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Recently, there has been great interest in connections between continuous-time dynamical systems and optimization methods, notably in the context of accelerated methods for smooth and unconstrained problems. In this paper we extend this…
Small random perturbations may have a dramatic impact on the long time evolution of dynamical systems, and large deviation theory is often the right theoretical framework to understand these effects. At the core of the theory lies the…
Recently, there has been an increasing interest in using tools from dynamical systems to analyze the behavior of simple optimization algorithms such as gradient descent and accelerated variants. This paper strengthens such connections by…
Averaging is an important method to extract effective macroscopic dynamics from complex systems with slow modes and fast modes. This article derives an averaged equation for a class of stochastic partial differential equations without any…
When a dynamical system is subject to a periodic perturbation, the averaging method can be applied to obtain an autonomous leading order "guiding system", placing the time dependence at higher orders. Recent research focused on…
A physical pendulum with variable point of suspension (and, as an outcome, variable inertia moment) is experimentally analysed. In particular, the period of the small oscillations as a function of position of the suspension point is…
A version of the Dynamical Systems Method (DSM) for solving ill-posed nonlinear equations with monotone operators in a Hilbert space is studied in this paper. An a posteriori stopping rule, based on a discrepancy-type principle is proposed…
We introduce a novel technique to find the asymptotic time behaviour of deterministic systems exhibiting anomalous diffusion. The procedure is tested for various classes of simple but physically relevant 1-D maps and possible relevance of…
Hydrodynamic turbulence is studied as a constrained system from the point of view of metafluid dynamics. We present a Lagrangian description for this new theory of turbulence inspired from the analogy with electromagnetism. Consequently it…
Discontinuous dynamical systems with grazing solutions are discussed. The group property, continuation of solutions, continuity and smoothness of motions are thoroughly analyzed. A variational system around a grazing solution which depends…
The use of complex networks for time series analysis has recently shown to be useful as a tool for detecting dynamic state changes for a wide variety of applications. In this work, we implement the commonly used ordinal partition network to…
This paper is a preliminary work to address the problem of dynamical systems with parameters varying in time. An idea to predict their behaviour is proposed. These systems are called \emph{transient systems}, and are distinguished from…
We introduce a new analytical method, which allows to find out chaotic dynamics in non-smooth dynamical systems. A simple mechanical system consisting of a mass and a dry friction element is considered as an example. The corresponding…
This work is concerned with the identification problem for what we call the perturbation term or error term in a parabolic partial differential equation, through its approximate periodic solutions. The observation is made over a subregion…
A simple non-autonomous scalar differential equation with delay, exponential decay, nonlinear negative feedback and a periodic multiplicative coefficient is considered. It is shown that stable slowly oscillating periodic solutions with the…
Many models of physics beyond the Standard Model include towers of particles whose masses follow an approximately periodic pattern with little spacing between them. These resonances might be too weak to detect individually, but could be…
We discuss the equation of motion of the driven pendulum and generalize it to arbitrary driving angle. The pendulum will oscillate about a stable angle other than straight down if the drive amplitude and frequency are large enough for a…
The small angle approximation often fails to explain experimental data, does not even predict if a plane pendulum's period increases or decreases with increasing amplitude. We make a perturbation ansatz for the Conserved Energy Surfaces of…
We review a dynamical approach to non-equilibrium MD (D-NEMD). We show how, using a proper simulation setup, is possible to treat interesting cases in which the initial condition is a stationary non-equilibrium state produced by a suitable…
We consider a discrete time dynamic system described by a difference equation with periodic coefficients and with additive stochastic noise. We investigate the possibility of the periodicity for the solution. In particular, we found…