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Multiscale phenomena that evolve on multiple distinct timescales are prevalent throughout the sciences. It is often the case that the governing equations of the persistent and approximately periodic fast scales are prescribed, while the…
The value of unknown parameters of multibody systems is crucial for prediction, monitoring, and control, sometimes estimated using a biased physics-based model leading to incorrect outcomes. Discovering motion equations of multibody systems…
Noise-induced transitions between metastable fixed points in systems evolving on multiple time scales are analyzed in situations where the time scale separation gives rise to a slow manifold with bifurcation. This analysis is performed…
We introduce a simple method to estimate the system parameters in continuous dynamical systems from the time series. In this method, we construct a modified system by introducing some constants (controlling constants) into the given…
The limiting stability of invariant probability measures of time homogeneous transition semigroups for autonomous stochastic systems has been extensively discussed in the literature. In this paper we initially initiate a program to study…
In this paper, we consider asymptotic behaviors of multiscale multivalued stochastic systems with small noises. First of all, for general, fully coupled systems for multivalued stochastic differential equations of slow and fast motions with…
Finding a reduction of complex, high-dimensional dynamics to its essential, low-dimensional "heart" remains a challenging yet necessary prerequisite for designing efficient numerical approaches. Machine learning methods have the potential…
In this paper we study model reduction of linear and bilinear quadratic stochastic control problems with parameter uncertainties. Specifically, we consider slow-fast systems with unknown diffusion coefficient and study the convergence of…
The estimation of static parameters in dynamical systems and control theory has been extensively studied, with significant progress made in estimating varying parameters in specific system types. Suppose, in the general case, we have data…
We construct a novel estimator for the diffusion coefficient of the limiting homogenized equation, when observing the slow dynamics of a multiscale model, in the case when the slow dynamics are of bounded variation. Previous research…
We introduce a simple stochastic system able to generate anomalous diffusion both for position and velocity. The model represents a viable description of the Fermi's acceleration mechanism and it is amenable to analytical treatment through…
Inspired by one--dimensional light--particle systems, the dynamics of a non-Hamiltonian system with long--range forces is investigated. While the molecular dynamics does not reach an equilibrium state, it may be approximated in the…
In a wide range of applications, the stochastic properties of the observed time series change over time. The changes often occur gradually rather than abruptly: the prop- erties are (approximately) constant for some time and then slowly…
A deterministic multiscale toy model is studied in which a chaotic fast subsystem triggers rare transitions between slow regimes, akin to weather or climate regimes. Using homogenization techniques, a reduced stochastic parametrization…
A new concept of {\em an evolution system of measures for stochastic flows} is considered. It corresponds to the notion of an invariant measure for random dynamical systems (or cocycles). The existence of evolution systems of measures for…
A key issue in dimension reduction of dissipative dynamical systems with spectral gaps is the identification of slow invariant manifolds. We present theoretical and numerical results for a variational approach to the problem of computing…
We present a fast method for nonlinear data-driven model reduction of dynamical systems onto their slowest nonresonant spectral submanifolds (SSMs). We use observed data to locate a low-dimensional, attracting slow SSM and compute a…
Estimating and quantifying uncertainty in unknown system parameters from limited data remains a challenging inverse problem in a variety of real-world applications. While many approaches focus on estimating constant parameters, a subset of…
Stochastic processes have found numerous applications in science, as they are broadly used to model a variety of natural phenomena. Due to their intrinsic randomness and uncertainty, they are, however, difficult to characterize. Here, we…
We propose a simple model for sample space reducing (SSR) stochastic process, where the dynamical variable denoting the size of the state space is continuous. In general, one can view the model as a multiplicative stochastic process, with a…