Subexponential instability in one-dimensional maps implies infinite invariant measure

Akimoto, Takuma; Aizawa, Yoshifusa

doi:10.1063/1.3470091

Cited by 29 publications

(50 citation statements)

References 33 publications

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“…The exponent α smaller than 1 implies a divergence of the mean. In dynamical systems, this divergent mean brings an infinite measure [4]. Thus, such distributional behavior of time-averaged observables is also shown in infinite ergodic theory [1,5].…”

Section: Introductionmentioning

confidence: 93%

Phase Diagram in Stored-Energy-Driven Lévy Flight

Akimoto

Miyaguchi

2014

J Stat Phys

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Phase diagram based on the mean square displacement (MSD) and the distribution of diffusion coefficients of the time-averaged MSD for the stored-energy-driven Lévy flight (SEDLF) is presented. In the SEDLF, a random walker cannot move while storing energy, and it jumps by the stored energy. The SEDLF shows a whole spectrum of anomalous diffusions including subdiffusion and superdiffusion, depending on the coupling parameter between storing time (trapping time) and stored energy. This stochastic process can be investigated analytically with the aid of renewal theory. Here, we consider two different renewal processes, i.e., ordinary renewal process and equilibrium renewal process, when the mean trapping time does not diverge. We analytically show the phase diagram according to the coupling parameter and the power exponent in the trapping-time distribution. In particular, we find that distributional behavior of time-averaged MSD intrinsically appears in superdiffusive as well as normal diffusive regime even when the mean trapping time does not diverge.

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Section: Introductionmentioning

confidence: 93%

Phase Diagram in Stored-Energy-Driven Lévy Flight

Akimoto

Miyaguchi

2014

J Stat Phys

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“…By substituting f (T k x) = log |T ′ (x k )|, we know that the a value of Lyapunov exponent of Boole transformation is zero. However, it is known that the dynamical system behaves sub-exponentially [7]. Namely, its orbital expansion rate ∆ grows ∆ ∼ exp(t 1 2 ).…”

Section: Ultra Fast Lyapunov Indicatormentioning

confidence: 99%

“…In addition, this new chaos indicator can firstly detect a sharp transition between Arnold diffusion and Chirikov diffusion. Then, we propose another new indicator which can find the index α about subexponential chaos whose orbital expansion rate ∆ grows ∆ ∼ exp(n α ), 0 < α < 1 occuring in the Boole transformation [7,10] and the S-unimodal function [11].…”

Section: Introductionmentioning

confidence: 99%

New Chaos Indicators for Systems with Extremely Small Lyapunov Exponents

Okubo¹,

Umeno²

2017

Chaos, Complexity and Transport

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We propose new chaos indicators for systems with extremely small positive Lyapunov exponents. These chaos indicators can firstly detect a sharp transition between the Arnold diffusion regime and the Chirikov diffusion regime of the Froeschlé map and secondly detect chaoticity in systems with zero Lyapunov exponent such as the Boole transformation and the S-unimodal function to characterize sub-exponential diffusions. PACS numbers: 45.05.+x, 46.40.Ff, 96.12.De

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“…This means that a slight modification of parameter α toward unity (the Boole transformations) from below causes a large effect on the value of Lyapunov exponent at the edge of 0 < α < 1. Furthermore, we can say that it is extremely difficult to numerically obtain the Lyapunov exponents near α = 1 and 0 < α < 1 because for α = 1 subexponential chaos [13,12] occurs and we cannot determine a Lyapunov exponent for finite calculation. According to Eq.…”

Section: Analytic Formula Of Lyapunov Exponents For the Generalized Bmentioning

confidence: 99%

Exact Lyapunov exponents of the generalized Boole transformations

Umeno

Okubo

2016

Prog. Theor. Exp. Phys.

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The generalized Boole transformations have rich behavior ranging from the mixing phase with the Cauchy invariant measure to the dissipative phase through the infinite ergodic phase with the Lebesgue measure. In this Letter, by giving the proof of mixing property for 0 < α < 1 we show an analytic formula of the Lyapunov exponents λ which are explicitly parameterized in terms of the parameter α of the generalized Boole transformations for the whole region α > 0 and bridge those three phase continuously. We found the different scale behavior of the Lyapunov exponent near α = 1 using analytic formula with the parameter α. In particular, for 0 < α < 1, we then prove an existence of extremely sensitive dependency of Lyapunov exponents, where the absolute values of the derivative of Lyapunov exponents with respect to the parameter α diverge to infinity in the limit of α → 0, and α → 1. This result shows the computational complexity on the numerical simulations of the Lyapunov exponents near α ≃ 0, 1.

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Subexponential instability in one-dimensional maps implies infinite invariant measure

Cited by 29 publications

References 33 publications

Phase Diagram in Stored-Energy-Driven Lévy Flight

Phase Diagram in Stored-Energy-Driven Lévy Flight

New Chaos Indicators for Systems with Extremely Small Lyapunov Exponents

Exact Lyapunov exponents of the generalized Boole transformations

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