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For one dimensional maps the trajectory scaling functions is invariant under coordinate transformations and can be used to compute any ergodic average. It is the most stringent test between theory and experiment, but so far it has proven…

Due to existence of periodic windows, chaotic systems undergo numerous bifurcations as system parameters vary, rendering it hard to employ an analytic continuation, which constitutes a major obstacle for its effective analysis or…

For appropriately chosen weights, temporal averages in chaotic systems can be approximated as a weighted sum of averages over reference states, such as unstable periodic orbits. Under strict assumptions, such as completeness of the orbit…

Periodic orbit theory allows calculations of long time properties of chaotic systems from traces, dynamical zeta functions and spectral determinants of deterministic evolution operators, which are in turn evaluated in terms of periodic…

We show that in a broad class of processes that show a $1/f^{\alpha}$ spectrum, the power also explicitly depends on the characteristic time scale. Despite an enormous amount of work, this generic behavior remains so far overlooked and…

Periodic orbits and cycles, respectively, play a significant role in discrete- and continuous-time dynamical systems (i.e. maps and flows). To succinctly describe their shifts when the system is applied perturbation, the notions of…

Periodic orbit theory is an effective tool for the analysis of classical and quantum chaotic systems. In this paper we extend this approach to stochastic systems, in particular to mappings with additive noise. The theory is cast in the…

Conventional notions of generalization often fail to describe the ability of learned models to capture meaningful information from dynamical data. A neural network that learns complex dynamics with a small test error may still fail to…

Many fluctuating systems consist of macroscopic structures in addition to noisy signals. Thus, for this class of fluctuating systems, the scaling behaviors are very complicated. Such phenomena are quite commonly observed in Nature, ranging…

A central challenge in neuroscience is understanding how neural system implements computation through its dynamics. We propose a nonlinear time series model aimed at characterizing interpretable dynamics from neural trajectories. Our model…

We show that scaling arguments are very useful to analyze the dynamics of periodically modulated noisy systems. Information about the behavior of the relevant quantities, such as the signal-to-noise ratio, upon variations of the noise…

We present a general mathematical framework for trajectory stratification for simulating rare events. Trajectory stratification involves decomposing trajectories of the underlying process into fragments limited to restricted regions of…

A physical (e.g. astrophysical, geophysical, meteorological etc.) data may appear as an output of an experiment or it may contain some sociological, economic or biological information. Whatever be the source of a time series data some…

This paper proposes stochastic models for the analysis of ocean surface trajectories obtained from freely-drifting satellite-tracked instruments. The proposed time series models are used to summarise large multivariate datasets and infer…

We present an optimization process to estimate parameters in systems of ordinary differential equations from chaotic time series. The optimization technique is based on a variational approach, and numerical studies on noisy time series…

Functional data are typically modeled as sample paths of smooth stochastic processes in order to mitigate the fact that they are often observed discretely and noisily, occasionally irregularly and sparsely. The smoothness assumption is…

Rare trajectories of stochastic systems are important to understand -- because of their potential impact. However, their properties are by definition difficult to sample directly. Population dynamics provides a numerical tool allowing their…

Unstable periodic orbits scar wave functions in chaotic systems. This also influences the associated spectra, that follow the otherwise universal Porter--Thomas intensity distribution. We show here how this deviation extend to other longer…

The extraction of signals from noise is a common problem in all areas of science and engineering. A particularly useful version is that of forecasting: determining a causal filter that estimates a future value of a hidden process from past…

It is a big challenge in the analysis of experimental data to disentangle the unavoidable measurement noise from the intrinsic dynamical noise. Here we present a general operational method to extract measurement noise from stochastic time…