Related papers: Correlation dimension and phase space contraction …
The dynamics of many-body systems can often be captured in terms of only a few relevant variables. Mathematical and numerical approaches exist to identify these variables by exploiting a separation of time scales between slow relevant and…
We study the pointwise dimension for a new class of projection measures on arbitrary fractal limit sets without separation conditions. We prove that the pointwise dimension exists a.e. for this class of measures associated to equilibrium…
The dynamical response theory is used to obtain an analytical expression for the exchange energy of a quantum wire for arbitrary polarization and width. It reproduces the known form of exchange energy for 1D electron gas in the limit of…
Extreme value statistics provides accurate estimates for the small occurrence probabilities of rare events. While theory and statistical tools for univariate extremes are well-developed, methods for high-dimensional and complex data sets…
In a smooth flow, the leading-order response of trajectories to infinitesimal perturbations in their initial conditions is described by the finite-time Lyapunov exponents and associated characteristic directions of stretching. We give a…
The computation of the entire Lyapunov spectrum for extended dynamical systems is a very time consuming task. If the system is in a chaotic spatio-temporal regime it is possible to approximately reconstruct the Lyapunov spectrum from the…
Information theory on a time-discrete setting in the framework of time series analysis is generalized to the time-continuous case. Considerations of the Roessler and Lorenz dynamics as well as the Ornstein-Uhlenbeck process yield for…
We provide appropriate tools for the analysis of dynamics and chaos for one-dimensional systems with periodic boundary conditions. Our approach allows for the investigation of the dependence of the largest Lyapunov exponent on various…
Neural networks have dramatically increased our capacity to learn from large, high-dimensional datasets across innumerable disciplines. However, their decisions are not easily interpretable, their computational costs are high, and building…
The problems of causality, modeling, and control for chaotic, high-dimensional dynamical systems are formulated in the language of information theory. The central quantity of interest is the Shannon entropy, which measures the amount of…
Verifying entanglement with experimental measurements requires that we take the limitations of experimental techniques into account, while still proving that the data obtained could not have been generated from a classical source. In the…
The principle of entropy increase is not only the basis of statistical mechanics, but also closely related to the irreversibility of time, the origin of life, chaos and turbulence. In this paper, we first discuss the dynamic system…
Intrinsic instability of trajectories characterizes chaotic dynamical systems. We report here that trajectories can exhibit a surprisingly high degree of stability, over a very long time, in a chaotic dynamical system. We provide a detailed…
An important theme in modern inverse problems is the reconstruction of time-dependent data from only finitely many measurements. To obtain satisfactory reconstruction results in this setting it is essential to strongly exploit temporal…
Whether a system is to be considered complex or not depends on how one searches for correlations. We propose a general scheme for calculation of entropies in lattice systems that has high flexibility in how correlations are successively…
This work presents numerical evidences that for discrete dynamical systems with one positive Lyapunov exponent the decay of the distance autocorrelation is always related to the Lyapunov exponent. Distinct decay laws for the distance…
We develop novel methods to compute auto-correlation functions, or power spectral densities, for chaotic dynamical systems generated by an inverse method whose starting point is an invariant distribution and a two-form. In general, the…
We introduce new machine-learning techniques for analyzing chaotic dynamical systems. The primary objectives of the study include the development of a new and simple method for calculating the Lyapunov exponent using only two trajectory…
We study the discrete-time, real valued bounded process $\{X_n, n\in {\mathbb N} \}$ defined by a second order recurrence relation $X_{n+2} = \varphi(X_n,X_{n+1})$. We obtain the decay of correlations under analytical hypotheses on $\varphi…
We investigate correlation time numerically in extremal self-organized critical models, namely, the Bak-Sneppen evolution and the Robin Hood dynamics. The (fitness) correlation time is the duration required for the extinction or mutation of…