Renormalization group structure for sums of variables generated by incipiently chaotic maps

Abstract

We look at the limit distributions of sums of deterministic chaotic variables in unimodal maps and find a remarkable renormalization group (RG) structure associated to the operation of increment of summands and rescaling. In this structure - where the only relevant variable is the difference in control parameter from its value at the transition to chaos - the trivial fixed point is the Gaussian distribution and a novel nontrivial fixed point is a multifractal distribution that emulates the Feigenbaum attractor, and is universal in the sense of the latter. The crossover between the two fixed points is explained and the flow toward the trivial fixed point is seen to be comparable to the chaotic band merging sequence. We discuss the nature of the Central Limit Theorem for deterministic variables.

14 pages, 5 figures, to appear in Journal of Statistical Mechanics

References in corpus (6)

• Central limit behavior of deterministic dynamical systems
• A closer look at time averages of the logistic map at the edge of chaos
• q-Deformed Statistical-Mechanical Property in the Dynamics of Trajectories en route to the Feigenbaum Attractor
• Dynamics towards the Feigenbaum attractor
• Incidence of nonextensive thermodynamics in temporal scaling at Feigenbaum points
• Comment on "Central limit behavior in deterministic dynamical systems"

Cited by in corpus (9)

• Probability densities for the sums of iterates of the sine-circle map in the vicinity of the quasi-periodic edge of chaos
• Generalized Huberman-Rudnick scaling law and robustness of $q$-Gaussian probability distributions
• Generalization of the possible algebraic basis of $q$-triplets
• Generalized Statistical Mechanics at the Onset of Chaos
• Sums of variables at the onset of chaos
• Scaling of distributions of sums of positions for chaotic dynamics at band-splitting points
• Sums of variables at the onset of chaos, replenished
• Pascal (Yang Hui) triangles and power laws in the logistic map
• Conduction at the onset of chaos