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Saddle fixed points are the centerpieces of complicated dynamics in a system. The one-dimensional stable and unstable manifolds of these saddle-points are crucial to understanding the dynamics of such systems. While the problem of sketching…
We address a numerical methodology for the computation of coarse-grained stable and unstable manifolds of saddle equilibria/stationary states of multiscale/stochastic systems for which a "good" macroscopic description in the form of…
Probabilistic solvers for ordinary differential equations (ODEs) provide efficient quantification of numerical uncertainty associated with simulation of dynamical systems. Their convergence rates have been established by a growing body of…
In modern power systems, the operating point, at which the demand and supply are balanced, may take different values due to changes in loads and renewable generation levels. Understanding the dynamics of stressed power systems with a range…
This article establishes the foundation for a new theory of invariant/integral manifolds for non-autonomous dynamical systems. Current rigorous support for dimensional reduction modelling of slow-fast systems is limited by the rare events…
In this paper, we first analyze the strong and weak convergence of projective integration methods for multiscale stochastic dynamical systems driven by $\alpha$-stable processes, which are used to estimate the effect that the fast…
The paper focuses on the numerical stability and accuracy of implicit time-domain integration (TDI) methods when applied for the solution of a power system model impacted by time delays. Such a model is generally formulated as a set of…
We introduce a new class of integrators for stiff ODEs as well as SDEs. These integrators are (i) {\it Multiscale}: they are based on flow averaging and so do not fully resolve the fast variables and have a computational cost determined by…
We present a new strategy for solving stiff ODEs with explicit methods. By adaptively taking a small number of stabilizing small explicit time steps when necessary, a stiff ODE system can be stabilized enough to allow for time steps much…
In this document, we deal with the stabilization problem of slow-fast systems (or singularly perturbed Ordinary Differential Equations) at a non-hyperbolic point. The class of systems studied here have the following properties: 1) they have…
Stably inverting a dynamic system model is the foundation of numerous servo designs. Existing inversion techniques have provided accurate model approximations that are often highly effective in feedforward controls. However, when the…
In many applications, the governing PDE to be solved numerically contains a stiff component. When this component is linear, an implicit time stepping method that is unencumbered by stability restrictions is often preferred. On the other…
Stiff systems of ordinary differential equations (ODEs) and sparse training data are common in scientific problems. This paper describes efficient, implicit, vectorized methods for integrating stiff systems of ordinary differential…
We address an iterative procedure that can be used to detect coarse-grained hyperbolic unstable equilibria (saddle points) of microscopic simulators when no equations at the macroscopic level are available. The scheme is based on the…
Many problems in science and engineering require an efficient numerical approximation of integrals or solutions to differential equations. For systems with rapidly changing dynamics, an equidistant discretization is often inadvisable as it…
High-index saddle dynamics (HiSD) is an effective approach for computing saddle points of a prescribed Morse index and constructing solution landscapes for complex nonlinear systems. However, for problems with ill-conditioned Hessians…
This article concerns the stability of a model for mass-spring systems with positive damping and negative stiness. It is well known that when the coefficients are frozen in time the system is unstable. Here we find conditions on the…
Inference for mechanistic models is challenging because of nonlinear interactions between model parameters and a lack of identifiability. Here we focus on a specific class of mechanistic models, which we term stable differential equations.…
A fast convergence in a fixed-time of solutions of nonlinear dynamical systems, for which special requirements are satisfied on the derivative of a quadratic function calculated along the solutions of the system, is proposed. The conditions…
In many applications, one needs to learn a dynamical system from its solutions sampled at a finite number of time points. The learning problem is often formulated as an optimization problem over a chosen function class. However, in the…