Logistic map trajectory distributions: Renormalization-group, entropy and criticality at the transition to chaos
LLogistic map trajectory distributions:Renormalization-group, entropy and criticality at thetransition to chaos
A. Diaz-Ruelas * , F. Baldovin † , A. Robledo ‡ Abstract
We study the evolution of the probability density of ensembles of iterates of the logistic map thatadvance towards and finally remain at attractors of representative dynamical regimes. We consider themirror families of superstable attractors along the period-doubling cascade, and of chaotic-band attrac-tors along the inverse band-splitting cascade. We examine also their common aperiodic accumulationpoint. The iteration time progress of the densities of trajectories is determined via the action of theFrobenius-Perron (FP) operator. As a difference with the study of individual orbits, the analysis of en-sembles of positions offers a viewpoint from which the nonlinear dynamical features of this iconic modelcan be better characterized in statistical-mechanical terms. The scaling of the densities along the con-sidered families of attractors conforms to a renormalization-group (RG) structure, while their entropiesare seen to attain extrema at the fixed points of the RG flows. Additionally, this entropy as a function ofthe map control parameter displays the characteristic features of an equation of state of a thermal systemundergoing a second-order phase transition. We discuss our results.
Already few decades ago it had been a common general commentary within the ComplexSystems community that observations of complex systems in nature appear to indicate, in thelanguage of nonlinear dynamics, that their conduct is as if they evolve at the ‘edge of chaos’.Likewise, the same community nowadays often shares the general commentary that the obser-vations of complex systems in nature seem to imply, in the language of statistical mechanics,that they thrive in a state of ‘criticality’. Interestingly, as we describe here, these two paradigmsappear to be equivalent at the transition to chaos displayed by the archetypal nonlinear dynam-ical model, the quadratic map. To see this we consider two families of attractors, the supercy-cles along the period-doubling cascade and the band-splitting (Misiurewicz) points along thechaotic-band cascade, together with their joint accumulation point at the transition to and out * Max Planck Institute for the Physics of Complex Systems, N¨othnitzer Str. 38, D-01187 Dresden, Germany ([email protected]) † INFN-Dipartimento di Fisica, Universit`a di Padova, Via Marzolo 8, I-35131 Padova, Italy, ([email protected]) ‡ Instituto de F´ısica, Universidad Nacional Aut´onoma de M´exico, Apartado Postal 20-364, M´exico 01000 D.F., Mexico,(robledo@fisica.unam.mx) a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b f chaos. With their invariant densities in hand (provided by the Frobenius-Perron method) afamiliar Renormalization Group picture appears, while the uncomplicated task of evaluatingtheir entropies is an opportunity to be taken. First of all, the fixed points are identified as en-tropy extrema. For period one the entropy vanishes reaching its minimum possible value, whilethe entropy for the single chaotic band attains the maximum value. The entropy for the non-trivial fixed point at the transition to chaos is maximum for all supercycles and minimum forall Misiurewicz points. Secondly, the entropy of the invariant densities grows monotonicallyfrom period one through all supercycles, and all Misiurewcz points to the final single-bandchaotic attractor. But most remarkably, when the collection of entropies for the two families ofattractors is viewed along the values of control parameter of the map the familiar pattern ap-pears of a statistical-mechanical two-phase system separated by a continuous phase transition,an equation of state containing a critical point. The conventional approach to study the dynamics of low-dimensional nonlinear systems, e.g., iterated mapsof the interval, is to look at the asymptotic properties of single orbits [1, 2, 3]. In contrast here we analyzeboth the transient and the asymptotic behavior of the probability distribution, or density, associated withensembles of orbits. This change of perspective is comparable to that present in the description of clas-sical normal diffusion, in which tracking the dynamics of single particles via the Langevin equation [4]is reformulated into a partial differential equation for the evolution of the probability density of finding aparticle at a specific position and time, i.e., the Fokker-Planck equation [5]. This substitution in object ofstudy helps to illuminate the inner workings behind statistical-mechanical descriptions [6]. Motivated bypotential statistical-mechanical insight gain, and as a difference with normal diffusion, here we look at theevolution of the densities of trajectories that take place in situations governed by nonlinear dynamics. Wechoose to examine the familiar period-doubling route to chaos in dissipative systems together with its com-panion sequence of chaotic-band splitting attractors [1, 2, 3]. For practical reasons a convenient setting forour planned investigation is the standard logistic map. Unless specifically stated, we describe the evolutionof densities of a uniformly distributed set of initial conditions along the interval of definition of the map. Asthe trajectories evolve towards the attractors the densities advance likewise to a final stage that reflects thedistinct visiting order of attractor positions or bands in unimodal maps [1, 2, 3]. As anticipated the knownself-affine properties displayed by these families of attractors [1, 2, 3] manifest also in their densities and wetake advantage of this to formulate an appropriate Renormalization Group (RG) transformation for whichthe density at the RG nontrivial fixed point corresponds to the transition into or out of chaos. The densitiesfor the trivial fixed points are those for period one and single band attractors. But also the entropies associ-ated with the densities can be readily determined and these are found to be extrema for the RG fixed points.The overall picture obtained is that of a statistical-mechanical system in the vicinity of a critical point.
Background recall.
The density of trajectories ρ t ( x ) at positions x ∈ I and iteration time t under theaction of a given map f ( x ) defined for the phase-space interval I can be constructed directly from an initialdensity by means of a linear operator approach, This operator, known as the transfer or Frobenius-Perron2perator [7], acts on arbitrary densities and drives them forward in time. It is defined by the action ρ t ( x ) = L ( t ) ρ ( x ) , (1)which is written in explicit form as ρ t ( x ) = (cid:90) I δ [ y − f ( t ) ( x )] ρ ( y ) dy , (2)where x = f ( t ) ( x ) is f ( x ) composed t times with itself. Eq. (2) is called the Frobenius-Perron equationand δ [ y − f ( t ) ( x )] is the singular kernel of its associated linear operator [6]. Even though the evolutionof trajectories under f ( x ) is nonlinear there is a linear relation between densities via the integral operation.Together with Eq. (2) we require (cid:90) f ( t ) ( I ) ρ t ( x ) dx = (cid:90) I ρ ( x ) dx , for arbitrary densities ρ ( x ) of initial conditions distributed over I . This is equivalent to the conservation ofthe total Lebesque measure of I under f ( x ) , or, in other words, the initial number of trajectories is preserved.We consider the relation above to be valid for the dissipative case, as employed here. [6, 8].In the following Section 2 we particularize the Frobenius-Perron approach to the logistic map, and in thenext Section 3 we present the resultant densities of ensembles of trajectories that first proceed towards andthen evolve within the superstable attractors, the band-splitting or Misiurewicz points, and their commonaccumulation point, the Feigebaum attractor [1, 2, 3]. We indicate there the development of a larger (thanconsecutive iteration t ) time scale, of the form τ = N n , N = , , . . . , with n fixed ( n indicates the order ofthe superstable orbit of period 2 n , or that of the Misiurewicz point when 2 n bands are about to appear. Thenwe look at the scaling of these densities as the accumulation point of the superstable cycles is approached.We confirm the stationary character of the densities in the larger time scale τ . In Section 4 the previousnumerical results are reproduced via a rescaling scheme that uses as starting input the smooth, invariant dis-tribution for the fully-developed single-band attractor at the end value of the map control parameter, knownas the Ulam density [1, 2, 3]. The rescaling procedure expresses the self-affinity that permeates through theproperties of the logistic and other quadratic maps and reproduces sequentially the invariant densities at theband-splitting points (and also those for the supercycles) in a quantitative way on the time scale τ . Next, inSection 5 we put together the properties of the two families of attractors studied, including their commonaccumulation point, into a Renormalization Group (RG) framework, such that the densities flow towardstwo trivial fixed points, period one for the periodic attractors and a single band for the chaotic attractors.The density for the accumulation point at the transition into or out of chaos corresponds to the nontrivialfixed point. Then we evaluate the (Shannon) entropies of the densities along the two families of attractorsand observe that the entropy attains extreme values at the RG fixed points [9, 10], while the overall shapeof the entropy and its derivative as a function of the map control parameter displays the characteristics ofan equation of state and the response function of a statistical-mechanical system undergoing a second orderphase transition. Finally, in Section 6 we summarize and discuss our results.3 A Fokker-Planck equation for the logistic map
The normal iterative procedure of generating trajectories x t , t = , , . . . , from the logistic map f µ ( x ) = − µ x , x ∈ [ − , ] , µ ∈ [ , ] , (3)starting from an initial condition x at fixed value of the control parameter µ , resembles the descriptionof fluid motion in the Lagrangian frame of reference where x t + = − µ x t plays the role of a Langevinequation [4]. The change of perspective when looking at fluid motion through an Eulerian frame of referenceinvolves the time evolution of the density of particles as represented by the Fokker-Planck equation [5]. Theparallel description to study the dynamics contained in the logistic map centers on the transition probabilityof trajectories between positions reached at consecutive iteration times. This is ρ t ( x ) = (cid:90) d x (cid:48) p ( x , t | x (cid:48) , t − ) ρ t − ( x (cid:48) ) , (4)with x = − µ x (cid:48) . In our example, the conditional transition probability is given by the Dirac delta function p ( x , t | x (cid:48) , t − ) = δ ( x − + µ x (cid:48) ) . (5)Therefore we have ρ t ( x ) = (cid:90) − d x (cid:48) δ ( x − + µ x (cid:48) ) ρ t − ( x (cid:48) ) , (6)which, after the change of variables y = − µ x (cid:48) for x (cid:48) ∈ [ , ] and z = − µ x (cid:48) for x (cid:48) ∈ [ − , ] becomes ρ t ( x ) = (cid:112) µ ( − x ) (cid:2) ρ t − (cid:0) ξ µ ( x ) (cid:1) + ρ t − (cid:0) − ξ µ ( x ) (cid:1)(cid:3) , (7)with ξ µ ( x ) ≡ (cid:112) ( − x ) / µ and where x ∈ [ − µ , ] and ρ t = x ∈ ( − , − µ ) . This is the Frobenius-Perron equation particularized to the logistic map when written for consecutive iteration times.There is an important difference between the familiar linear Fokker-Planck equation in fluid motion ordiffusion problems and the equation we obtained for the logistic map, Eq. (7), and this is that the functionalinverse of the logistic map is not unique. Therefore, the “backwards” companion equation to Eq. (7),analogous to reverse time in the Fokker-Planck equation, is obtained by inserting in ρ t − ( x ) = (cid:90) − µ d x (cid:48) p ( x , t − | x (cid:48) , t ) ρ t ( x (cid:48) ) (8)the backward propagation of the probability density, with x = ± ξ (cid:48) µ , ξ (cid:48) µ = (cid:112) ( − x (cid:48) ) / µ . Explicitly, theconditional transition probability in Eq.(4) is a sum of two Dirac deltas p ( x , t − | x (cid:48) , t ) = δ (cid:16) x + ξ (cid:48) µ (cid:17) + δ (cid:16) x − ξ (cid:48) µ (cid:17) . (9)4ence we have the expression ρ t − ( x ) = (cid:90) − µ d x (cid:48) (cid:20) δ (cid:16) x + ξ (cid:48) µ (cid:17) + δ (cid:16) x − ξ (cid:48) µ (cid:17)(cid:21) ρ t ( x (cid:48) ) , (10)that yields ρ t − ( x ) = (cid:40) µ x ρ t ( − µ x ) , x ∈ [ , ] − µ x ρ t ( − µ x ) , x ∈ [ − , ] , (11)which is already normalized. The family of superstable attractors or supercycles [1, 2] of the logistic map, and in general unimodal maps,has become a standard choice when describing dynamical properties along the period-doubling cascade.The rapid convergence of trajectories into these attractors (exponential of an exponential decay rate [11])was a convenient option in the early studies that revealed basic properties and defined key quantities, suchas the so-called diameters [1, 2], and this in turn have stimulated many subsequent developments throughtheir use. We select this family of attractors to determine densities of trajectories via the Frobenius-Perronmethod. The control parameter value for the supercycle of period 2 n is denoted S n , n = , , , . . . In Fig. 1 we show numerical results for the solutions of the Frobenius-Perron Eq. (7) when µ = S at early and moderately large iteration times for an initially uniform distribution of initial positions in theinterval [ − , ] . As observed there, only one iteration is sufficient to wipe out uniformity and concentratethe trajectories near x =
1. A second iteration divides the trajectories into two groups around two positionsof the attractor with the formation of one central gap. Soon after the trajectories divide into four groupslocated close to the four attractor positions separated by two new gaps. The total of three gaps containthe three repellor positions present for period-four attractors. For all subsequent number of iterations theheights of the four peaks that express the populations of the four groups of trajectories alternate locationsaccording to the fixed order of visits of attractor positions in unimodal map dynamics [1]. For large iterationtime t the density is invariant when observed at multiples of 2 t .Similarly, in Fig. 2 we show numerical results for the solutions of Eq. (7) when µ = S at early and largeiteration times for an initially uniform distribution of initial positions in the interval [ − , ] . As seen before,uniformity is sharply erased at the first iteration and trajectories cluster around x =
1. Trajectories splitat the second iteration into two groups centered around two attractor positions and create one central gap.Four groups of trajectories now around four attractor positions appear at four iterations while the centralgap divides into three gaps. Lastly, eight groups of trajectories are formed on the eight attractor positionsat iteration eight separated by seven gaps. The final sharp delta peaks that form the attractor separatedby empty intervals are reached continuously for larger iteration times. As before, the heights of the eightpeaks, that correspond to the populations of the eight groups of trajectories, alternate locations according to5 igure 1: Evolution of an initially uniform density of positions in the interval [ − , ] when µ = S . (a) First iteration, thedensity accumulates sharply around x =
1. (b) Second iteration, another peak forms around x ∼ − .
3. (c) Fourth iteration, thereare four peaks centered on the final attractor points x ∼ − . , , . ,
1. (d) Thirty two iterations, we observe already a goodapproximation of the final density consisting of four delta functions.Figure 2: Evolution of an initially uniform density of positions in the interval [ − , ] when µ = S . (a) 1st iteration, the densityrises steeply around x =
1. (b) 2nd iteration, a second peak forms around x ∼ − .
3. (c) 4th iteration, two more peaks developbetween the former two. (d) 8th iteration, four additional peaks appear, there are eight peaks centered on the final attractor points x ∼ − . , − . , , . , . , . , . ,
1. (e) 16th iteration, the gaps between the attractor positions develop further. (f) 256thiteration, we observe already a good approximation of the final density consisting of eight delta functions. t thedensity appears invariant when observed at multiples of 2 t .For (all) larger periods the solutions of Eq. (7) at µ = S n , n = , . . . , show a parallel iteration timedevelopment. This is to recapitulate the formation of 2 k , k = , , , , . . . , n , peaks and the, 2 k − , k = , , , , . . . , n −
1, gaps between them sequentially at times t = k , k = , , , , . . . , n . The large time t → ∞ density is the sum of 2 n delta functions centered at the attractor positions x i , i = , . . . , n . The densityvanishes in the intervals between these positions that contain the 2 n − n attractor positions according to the prescribed order of visits in the dynamicsof unimodal maps [1, 2, 3]. The density is invariant when the alternative time scale τ = N n , N = , , . . . ,is adopted. The two time scales t and τ diverge from each other exponentially as n → ∞ . The family of chaotic-band attractors where bands are on the point to split, known also as Misiurewiczpoints [3] is a convenient option introduced here to determine densities of chaotic trajectories via theFrobenius-Perron method. The band-splitting sequence for µ > µ ∞ is the chaotic equivalent to the period-doubling supercycles for µ < µ ∞ . For recent developments assisted through their use, see for example[12, 13]. The determination of the band widths at the n th Misiurewicz point is facilitated by the circum-stance that the set of band edge points, that we denote by { e n , k } ∈ [ − µ , ] , k = , . . . , ∗ n − , corre-spond to positions of the the trajectory initiated at the position of the map x = i.e. , e n , k = f kM n ( ) , where M n , n = , , , . . . is the control parameter value for the 2 n -band Misiurewicz point. Orbits initiated atthe position x = µ = M n are eventually periodic [14], which means they posses a transient called preperiod after which they are periodic. In the former, 3 ∗ n − is the sum of the preperiod q = n and theperiod p = n − of each M n . Therefore e n , = , e n , = − M n , e n , = − M n e n , = − M n ( − M n ) , andso on. The positions of the points e n , k correspond to the peaks of the density in the final panels in Figs. 3and 4. Following the same procedure for arbitrary values of µ defines the polynomials P n ( µ ) = − µ P n − ( µ ) , P = , µ ∈ [ , ] (12)of order 2 n −
1, that are sometimes referred to as shade curves or critical polynomials [14], that we will callhere simply µ -polynomials or µ -curves. They conform the loci of all the band edges, and supercycle andperiodic window’s attracting positions (see Fig. 7), a fact that will be employed in the construction of theRenormalization-Group (RG) transformation.Fig. 3 shows numerical results for the solutions of the Frobenius-Perron Eq. (7) when µ = M at earlyand larger iteration times for an initially uniform distribution of initial positions in the interval [ − , ] .Again, only one iteration is sufficient to wipe out uniformity and concentrate the trajectories near x =
1. Ata second iteration appears a second peak that gives the distribution a single u -shaped form. At the fourthiteration the trajectories are divided into two groups forming u -shaped densities separated by one centralgap. A few more iterations lead to a density that approximates the final form for this chaotic band attractor.For large iteration time t the density is invariant when observed at multiples of 2 t .7 igure 3: Evolution of an initially uniform density of positions in the interval [ − , ] when µ = M . (a) First iteration, the densityaccumulates sharply around x =
1. (b) Second iteration, another peak forms around x ∼ − .
4. (c) Fourth iteration, there arefour peaks delineating the edges of the two final bands separated by a gap. (d) Thirty two iterations, we observe already a goodapproximation of the final density that shows the twin u -shaped form at each of the two bands.Figure 4: Evolution of an initially uniform density of positions in the interval [ − , ] when µ = M . (a) 1st iteration, the densityrises steeply around x =
1. (b) 2nd iteration, a second peak forms around x ∼ − .
4. (c) 4th iteration, two more peaks developbetween the former two forming two bands separated by a central gap. (d) 8th iteration, four additional peaks appear, leading tofour bands separated by three gaps. (e) 16th iteration, the gaps between the attractor positions develop further while four newpeaks give the densities the characteristic Misiurewicz-point repeated twin u -shaped form. (f) 32nd iteration, we observe alreadya good approximation of the final density. µ = M at early andlarge iteration times for an initially uniform distribution of initial positions in the interval [ − , ] . Again, thesame sequential pattern is observed, fast departure from uniformity with clustering first at x =
1, then at theother edge of the main band, central gap formation, and splitting of the two bands. Next, the same eventsleading now to three gaps separating four bands, each exhibiting the characteristic double u -shaped densityof the Misiurewicz points. For large iteration time t the density is invariant when observed at multiples of2 t . The evolution via sequential gap formation of uniformly distributed ensemble of trajectories towards su-percycle and Misiurewicz point attractors display a ‘recapitulation’ property [13, 15, 16], i.e. progressiontowards 2 n -periodic or 2 n -band chaotic attractors repeats successively that towards those attractors with2 k , k = , , , ..., n −
1. As described above, we have seen that this property appears reflected in the timedevelopment of their densities. When µ = µ ∞ recapitulation never ends and the attractor becomes a multi-fractal set while the density becomes an infinite set of delta functions placed at the attractor positions (seeFig. 9 below). At the accumulation point the difference between the two time scales t and τ diverges andadvancing by successive iterations does not reach the complete invariant density. A different option is toplace a uniform distribution on the multifractal attractor (say one initial condition on each point in Fig. 9)and add a position at infinity, x = x ∞ , with the rule f µ ∞ ( x ∞ ) =
0. This distribution remains invariant in theiteration time scale t . µ = The invariant density of the Ulam map f ( x ) = − x has been known for some time [1, 2, 3]. For thelogistic map in the fully chaotic regime µ = ρ t ( x ) = ρ t + ( x ) is the u -shapedfunction ρ ( x ) = π √ − x , (13)as it satisfies ρ ( x ) = (cid:112) µ ( − x ) (cid:34) ρ (cid:32)(cid:115) − x µ (cid:33) + ρ (cid:32) − (cid:115) − x µ (cid:33)(cid:35) , (14)or ρ ( x ) = ρ (cid:16)(cid:113) − x µ (cid:17)(cid:112) µ ( − x ) , (15)9ince Eq. (13) is symmetric around x = ρ ( − x ) = ρ ( x ) . Now, that we have determined numerically the invariant densities (in the large time scale τ = N n , N = , , . . . ) at the Misiurewicz points we will reproduce them quantitatively by means of a scaling argument.Consider the affine transformation y ≡ bx + c , < b < , − µ < c < ρ X ( x ) , we have ρ Y ( y ) = (cid:90) d x ρ X ( x ) δ ( y − bx − c ) (17) = b ρ X (cid:18) y − cb (cid:19) . When ρ X ( x ) is the Ulam invariant density (Eq. (13)) we obtain ρ ( x ) = π (cid:112) b − ( x − c ) . (18)Considering that at a Misiurewicz point each chaotic band splits into two new bands, and that correspond-ingly the invariant density duplicates the number of u -shaped elements in it, then we assume that each u -shaped element at M n gets a proportion of 2 − n of the total measure at µ =
2. So, at the first Misiurewiczpoint the scaling ansatz gives ρ M ( x ) = (cid:16) π (cid:113) b , − ( x − c , ) (cid:17) − , x ∈ [ e , , e , ] (cid:16) π (cid:113) b , − ( x − c , ) (cid:17) − , x ∈ [ e , , e , ] (19)where we introduce the notation c n , i , b n , i with n indicating the generation of the band-splitting cascade,and the second index denotes the i th u -shaped density element i = min ( l , s ) where l , s = , . . . , ∗ n − arethe indices of the corresponding edges e n , l , e n , s of its support, that we denote by U n , i = [ e n , l , e n , s ] followingthe same definition of its indices as above. In this way c n , i = ( e n , l + e n , s ) /
2. The contraction parameters b n , i = | e n , l − e n , s | / M n as theFeigenbaum attractor is approached when n → ∞ . The i th density element of the measure at the n -thMisiurewicz point, with support U n , i has the form ρ n , i ( x ) = n π (cid:113) b n , i − ( x − c n , i ) . (20)10 igure 5: Invariant density for the first Misiurewicz point M . Left panel, numerically determined from the Frobenius-Perronequation. Right panel, obtained from scaling and duplication of the Ulam density in Eq.(13) according to Eq. (19).Figure 6: Invariant density for the second Misiurewicz point M . Left panel, numerically determined from the Frobenius-Perronequation. Right panel, obtained from scaling and duplication of the density in Eq.(19) according to the general expression in Eq.(20). b n , i − ( x − c n , i ) ≥ x ∈ U n , i . In Fig. 5 (Fig. 6) we show the agreement between the numericallydetermined and the scaled and duplicated invariant densities for the first (second) Misiurewicz point as givenby Eq. (20). As it will be highlighted in the next section, the Feigenbaum point µ ∞ can be interpreted as the nontrivialfixed point of a discrete RG transformation. The action of the transformation maps µ (cid:54) = µ ∞ towards one ofthe trivial fixed points: either towards µ = µ ∈ { S n } or µ = µ = { M n } ). For the sequence { M i } ofMisiurewicz points the direction of the RG flow is M n → M n − and thus, it is given by the inverse of ourself-affine transformation Y ( x ) = β n , x + γ n , (21)with Y ( x ) = y − ( x ) , β n , = / b n , and γ n , = − c n , / b n , . For simplicity and clarity in the derivation, letus start by focusing on the intervals U n , = [ e n , q + , e n , = ] (with q = n the preperiod of M n , see Sec.3.2). The RG transformation maps the interval U n , onto U n − , through Eq. (21). In order to realize thismapping, notice first that all the intervals U n , share the same boundary at x = = e n , . Mapping this edgewith Eq. (21) e n + , → e n , , gives the relation γ n , = − β n , . (22)For mapping the edges e n , q + it is illustrative to write them first in terms of the µ -polynomials defined byEq. (12) (see also Fig. 7), evaluated at µ n = M n e n , q + = P q ( µ n ) , with q = n . The mapping e n , q + → e n , q ∗ + (with q ∗ = n − ) corresponds then to P q ( µ n ) → P q ∗ ( µ n − ) orequivalently to the equation Y ( P q ( µ n )) = P q ∗ ( µ n − ) , whose solution for β n , is β n , = − P q ∗ ( µ n − ) − P q ( µ n ) . (23)Notice that Eq. (28) corresponds to the ratio of the lengths | · | of successive intervals | U n − , | / | U n , | and,we have obtained it through a RG argument. The asymptotic value lim n → ∞ β n , = β ∞ , is estimated withEq. (28) up to the 6th Misiurewicz point as β , = ( − P ( M )) / ( − P ( M )) ∼ . ... .This value has a discrepancy of only 0 . α = . α = − . . . . the universal Feigenbaum constant[17] giving the localscaling around the maximum of all quadratic unimodal maps at the accumulation point of the period-doubling scenario.In Fig. (8 we show the monotone convergence of the numerical estimate of β n , to α as given by Eq.(28). 12 igure 7: Bifurcation diagram of the logistic map with superimposed µ -polynomials f n µ ( ) = P n ( µ ) = − µ P n − ( µ ) , P = n = , , , ,
4, outlining the bifurcation diagram. We indicate with vertical lines the intervals U n , i for n = , , i = , , , µ ∈ { M n } . The dotted vertical line indicates µ ∞ = . β n , to the value β ∞ , = α = . . . . , indicated with adashed horizontal line. The solid circles correspond to the first 6 control parameter values at Misiurewicz points. The line joiningthe points is only a guide to the eye.
13n a similar way, now we obtain the parameter β n , q for the symmetric intervals centered at x = U n , q =[ e n , q ∗ + q , e n , q ] , with e n , q > , e n , q ∗ + q < q ∗ = n − , q = n . This time γ n , q =
0, and by symmetry e n , q = − e n , q ∗ + q hence | U n , q | = e n , q . The edge e n , q can be written in terms of the µ -polynomials simply as e n , q = P q − ( µ n ) . With this, solving the RG equation Y ( P q − ( µ n )) = P q ∗ − ( µ n − ) for β n , q gives β n , q = P q ∗− ( µ n − ) P q − ( µ n ) (24)By performing the corresponding numerical evaluations of Eq. (24) for µ n = { M , M , ..., M } we get β , = − . . α . Fig. 8 shows thevalues of the transformation coefficients β n , for the first few M n as they converge to the limiting valuelim n → ∞ β n , = α . Our arguments for the derivation of the RG transformation and the numerical evidencefor the convergence of β n , → α and β n , q → α as n is increased, enables us to conclude with confidence,that in the limit n → ∞ β ∞ , q = α (25) β ∞ , = α (26)The asymptotic values given above are naturally expected from the local scaling at µ ∞ around x = x = β ∞ , q and β ∞ , , respectively, as given by the (recirpocal of) Feigenbaum’s universal trajectoryscaling function[1, 18] 1 / σ ( x = ) = α , indicating the most crowded region of the multifractal attractor,and 1 / σ ( x = ) = α being the sparsest. This direct connection with the function 1 / σ ( x ) provides an evenmore interesting interpretation to the asymptotic values β ∞ , i and invites its reformulation as the function β ( x ) = / σ ( x ) in terms of the continuous variable x = i / p with p = n − the period of M n , just in the wayit is done for 1 / σ ( x ) , thus providing a new way to obtain 1 / σ ( x ) approaching form µ > µ ∞ . The designedRG transformation works also for µ < µ ∞ , i.e. , at the sequence of supercycle attractors µ ∈ { S n } . Instead of following the customary analytical format for the functional composition Renormalization Group(RG) procedure applied to the period doubling cascade [1] we follow a graphical representation that facil-itates its extension to the collection of the invariant densities we have already determined. Then we lookat the entropy associated with them, and after this we remark on a statistical-mechanical critical point per-spective of the Feigembaum accumulation point and its neighborhood.In Fig. 9 we show the absolute values of positions in logarithmic scales of the first 1000 iterations forthe trajectory initiated at x when µ = µ ∞ . We observe in this figure that the positions appear arrangedinto horizontal bands separated by gaps, all bands of equal widths and all gaps of equal widths (seen moreclearly defined for large t ). The top band of positions contain 1 / / n -th band 2 − n positions. The RG transformation is: i) Eliminate the topband, all positions with odd iteration times. This is half of the multifractal attractor. ii) Then shift theremaining positions to the left a distance ln 2 and up a distance ln α . The result is that one recovers the14 igure 9: Trajectory with initial condition x = µ ∞ in absolute values and logarithmic scales. Fromthis visualization of this orbit it is straightforward to see how the absolute values of the positions group: All the odd iteratesgroup to form the top band around x = { k } . See text. same figure when µ = µ ∞ . Repeat the operation any number of times. This is the nontrivial fixed point.Elimination of the top band is the same as functional composition, so that the operations above correspondto the original RG. If µ is less than µ ∞ , say we are at a supercycle, then the repeated RG operations leadto period one, one of the trivial fixed points. If µ is greater than µ ∞ , say we are at a Misiurewicz point,then the repeated RG operations lead to one chaotic band, the other trivial fixed point.The RG in our plan isto do the same geometrical operations with the invariant densities obtained from the Frobenius-Perron, or,equivalently, from the self-affine property. The (only) relevant variable (in RG language) is the difference ∆ µ ≡ µ − µ ∞ , and it is similar to the temperature distance to the critical point in thermal systems. The RGrelevant variables need to be set to zero in order to reach the nontrivial RG fixed point, the accumulationpoint at µ ∞ . Otherwise the transformation flows towards the trivial RG fixed points, in our case period oneor one chaotic band. The renormalization scheme operating on the invariant distributions at Misiurewicz points consists on fold-ing each pair of adjacent u -shaped elements into one u -shaped unit followed by elimination of the gapsbetween them. Fig. 6 for M becomes Fig. 5 for M under this transformation. The RG transformationworks inversely with respect to the affine transformation in the previous section. The RG transformationfor the (multi-delta function) invariant distributions at the supercycle attractors consists of merging pairs oftheir latest generation of delta functions into single ones, therefore eliminating the gaps between them andresulting into the invariant distribution of the previous supercycle. Fig. 2(f) for S becomes Fig. 1(d) for S under this transformation. Recall that the distributions for Misiurevicz Points M k and supercycle points15 k are invariant in the time scale τ = N k , N = , , . . . , k fixed, but show a cyclical pattern along iterationtimes t , one cycle covered through t = N k , N k + , N k + , . . . , N k + k , N fixed. This, of course, afterthe transient behavior is over and only the asymptotic solution of the FP equation is observed. Figure 10: Flow diagram of the renormalization group (RG) transformation. The RG transformation applied to any Misiurewiczpoint M k densities ( ∆ µ >
0) leads to the trivial fixed point that represents the single band Ulam distribution, denoted by theright full circle. On the other hand, the RG transformation applied to any supercycle S k density ( ∆ µ <
0) ends up at the trivialfixed point represented by the single delta function for period one, denoted by the left full circle. The non-trivial fixed pointcorresponds to the density made of an infinite set of delta functions located each at the positions shown in Fig. 9, denoted thecentral full circle ( ∆ µ = The Shannon entropy S µ k = − (cid:90) − dx ρ µ k ln ρ µ k ( x ) , (27)associated with the invariant densities ρ µ k at the families of supercycle attractors µ k = S k and Misiurewiczpoints µ k = M k we have determined can be readily obtained. These are shown in Fig. 11 as a functionof the control parameter distance to the accumulation point µ = µ ∞ . There we see behavior reminiscentof the entropy below, at, and above the critical temperature of a thermal system presenting ordered anddisordered phases separated by a phase transition. However here we are following the entropy of ensemblesof positions between two distinctive behaviors: The regular motion associated with the period-doublingcascade and irreguSolution of the RG eq gives: β n , = − P q ∗ ( µ n − ) − P q ( µ n ) . (28)by the former arguments and numerical evidence we concludelim n → ∞ β n , = α (29)lar motion associated with the chaotic band-splitting cascade. The entropy presents a sudden increase at thetransition from periodic motion to chaos. The logistic map on its route to chaos by either period-doublingor band splitting out of chaos can be viewed as a macroscopic system approaching a phase transition by asuccession of equilibrium states.If we recall that the RG trivial fixed points are the period-one supercycle and the single-band chaoticUlam attractor and that the RG nontrivial fixed point is their common accumulation point at µ ∞ we noticethat these points are also entropy extrema. The entropy at µ ∞ is maximum with respect to all periodicattractors and a minimum with respect to all chaotic attractors. This evidence reaffirms the claim advanced16 igure 11: Entropy vs control parameter value. The red, dotted vertical line represents the control parameter value µ ∞ of thetransition to chaos, whereas each dot corresponds to a member of the sequences of either superstable orbits (left to the dottedline) or the band-splitting cascade (to the right of the dotted line).Figure 12: Susceptibility χ calculated according to χ = µ − | ∂ µ / ∂ S | vs reduced distance in control parameter ∆ µ / µ ∞ . Thecharacteristic divergence at a critical point is clearly appreciated at the transition to chaos ∆ µ =
17n past works [19], that the fixed points of the RG approach are always related to entropy extrema, with theall-important nontrivial fixed point as saddle point. See Fig. (10).To visualize further the parallelism with a thermal system we calculated from the data in Fig. 11 thequantity that would correspond to a specific heat or susceptibility, χ = µ − (cid:12)(cid:12)(cid:12)(cid:12) ∂ S ∂ µ (cid:12)(cid:12)(cid:12)(cid:12) . (30)In Fig. 12 we can appreciate how χ , as defined above, diverges at the onset of chaos. We examined the properties of the logistic map through the use of the Frobenius-Perron (FP) equation. Wesolved this equation numerically for both the first dozen supercycles along the period-doubling cascade andthe first dozen Misiurewicz points along the chaotic band-splitting cascade of attractors. In both cases weobserved a fast convergence to the final dynamical cyclical repetition. As a starting point we chose eachtime a uniform distribution of initial conditions in the interval of definition of the map. As expected whenworking with supercycles we observed very fast approach of the FP densities to their limiting form fortrajectories inside the attractors. But this was also the case for the Misiurewicz points. We would expect tohave observed a distinctively slower approach for the case of the pitchfork bifurcation points, while fromour current experience we cannot indicate what family of chaotic attractors, if any, would exhibit slowapproach.Only the Ulam distribution for the chaotic single band when µ = t . All the distribu-tions for Misiurewicz points M n and supercycles S n become invariant on the consecutive cycle time scale τ = N n , N = , , . . . , n fixed. The two scales diverge from each other when the accumulation point of bothfamilies of attractors is approached, and so, the observation of an invariant density becomes increasinglyunreachable. But this is not necessarily so if the set of initial conditions for the ensemble of trajectories issuitably chosen. A uniform distribution of initial conditions placed only in the attractor positions leads toan invariant density in the time scale t .The invariant distributions obtained numerically from the FP equation were show to be quantitativelyreproduced via a self-affine transformation with mirror duplication of either the Ulam distribution for allMisiurewicz points or of a single delta function for all supercycles. In addition to this a RenormalizationGroup (RG) transformation on the invariant densities defined as the reverse self-affine transformation withmerging of mirror elements was introduced such that the RG flows towards two trivial fixed points thatrepresent the distributions of a single fully-chaotic band and of a single periodic point. The nontrivial fixedpoint distribution is an infinite set of delta functions with multifractal support, the Feigenbaum attractor.The only relevant variable is the control parameter distance to the period-doubling accumulation point, ∆ µ ≡ µ − µ ∞ .The (Shannon) entropies S for the distributions of the supercycles and the Misiurewicz points weredetermined and examined as a function of ∆ µ . The outcome bears a strong resemblance with a critical18sotherm in a typical thermal system undergoing a continuous phase transition. To affirm further this sim-ilarity the quantity that would correspond to a response function, χ = µ − | ∂ µ / ∂ S | , was also calculatedand was confirmed to display its characteristic divergence at a critical point. The families of attractors thatform the bifurcation diagram of the quadratic maps show basically two behaviors, periodic and chaotic, ap-pearing along two main cascades with increasing period or number of chaotic bands. They share the sameaccumulation point. These types of attractors represent the two ‘phases’ separated by a ‘critical’ point. Itis well-known that this feature is repeated an infinite number of times within the ‘periodic windows’ in thefractal bifurcation diagram.AR acknowledges support from DGAPA-UNAM-IN106120 and Ciencia-de-Frontera-CONACyT-39572(Mexican Agencies). References [1] H.G. Schuster.
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