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Non-stationarity is a fundamental challenge in multivariate long-term time series forecasting, often manifested as rapid changes in amplitude and phase. These variations lead to severe distribution shifts and consequently degrade predictive…

Sensitivity analysis is a classical and fundamental tool to evaluate the role of a given parameter in a given system characteristic. Because the phase response curve is a fundamental input--output characteristic of oscillators, we developed…

Intractable phase dynamics often challenge our understanding of complex oscillatory systems, hindering the exploration of synchronisation, chaos, and emergent phenomena across diverse fields. We introduce a novel conceptual framework for…

The parameterization method (PM) provides a broad theoretical and numerical foundation for computing invariant manifolds of dynamical systems. PM implements a change of variables in order to represent trajectories of a system of ordinary…

The article contains an overview over locally stationary processes. At the beginning time varying autoregressive processes are discussed in detail - both as as a deep example and an important class of locally stationary processes. In the…

The content of this contribution is based on the course on numerical analysis techniques for non-linear dynamics. After introducing basic concepts as the visual analysis of trajectories in phase space and the importance of the nature of…

In this paper, we consider the uniform estimate for the oscillatory integral with stationary phase, which was previously studied by Alazard-Burq-Zuily. We significantly reduce the order of required regularity condition on the phase and…

Investigating the spectral properties of the neural covariates that underlie spiking activity is an important problem in systems neuroscience, as it allows to study the role of brain rhythms in cognitive functions. While the spectral…

We propose predictive information, that is information between a long past of duration T and the entire infinitely long future of a time series, as a universal order parameter to study phase transitions in physical systems. It can be used,…

Cyclic spectral analysis is a signal processing technique designed to deal with stochastic signals whose statistics vary periodically with time. Pulsar radio emission is a textbook example of this signal class, known as cyclostationary…

We develop a technique for the multivariate data analysis of perturbed self-sustained oscillators. The approach is based on the reconstruction of the phase dynamics model from observations and on a subsequent exploration of this model. For…

An emerging way to deal with high-dimensional non-euclidean data is to assume that the underlying structure can be captured by a graph. Recently, ideas have begun to emerge related to the analysis of time-varying graph signals. This work…

In this paper, it is proposed a quantization procedure for the one-dimensional harmonic oscillator with time-dependent frequency, time-dependent driven force, and time-dependent dissipative term. The method is based on the construction of…

The purpose of this study is to provide a new methodology of how one can consistently estimate a change-point in time series data. In contrast with previous studies, the suggested methodology employs only the empirical spectral density and…

Phase reduction is a well-established technique used to analyze the timing of oscillations in response to weak external inputs. In the preceding decades, a wide variety of results have been obtained for weakly perturbed oscillators that…

This paper discusses a fast algorithm for analyzing the phases of a multi-tone phase calibration signal. The multi-tone method is superior to its single-tone counterpart. The PCal signal is a wide-band frequency comb derived from an…

The effect of a temporal modulation at three times the critical frequency on a Hopf bifurcation is studied in the framework of amplitude equations. We consider a complex Ginzburg-Landau equation with an extra quadratic term, resulting from…

A method is presented for investigating the periodic signal content of time series in which a number of signals is present, such as arising from the observation of multiperiodic oscillating stars in observational asteroseismology. Standard…

We obtain a formula for the statistical distribution of instantaneous frequencies in the Kuramoto-Sakaguchi model. This work is based on the Kuramoto-Sakaguchi's theory of globally coupled phase oscillators, which we review in full detail…

Circular variables such as phase or orientation have received considerable attention throughout the scientific and engineering communities and have recently been quite prominent in the field of neuroscience. While many analytic techniques…