Related papers: Trace formulas for stochastic evolution operators:…
Periodic orbits for the classical $\phi^4$ theory on the one dimensional lattice are systematically constructed by extending the normal modes of the harmonic theory, for periodic, fixed and free boundary conditions. Through the process, we…
Long periodic orbits constitute a serious drawback in Gutzwiller's theory of chaotic systems, and then it would be desirable that other classical invariants, not suffering from the same problem, could be used in the quantization of such…
The main purpose of these lectures is to discuss briefly recent methods of calculation of statistical properties of quantum eigenvalues for chaotic systems based on semi-classical trace formulas. Under the assumption that periodic orbit…
We consider contractive systems whose trajectories evolve on a compact and convex state-space. It is well-known that if the time-varying vector field of the system is periodic then the system admits a unique globally asymptotically stable…
Numerical stochastic perturbation theory is a powerful tool for estimating high-order perturbative expansions in lattice field theory. The standard algorithms based on the Langevin equation, however, suffer from several limitations which in…
Turbulent dynamical systems characterized by both a high-dimensional phase space and a large number of instabilities are ubiquitous among many complex systems in science and engineering. The existence of a strange attractor in the turbulent…
We have developed a semiclassical theory of short periodic orbits to obtain all quantum information of a bounded chaotic Hamiltonian system. If T_1 is the period of the shortest periodic orbit, T_2 the period of the next one and so on, the…
We show that a recently proposed numerical technique for the calculation of unstable periodic orbits in chaotic attractors is capable of finding the least unstable periodic orbits of any given order. This is achieved by introducing a…
We propose a novel framework for approximating the statistical properties of turbulent flows by combining variational methods for the search of unstable periodic orbits with resolvent analysis for dimensionality reduction. Traditional…
We study the metastability properties of a simple prototypical bistable system using the formalism of the Koopman operator. Instead of studying noise-induced transitions by following the trajectories of the system, we track them by studying…
Level fluctuations in quantum system have been used to characterize quantum chaos using random matrix models. Recently time series methods were used to relate level fluctuations to the classical dynamics in the regular and chaotic limit. In…
A stochastic subgrid-scale parameterization based on the Ruelle's response theory and proposed in Wouters and Lucarini [2012] is tested in the context of a low-order coupled ocean-atmosphere model for which a part of the atmospheric modes…
This paper introduces two new algorithms to accurately estimate the process noise covariance of a discrete-time Kalman filter online for robust orbit determination in the presence of dynamics model uncertainties. Common orbit determination…
Stochastic systems are used to model a variety of phenomena in which noise plays an essential role. In these models, one potential goal is to determine if noise can induce transitions between states, and if so, to calculate the most…
We consider the problem of mining signal temporal logical requirements from a dataset of regular (good) and anomalous (bad) trajectories of a dynamical system. We assume the training set to be labeled by human experts and that we have…
The quantum baker's map is the quantization of a simple classically chaotic system, and has many generic features that have been studied over the last few years. While there exists a semiclassical theory of this map, a more rigorous study…
This paper presents a new boundary-value problem formulation for quantifying uncertainty induced by the presence of small Brownian noise near transversally stable periodic orbits (limit cycles) and quasiperiodic invariant tori of the…
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…
Oscillations in the probability density of quantum transitions of the eigenstates of a chaotic Hamiltonian within classically narrow energy ranges have been shown to depend on closed compound orbits. These are formed by a pair of orbit…
This work investigates a three-dimensional slow-fast stochastic system with quadratic nonlinearity and additive noise, inspired by fluid dynamics. The deterministic counterpart exhibits a periodic orbit and a slow manifold. We demonstrate…