of the chaos-periodicity transition near α1 , F grows in rapid fashion, while C maintains rather high values before such control-value. In the transition “chaos 2” → “chaos 3”, it is C the quantity rapidly growing at α2 , and keeping the large values there attained. Note also how the 1st. period doubling is as complex as that the onset of periodicity. Chaotic dynamic is a low Fisher phenomenon except during intermittencies near α1 . “Chaos 3” is more complex than
G.L. Ferri et al. / Physics Letters A 373 (2009) 2210–2214
2213
Fig. 3. (a) Bifurcation diagram for the logistic map near a transition among two kinds of chaotic dynamics (α2 = 3.856807); (b) Complexity calculated using three different kinds of disequilibrium, namely, Jensen–Shannon, Wooters, and Euclidean, respectively. As we see, the JS-Complexity exhibits a larger jump than the other two at the transition between chaotic attractors. Moreover, the Euclidean one is not able to “see” such type of change; (c) Neither FIM nor the variance of the probability distribution can detect these transitions. Only the complexity does the job.
Fig. 4. Ikeda map. The JS-complexity detects changes in the dynamic before we can appreciate them in the bifurcation diagram. FIM only detects periodic windows.
part of z. We choose the parameter values a = 0.85, b = 0.9, and
Table 1 Chaos 1
Chaos 2
Chaos 3
Period doubling
Periodic
F low C = C 1 0.25
F medium 0 C < C1
F minimum C rather high
F high C medium-high
F maximum C =0
“chaos 2”, etc. The following Table 1 summarizes the properties of the various dynamical regimes of Fig. 2. In Fig. 3(a) we plot the bifurcation diagram for the logistic map in the vicinity of a transition among two kinds of chaotic dynamics (α2 = 3.856807 . . .). The complexities are now evaluated by recourse to three disequilibrium-forms. It is clear that the JS one yields the best results (Fig. 3(b)). We also compare what the Fisher measure is able to discern in relation to what one can “see” using as the discriminating-tool the variance of the pertinent probability distribution. The later is seen to fail in miserable fashion (Fig. 3(c)). iη Ikeda’s map. It reads [9] zn+1 = a − bzn exp (i κ − 1+|z |2 ); z = n
x + iy ∈ C. The ensuing time series is constructed with the real
κ = 0.4. The control parameter is η, that varies here between 7.22 and 7.5, traversing the critical point ηc = 7.26884894 . . . [9]. The equilibrium PDF is reached after evaluation of 103 values of η . For each of them we iterate the map 105 times, after discarding a transient of 104 iterations. In this way we build an histogram of 103 bins of equal size in the interval [−1.4, 1.1] so as to find our PDF. The ensuing dynamics is chaotic throughout, but at ηc = 7.26884894 . . . one detects a sudden growth of the attractor’s size, an interior crisis. For η < ηc , the attractor occupies the range −0.2 till +0.6, approximately. As η crosses ηc one encounters orbit-“bursts” that take it outside this range. The burst-frequency augments as η continues growing. Intermittencies are also detected in this map for the appreciable η -range that one needs to traverse while the new attractor begins to establish itself. The associated bifurcation diagram is depicted in Fig. 4, together with Fisher’s measure F and the statistical complexity C corresponding to the equilibrium PDF. F does not distinguishes between differ-
2214
G.L. Ferri et al. / Physics Letters A 373 (2009) 2210–2214
ent chaotic regimes, but C , instead, clearly “detects” the presence of ηc . C is high in the transient zone between the two attractors, and diminish as the second one stabilizes itself. For η 7.32 and η 7.36 one encounters F -peaks, indicative of periodic windows, that are not accompanied by a diminution of C , that even peaks there. This is due to the fact that periodicity ensues after a very long transient. For some η -values in the interior of the window a periodic orbit is not reached after then thousand iterations. Thus, whenever F and C frow simultaneously we are detecting a narrow periodic window within a large transient zone. 4. Conclusions We studied dynamical systems here so as to estimate the power of the Fisher-measure-Complexity conjunction as a tool for their analysis, with results showing that a rather powerful weapon for such research is thereby being added. Although FIM by itself does distinguishes chaos from periodicity [11], it is unable to detect intermittency or discriminate between distinct chaotic regimes. Dramatic improvements are seen when the statistical complexity C enters the scene. It vanishes for periodic dynamics, is small for chaotic ones, and high for intermittency (or interior crisis). As far as we know, ours is the only approach that allows for distinguishing between different kinds of chaos without explicit knowledge of the map’s structure. The plane Fisher-Complexity permits a clear visu-
alization of the nonlinear phenomena, locating at different places different kinds of chaos and n-period augmentations. Acknowledgement F. Pennini thanks FONDECYT-support, grant 1080487. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
R. Frieden, Science from Fisher Information, Cambridge Univ. Press, 2004. F. Pennini, A. Plastino, Phys. Rev. E 69 (2004) 057101. F. Pennini, A.R. Plastino, A. Plastino, Physica A 258 (1998) 446. M.J.W. Hall, Phys. Rev. A 62 (2000) 012107. R. López-Ruiz, H.L. Mancini, X. Calbet, Phys. Lett. A 209 (1995) 321. J.P. Crutchfield, K. Young, Phys. Rev. Lett. 63 (1989) 105. C. Vignat, J.-F. Bercher, Phys. Lett. A 312 (2003) 27. M. Fuentes, H. Larrondo, M.T. Martin, A. Plastino, O.A. Rosso, Phys. Rev. Lett. 99 (2007) 154102. E. Ott, Chaos in Dynamical Systems, Cambridge Univ. Press, N.Y., 1993; J.C. Sprott, Chaos and Time Series Analyis, Oxford Univ. Press, Oxford, 2004. A.H. Nayfeh, B. Balachandran, Applied Nonlinear Dynamics, Wiley, NY, 1998. M.T. Martin, J. Perez, A. Plastino, Physica A 291 (2001) 523. M.T. Martin, A. Plastino, O.A. Rosso, Physica A 369 (2006) 439. K. Sografos, K. Ferentinos, T. Papaiannous, Canad. J. Stat. 14 (2008) 355. M. Madiman, O. Johnson, I. Kontoyiannis, IEEE Int. Symp. Inform. Th., Nice, June 2007. C.E. Shannon, The Mathematical Theory of Communication, University of Illinois Press, IL, USA, 1949. P.W. Lamberti, M.T. Martin, A. Plastino, O.A. Rosso, Physica A 334 (2004) 119. A. Majtey, P.W. Lamberti, M.T. Martin, A. Plastino, Eur. Phys. J. D 32 (2005) 413.