arXiv subjects

Ken-ichi Okubo

Publications and source records attributed to Ken-ichi Okubo.

10 recordsLinked to original sources

arXiv2026

Empirical One-Step Conditional Entropy in Infinite Ergodic Systems: Vanishing Entropy Rate, Sparse-Transition Scaling, and Mittag-Leffler Fluctuations

Empirical entropy rates are widely used to quantify unpredictability from symbolic or time-series data, yet their interpretation is subtle in weakly chaotic dynamics, where ordinary Lyapunov exponents vanish and invariant measures are infinite. We address this issue by studying the empirical one-step conditional entropy for the fixed finite partitions considered below in one-dimensional intermittent maps with infinite invariant measures. For the modified Bernoulli map and the Boole transformation in the infinite-measure weak-chaos regime, we prove that this per-step empirical entropy converges to zero. Thus, the usual entropy-rate normalization becomes asymptotically blind to subexponential instability. The finite-time information sum, however, remains informative. Rare transitions between long laminar phases occur on the return-sequence scale, and their empirical self-information contributes an additional logarithmic factor. Under the stated regularity and moment assumptions, this mechanism yields a two-term estimate for the ensemble mean decay, supported by numerical simulations. Although the raw entropy rate vanishes, self-normalized fluctuations remain nontrivial and are numerically consistent with normalized Mittag-Leffler laws. A comparison with generalized Lyapunov sums shows that the corresponding information sum is not a Krengel entropy estimator, but a computable, partition-dependent finite-time measure of sparse symbolic transitions. These results clarify what empirical Markov entropy can, and cannot, measure in infinite-measure weak chaos.

nlin.CD↗

arXiv2024

Anosov Properties of a Symplectic Map with Time-Reversal Symmetry

This study presents a specific symplectic map, derived from a Hamiltonian, as a model that exhibits time-reversal symmetry on a microscopic scale. Based on the analysis, any initial density function, defined almost everywhere, converges to a uniform distribution in terms of mixing (irreversible behavior) on a macroscopic level. Furthermore, we established that this mixing invariant measure is a unique equilibrium state, unique SRB measure, and physical measure. Additionally, through analytical proof, we have shown that the Kolmogorov-Sinai entropy, representing the average information gain per unit time is positive. This was achieved by validating the Pesin's formula and demonstrating that the critical exponent of the Lyapunov exponent is $1/2$.

nlin.CD↗

arXiv2023

Spatial-photonic Ising machine by space-division multiplexing with physically tunable coefficients of a multi-component model

This paper proposes a space-division multiplexed spatial-photonic Ising machine (SDM-SPIM) that physically calculates the weighted sum of the Ising Hamiltonians for individual components in a multi-component model. Space-division multiplexing enables tuning a set of weight coefficients as an optical parameter and obtaining the desired Ising Hamiltonian at a time. We solved knapsack problems to verify the system's validity, demonstrating that optical parameters impact the search property. We also investigated a new dynamic coefficient search algorithm to enhance search performance. The SDM-SPIM would physically calculate the Hamiltonian and a part of the optimization with an electronics process.

physics.optics↗

arXiv2023

Low-rank combinatorial optimization and statistical learning by spatial photonic Ising machine

The spatial photonic Ising machine (SPIM) [D. Pierangeli et al., Phys. Rev. Lett. 122, 213902 (2019)] is a promising optical architecture utilizing spatial light modulation for solving large-scale combinatorial optimization problems efficiently. The primitive version of the SPIM, however, can accommodate Ising problems with only rank-one interaction matrices. In this Letter, we propose a new computing model for the SPIM that can accommodate any Ising problem without changing its optical implementation. The proposed model is particularly efficient for Ising problems with low-rank interaction matrices, such as knapsack problems. Moreover, it acquires the learning ability of Boltzmann machines. We demonstrate that learning, classification, and sampling of the MNIST handwritten digit images are achieved efficiently using the model with low-rank interactions. Thus, the proposed model exhibits higher practical applicability to various problems of combinatorial optimization and statistical learning, without losing the scalability inherent in the SPIM architecture.

cond-mat.dis-nn↗

arXiv2022

Universal Critical Behavior of Transition to Chaos: Intermittency Route

The robustness of the universality class concept of the chaotic transition was investigated by analytically obtaining its critical exponent for a wide class of maps. In particular, we extended the existing one-dimensional chaotic maps, thereby generalising the invariant density function from the Cauchy distribution by adding one parameter. This generalisation enables the adjustment of the power exponents of the density function and superdiffusive behavior. We proved that these generalised one-dimensional chaotic maps are exact (stronger condition than ergodicity) to obtain the critical exponent of the Lyapunov exponent from the phase average. Furthermore, we proved that the critical exponent of the Lyapunov exponent is $\frac{1}{2}$ regardless of the power exponent of the density function and is thus universal. This result can be considered as rigorous proof of the universality of the critical exponent of the Lyapunov exponent for a countably infinite number of maps.

nlin.CD↗

arXiv2020

Infinite Ergodicity that Preserves the Lebesgue Measure

We proved that for the countably infinite number of one-parameterized one dimensional dynamical systems, they preserve the Lebesgue measure and they are ergodic for the measure (infinite ergodicity). Considered systems connect the parameter region in which dynamical systems are exact and the parameter region in which systems are dissipative, and correspond to the critical points of the parameter in which weak chaos occurs (the Lyapunov exponent converges to zero). These results are the generalization of the work by R. Adler and B. Weiss. We show that the distributions of normalized Lyapunov exponent for these systems obey the Mittag-Leffler distribution of order $1/2$ by numerical simulation.

nlin.CD↗

arXiv2017

Universality of the Route to Chaos -- Exact Analysis

The universality of the route to chaos is analytically proven for countably infinite number of maps by proposing the Super Generalized Boole (SGB) transformations. As one of the route to chaos, intermittency route was studied by Pomeau and Manneville numerically. They conjectured the universality in Type 1 intermittency, that the critical exponent of the Lyapunov exponent is $1/2$ in Type 1 intermittency. In order to prove their conjecture, we showed that for certain parameter ranges, the SGB transformations are exact and preserve the Cauchy distribution. Using the property of exactness, we proved that the critical exponent is $1/2$ for countably infinite number of maps where Type 1 intermittency occurs.

nlin.CD↗

arXiv2016

Canonical superdiffusion and energy fluctuation divergence

We propose a noble physical model obtained from a Hamiltonian with periodic potential. This model is canonical, reversible and brings about chaotic superdiffusion with energy fluctuation divergence. The analytical formula of invariant density can be obtained in some parameter range. In the range it is proved that the map is Anosov diffeomorphism and the invariant measure is a SRB measure. We calculate the analytical formula of Lyapunov exponent.

nlin.CD↗

arXiv2016

Exact Lyapunov exponents of the generalized Boole transformations

The generalized Boole transformations have rich behavior ranging from the \textit{mixing} phase with the Cauchy invariant measure to the \textit{dissipative} phase through the \textit{infinite ergodic} phase with the Lebesgue measure. In this Letter, by giving the proof of mixing property for $0<α<1$ we show an \textit{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 \textit{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 $α\to 0$, and $α\to 1$. This result shows the computational complexity on the numerical simulations of the Lyapunov exponents near $α\simeq$ 0, 1.

nlin.CD↗

arXiv2015

New chaos indicators for systems with extremely small Lyapunov exponents

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.

nlin.CD↗