Ensemble Averages and Nonextensivity at the Edge of Chaos of One-Dimensional Maps

2004/01/15 by Garín F. Janampa Añaños, Garin F. J Ananos, Constantino Tsallis · 3 citations
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #CHAOS (operating system) #Complex Systems and Time Series Analysis #Computer science #Condensed matter physics #Edge of chaos #Enhanced Data Rates for GSM Evolution #Mathematics #Physics #Statistical Mechanics and Entropy #Statistical physics #cond-mat.stat-mech

paper · pdf · doi:10.1103/physrevlett.93.020601

5 pages, 5 figures

arxiv created 2004/01/15 · openalex publication_date 2004/07/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Ensemble averages of the sensitivity to initial conditions \ensuremathξ(t) and the entropy production per unit of time of a new family of one-dimensional dissipative maps, x_t+1=1\ensuremath-ae^\ensuremath-1/|x_t|^z(z>0), and of the known logisticlike maps, x_t+1=1\ensuremath-a|x_t|^z(z>1), are numerically studied, both for strong (Lyapunov exponent \ensuremathλ₁>0) and weak (chaos threshold, i.e., \ensuremathλ₁=0) chaotic cases. In all cases we verify the following: (i) both \ensuremath⟨ln_q\ensuremathξ\ensuremath⟩ [ln_qx\ensuremath≡(x^1\ensuremath-q\ensuremath-1)/(1\ensuremath-q); ln₁x=lnx] and \ensuremath⟨S_q\ensuremath⟩ [S_q\ensuremath≡\phantom\rule0ex0ex(1\ensuremath-\ensuremath∑_ip_i^q)/(q\ensuremath-1); S₁=\ensuremath-\ensuremath∑_ip_ilnp_i] linearly increase with time for (and only for) a special value of q, q_sen^av, and (ii) the slope of \ensuremath⟨ln_q\ensuremathξ\ensuremath⟩ and that of \ensuremath⟨S_q\ensuremath⟩ coincide, thus interestingly extending the well known Pesin theorem. For strong chaos, q_sen^av=1, whereas at the edge of chaos q_sen^av(z)<1.

Citations

• Nonextensive Pesin identity: Exact renormalization group analytical results for the dynamics at the edge of chaos of the logistic map
• Circular-like maps: sensitivity to the initial conditions, multifractality and nonextensivity
• CHARACTERISTIC LYAPUNOV EXPONENTS AND SMOOTH ERGODIC THEORY
• Nonequilibrium Probabilistic Dynamics of the Logistic Map at the Edge of Chaos

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