Related papers: Space-split algorithm for sensitivity analysis of …
We present a computable reformulation of Ruelle's linear response formula for chaotic systems. The new formula, called Space-Split Sensitivity or S3, achieves an error convergence of the order ${\cal O}(1/\sqrt{N})$ using $N$ phase points.…
This paper presents the space-split sensitivity or the S3 algorithm to transform Ruelle's linear response formula into a well-conditioned ergodic-averaging computation. We prove a decomposition of Ruelle's formula that is differentiable on…
Ruelle gave a formula for linear response of transitive Anosov diffeomorphisms. Recently, practically computable realizations of Ruelle's formula have emerged that potentially enable sensitivity analysis of certain high-dimensional chaotic…
It is well-known that linearized perturbation methods for sensitivity analysis, such as tangent or adjoint equation-based, finite difference and automatic differentiation are not suitable for turbulent flows. The reason is that turbulent…
Parametric derivatives of statistics are highly desired quantities in prediction, design optimization and uncertainty quantification. In the presence of chaos, the rigorous computation of these quantities is certainly possible, but…
Chaotic dynamical systems such as turbulent flows are characterized by an exponential divergence of infinitesimal perturbations to initial conditions. Therefore, conventional adjoint/tangent sensitivity analysis methods that are successful…
In this work, we investigate scale invariance in the temporal evolution and chaotic regime of discrete dynamical systems. By exploiting the close interrelation between scaling and inversion transformations, we formulate scale symmetry in…
Many nonlinear dynamical systems can be written as Lure systems, which are described by a linear time-invariant system interconnected with a diagonal static sector-bounded nonlinearity. Sufficient conditions are derived for the global…
This paper presents new sufficient conditions for convergence and asymptotic or exponential stability of a stochastic discrete-time system, under which the constructed Lyapunov function always decreases in expectation along the system's…
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…
In the study of chaotic behaviour of systems of many hard spheres, Lyapunov exponents of small absolute value exhibit interesting characteristics leading to speculations about connections to non-equilibrium statistical mechanics. Analytical…
Low-dimensional chaotic systems such as the Lorenz-63 model are commonly used to benchmark system-agnostic methods for learning dynamics from data. Here we show that learning from noise-free observations in such systems can be achieved up…
The computation of the entire Lyapunov spectrum for extended dynamical systems is a very time consuming task. If the system is in a chaotic spatio-temporal regime it is possible to approximately reconstruct the Lyapunov spectrum from the…
Complex chaotic dynamics, seen in natural and industrial systems like turbulent flows and weather patterns, often span vast spatial domains with interactions across scales. Accurately capturing these features requires a high-dimensional…
We develop the no-propagate algorithm for sampling the linear response of random dynamical systems, which are non-uniform hyperbolic deterministic systems perturbed by noise with smooth density. We first derive a Monte-Carlo type formula…
This paper develops a new approach to the estimation of the degree of boundedness or stability of multidimensional nonlinear systems with time-dependent nonperiodic coefficients-an essential task in various engineering and natural science…
The Ruelle resonances of a dynamical system are spectral data describing the precise asymptotics of correlations. We classify them completely for a class of chaotic two-dimensional maps, the linear pseudo-Anosov maps, in terms of the action…
Finding and sampling rare trajectories in dynamical systems is a difficult computational task underlying numerous problems and applications. In this paper we show how to construct Metropolis- Hastings Monte Carlo methods that can…
Discrete numerical methods with finite time-steps represent a practical technique to solve initial-value problems involving nonlinear differential equations. These methods seem particularly useful to the study of chaos since no analytical…
An assumption of smooth response to small parameter changes, of statistics or long-time averages of a chaotic system, is generally made in the field of sensitivity analysis, and the parametric derivatives of statistical quantities are…