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Invariant manifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems. The theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems, and for…
Invariant manifolds are important constructs for the quantitative and qualitative understanding of nonlinear phenomena in dynamical systems. In nonlinear damped mechanical systems, for instance, spectral submanifolds have emerged as useful…
The aim of this work is to establish the existence of invariant manifolds in complex systems. Considering trajectory curves integral of multiple time scales dynamical systems of dimension two and three (predator-prey models, neuronal…
The use of 3D printing, or additive manufacturing, has gained significant attention in recent years due to its potential for revolutionizing traditional manufacturing processes. One key challenge in 3D printing is managing energy…
We use a modification of the parameterization method to study invariant manifolds for difference equations. We establish existence, regularity, smooth dependence on parameters and study several singular limits, even if the difference…
Invariant manifolds facilitate the understanding of nonlinear stochastic dynamics. When an invariant manifold is represented approximately by a graph for example, the whole stochastic dynamical system may be reduced or restricted to this…
This is the first in a series of papers where we will derive invariants of three-manifolds and framed knots in them from the geometry of a manifold pseudotriangulation put in some way in a four-dimensional Euclidean space. Thus, the…
We propose a hybrid physics-informed machine learning framework to approximate invariant manifolds (IMs) of discrete-time dynamical systems driven by exogenous autonomous dynamics (exosystems). Such systems appear in applications ranging…
Accurate physical modeling with 3D-printing techniques could lead to new approaches to study structure and dynamics of biological systems complementing computational methods. Computational biology has become an important part of research…
Random invariant manifolds are geometric objects useful for understanding complex dynamics under stochastic influences. Under a nonuniform hyperbolicity or a nonuniform exponential dichotomy condition, the existence of random pseudo-stable…
Airfoil shape design is a classical problem in engineering and manufacturing. In this work, we combine principled physics-based considerations for the shape design problem with modern computational techniques using a data-driven approach.…
Since its foundations, more than one hundred years ago, the field of structural biology has strived to understand and analyze the properties of molecules and their interactions by studying the structure that they take in 3D space. However,…
Invariant manifolds are the skeleton of the chaotic dynamics in Hamiltonian systems. In Celestial Mechanics, for instance, these geometrical structures are applied to a multitude of physical and practical problems, such as to the…
We develop a geometric version of the inverse problem of the calculus of variations for discrete mechanics and constrained discrete mechanics. The geometric approach consists of using suitable Lagrangian and isotropic submanifolds. We also…
We introduce a novel neural network-based computational pipeline as a representation-agnostic slicer for multi-axis 3D printing. This advanced slicer can work on models with diverse representations and intricate topology. The approach…
Three-dimensional (3D) printed preoperative planning models serve a critical role in the success of many medical procedures. However, many of these models do not portray the patient's complete anatomy due to their monolithic and static…
A new computational technique based on the symbolic description utilizing kneading invariants is proposed and verified for explorations of dynamical and parametric chaos in a few exemplary systems with the Lorenz attractor. The technique…
The dynamics of plasmas are governed by a set of non-linear differential equations which remain challenging to solve directly for large 2D and 3D problems. Here we investigate how tensor networks could be applied to plasmas described by the…
Predicting the evolution of systems that exhibit spatio-temporal dynamics in response to external stimuli is a key enabling technology fostering scientific innovation. Traditional equations-based approaches leverage first principles to…
Three-dimensional (3D) printing is a powerful development tool both in industry, as well as in biomedical research. Additive-lathe 3D printing is an emerging sub-class of 3D printing whereby material is layered outward from the surface of a…