q-deformed statistical-mechanical structure in the dynamics of the Feigenbaum attractor

We show that the two complementary parts of the dynamics associated to the Feigenbaum attractor, inside and towards the attractor, form together a q-deformed statisticalmechanical structure. A time-dependent partition function produced by summing distances between neighboring positions of the attractor leads to a q-entropy that measures the ratio of ensemble trajectories still away at a given time from the attractor (and the repellor). The values of the q-indexes are given by the attractor’s universal constants, while the thermodynamic framework is closely related to that first developed for multifractals. The Feigenbaum attractor, an icon of the historical developments in the theory of nonlinear dynamics [1], is getting renewed attention [2]. This is because it offers a convenient model system to explore features that might reflect those of statistical mechanical systems under conditions of phase space mixing and ergodicity breakdown. It therefore offers insights on the limits of validity of ordinary statistical mechanics. Recently a thorough description with newly revealed features has been given [3] [4] of the intricate dynamics that takes place both inside and towards this famous multifractal attractor. Here we show that these two types of dynamics are related to each other in a statistical-mechanical fashion, i.e. the dynamics at the attractor provides the ‘microscopic configurations’ in a partition function while the approach to the attractor is economically described by an entropy obtained from it. As we show below, this property conforms to a q-deformation [5] [6] of the ordinary exponential (Boltzmann) weight statistics. Furthermore, this novel statistical-mechanical property arises in many other cases, some very familiar, that involve multifractal attractors with vanishing Lyapunov exponents. We have already uncovered [7] the prerequisites for the quasiperiodic transition to chaos (specifically along the golden mean) to be another example of the same kind of q-statistics [8]. Features in problems that relate to the period-doubling or the quasiperiodic routes to chaos (with general nonlinearities differing from the usual quadratic or cubic) acquire thermodynamic q-deformed structures. These include, for instance, phase transitions in spin or plaquette models of systems with many degrees of freedom, when built-in scaling properties lead to iteration techniques, and their singular behavior appears linked to Mandelbrot-like sets [9]. The broader issue of incidence of q-statistical properties in the complex dynamics of Julia sets associated to neutral or indeferent fixed points is an interesting question to go in for. Trajectories within the Feigenbaum attractor show self-similar temporal structures, they preserve memory of their previous locations and do not have the mixing property of chaotic XI Latin American Workshop on Nonlinear Phenomena IOP Publishing Journal of Physics: Conference Series 246 (2010) 012025 doi:10.1088/1742-6596/246/1/012025 c © 2010 IOP Publishing Ltd 1 trajectories [10]. The fluctuating sensitivity to initial conditions has the form of infinitely many interlaced q-exponential functions that fold into a single one with use of a two-time scaling property [3] [6] [7]. More precisely, there is a hierarchy of such families of interlaced q-exponentials; an intricate (and previously unknown) state of affairs that befits the rich scaling features of a multifractal attractor. Furthermore, the entire dynamics is made of a family of pairs of Mori’s dynamical q-phase transitions [10] [3] [6] [7]. On the other hand, the process of convergence of trajectories into the Feigenbaum attractor is governed by another unlimited hierarchy feature built into the preimage structure of the attractor and its counterpart repellor [4]. The overall rate of approach of trajectories towards the attractor (and repellor) is conveniently measured by the fraction of (fine partition) bins Wt1 still occupied at time t1 by an ensemble of trajectories with initial positions uniformly distributed over phase space [11] [4]. For the first few time steps the rate Wt1 remains constant, Wt1 ! ∆, 1 ≤ t1 ≤ t0, t0 = O(1) [12], after which a power-law decay with log-periodic modulation sets in, a signature of discrete-scale invariance [13]. This property of Wt1 is explained in terms of a sequential formation of gaps in phase space, and its self-similar features are seen to originate in the ladder feature of the preimage structure [4]. The rate Wt1 was originally presented in Ref. [11] where the power law exponent φ in Wt1 ! ∆ h ( ln t lnΛ ) t−φ, t = t1 − t0, (1) was obtained numerically. Above, h(x) is a periodic function with h(1) = 1, and Λ is the scaling factor between the periods of two consecutive oscillations [14]. We proceed now to demonstrate the connection between the aforementioned dynamical properties. We recall [1] the definition of the interval lengths or diameters dn,m that measure the bifurcation forks that form the period-doubling cascade sequence in unimodal maps, here represented by the logistic map fμ(x) = 1− μx2, −1 ≤ x ≤ 1, 0 ≤ μ ≤ 2. These quantities are measured when considering the superstable periodic orbits of lengths 2n, n = 1, 2,...; i.e. the 2n-cycles that contain the point x = 0 at μn < μ∞, where μ∞ = 1.401155189... is the value of the control parameter μ at the period-doubling accumulation point [15]. The positions of the limit 2∞-cycle constitute the Feigenbaum attractor. The dn,m in these orbits are defined (here) as the (positive) distances of the elements xm, m = 0, 1, ..., 2n−1 − 1, to their nearest neighbors f (2 n−1) μn (xm), i.e. dn,m ≡ ∣∣f (m+2 n−1) μn (0)− f (m) μn (0) ∣∣. For large n, dn,0/dn+1,0 ! α, where α is Feigenbaum’s universal constant α ! 2.5091. Innermost to our arguments is the following comprehensive property: Time evolution at μ∞ from t = 0 up to t → ∞ traces the period-doubling cascade progression from μ = 0 up to μ∞. Not only is there a close resemblance between the two developments but also asymptotic quantitative agreement. For example, the trajectory inside the Feigenbaum attractor with initial condition x0 = 0, takes positions xt such that the distances between nearest neighbor pairs of them reproduce the diameters dn,m defined from the supercycle orbits with μn < μ∞. See Fig. 1. This property has been central to obtain rigorous results for the fluctuating sensitivity to initial conditions ξt(x0) within the Feigenbaum attractor, as separations at chosen times t of pairs of trajectories originating close to x0 can be obtained as diameters dn,m [3] [6]. Further, the complex dynamical events that fix the decay rate Wt1 can be understood in terms of the correlation between time evolution at μ∞ from t = 0 up to t→∞ and the ‘static’ perioddoubling cascade progression from μ = 0 up to μ∞. As shown recently [16] each doubling of the period (obtained through the shift μn → μn+1) introduces additional elements in the hierarchy of the preimage structure and in the family of sequentially-formed phase space gaps in the finite period cycles. The complexity of these added elements is similar to that of the total period 2n system. Also, the shift μn → μn+1 increases in one unit the number of undulations in the XI Latin American Workshop on Nonlinear Phenomena IOP Publishing Journal of Physics: Conference Series 246 (2010) 012025 doi:10.1088/1742-6596/246/1/012025