Soya Shinkai

The Lempel-Ziv Complexity of Non-Stationary Chaos in Infinite Ergodic Cases

The large deviation properties of the Lempel-Ziv complexity are studied using a onedimensional non-hyperbolic chaos map called the “modified Bernoulli map”, where the transition between stationary and non-stationary chaos is clearly observed. The upper limit of the Lempel-Ziv complexity in the non-stationary regime is theoretically evaluated, and the relationship between the algorithmic complexity and the Lempel-Ziv complexity is discussed. Non-stationary processes are universal phenomena in non-hyperbolic systems, and they are usually characterized by an infinite ergodic measure and intrinsic long time tails, such as 1/f ν spectral fluctuations. It is shown that the Lempel-Ziv complexity obeys universal scaling laws and that the Lempel-Ziv complexity has the L 1 -function property, which guarantees the Darling-Kac-Aaronson theorem for an infinite ergodic system. The most striking result is that the maximum diversity appears at the transition point from stationary chaos to non-stationary chaos where the exact 1/f spectral process is generated.

1/f spectrum and 1-stable law in one-dimensional intermittent map with uniform invariant measure and Nekhoroshev stability

We investigate ergodic-theoretical quantities and large deviation properties of one-dimensional intermittent maps, that have not only an indifferent fixed point but also a singular structure such that the uniform measure is invariant under the mapping. The probability density of the residence time and the correlation function are found to behave polynomially: f(m) ∼ m (�+1) and C(�) ∼ � (� 1) (� > 1). Using the Doeblin-Feller theorems in probability theory, we derive the conjecture that the rescaled fluctuations of the time average of some observable functions obey the stable distribution with the exponent 1 < � ≤ 2. Some exponents of the stable distribution are precisely determined by numerical simulations, and the conjecture is verified numerically. The polynomial decay of large deviations is also discussed, and it is found that the entropy function does not exist, because the moment generating function of the stable distribution can not be defined.We investigate ergodic properties of a one-dimensional intermittent map that has not only an indifferent fixed point but also a singular structure such that a uniform measure is invariant under mapping. The most striking aspect of our model is that stagnant motion around the indifferent fixed point is induced by the log-Weibull law, which is derived from Nekhoroshev stability in the context of nearly-integrable Hamiltonian systems. Using renewal analysis, we derive a logarithmic inverse power decay of the correlation function and a

Ergodic properties of the Log‐Weibull map with an infinite measure

1/\omega

Formal model of embodiment on abstract systems: from hierarchy to heterarchy

-like power spectral density. We also derive the so-called 1-stable law as a component of the time-average distribution of a simple observable function. This distributional law enables us to calculate a logarithmic inverse power law of large deviations. Numerical results confirm these analytical results. Finally, we discuss the relationship between the parameters of our model and the degrees of freedom in nearly-integrable Hamiltonian systems.

Dissipative Heat Decomposition in Stochastic Energetics: Implication of the Instantaneous Diffusion Coefficient in Nonequilibrium Steady States

A one‐dimensional map with an infinite measure is investigated. The motions generated by the map are very sticky around indifferent fixed points. Here we report three theoretical results: First, the residence time distribution around the points is the log‐Weibull one which is same as a universal law in Hamiltonian systems. Secondly, the Darling‐Kac‐Aaronson theorem can be applied with the order α = 0. Finally, the power spectrum density of the orbits reveals the strongest non‐stationarity such as f−2 spectral fluctuations.

The Lempel-Ziv complexity in infinite ergodic systems

An embodiment of a simple system, such as a one-dimensional map system, derived from heterarchical duality is discussed. We formalized two pairs of heterarchical layers induced by the indefiniteness of the environment and inconsistency between parts and wholeness by using category theory and applied its construction to a logistic map. From the analysis of its behavior, we universally observed 1/f spectrum for orbits and the fractal-like behavior in the dynamics of return maps. For the coupling map system, the parameter region with an on-off intermittency was clearly extended. Finally, we discuss the relationship between this model and the recent interest in morphological computations and search for a way to deal theoretically with the concept of adaptability.

Log-Weibull Law in Fermi-Pasta-Ulam Oscillator Chain : Induction Time as Nekhoroshev Stability Time (Special Issue on New Challenges in Complex Systems Science)

We give a decomposition expression for dissipative heat using the instantaneous diffusion coefficient in a nonequilibrium steady state. The dissipative heat can be expressed using three diffusion coefficients: instantaneous, equilibrium, and drift. An experimental application of the decomposition expression permits us to evaluate the heat dissipation rate from single-trajectory data only. We also numerically demonstrate this method.