Related papers: Validated Computation of Heteroclinic Sets
Understanding the topological structure of phase space for dynamical systems in higher dimensions is critical for numerous applications, including the computation of chemical reaction rates and transport of objects in the solar system. Many…
We investigate error bounds for numerical solutions of divergence structure linear elliptic PDEs on compact manifolds without boundary. Our focus is on a class of monotone finite difference approximations, which provide a strong form of…
Homoclinic snaking is a widespread phenomenon observed in many pattern-forming systems. Demonstrating its occurrence in non-perturbative regimes has proven difficult, although a forcing theory has been developed based on the identification…
Establishing Lipschitz stability estimates is crucial for ensuring the mathematical robustness of neural network (NN) approximations in machine learning (ML)-based parameter estimation, particularly in physics-informed settings. In this…
This paper presents new methods for set-valued state estimation of nonlinear discrete-time systems with unknown-but-bounded uncertainties. A single time step involves propagating an enclosure of the system states through the nonlinear…
We revisit the problem of computing (robust) controlled invariant sets for discrete-time linear systems. Departing from previous approaches, we consider implicit, rather than explicit, representations for controlled invariant sets.…
In this contribution, the optimal stabilization problem of periodic orbits is studied via invariant manifold theory and symplectic geometry. The stable manifold theory for the optimal point stabilization case is generalized to the case of…
In this work, we only use data on the unstable manifold to locate the partition boundaries by checking folding points at different levels, which practically coincide with homoclinic tangencies (HTs). The method is then applied to the…
Let H be a 3-uniform hypergraph with N vertices. A tight Hamilton cycle C \subset H is a collection of N edges for which there is an ordering of the vertices v_1, ..., v_N such that every triple of consecutive vertices {v_i, v_{i+1},…
Compact sets in constructive mathematics capture our intuition of what computable subsets of the plane (or any other complete metric space) ought to be. A good representation of compact sets provides an efficient means of creating and…
To verify the correct operation of systems, engineers need to determine the set of configurations of a dynamical model that are able to safely reach a specified configuration under a control law. Unfortunately, constructing models for…
Given two arbitrary closed sets in Euclidean space, a simple transversality condition guarantees that the method of alternating projections converges locally, at linear rate, to a point in the intersection. Exact projection onto nonconvex…
We introduce a multi-fidelity estimator of covariance matrices that employs the log-Euclidean geometry of the symmetric positive-definite manifold. The estimator fuses samples from a hierarchy of data sources of differing fidelities and…
By means of an updated renormalization method, we construct asymptotic expansions for unstable manifolds of hyperbolic fixed points in the double-well map and the dissipative H\'enon map, both of which exhibit the strong homoclinic chaos.…
Stable fold maps are fundamental tools in studying a generalized theory of the theory of Morse functions on smooth manifolds and its application to geometry of the manifolds. It is important to construct explicit fold maps systematically to…
We present a method for the steady state optimization of nonlinear delay differential equations. The method ensures stability and robustness, where a system is called robust if it remains stable despite uncertain parameters. Essentially, we…
The intersection of numerical analysis and machine learning, particularly in the domain of Neural ODEs and Physics-Informed Neural Networks (PINNs), relies heavily on discrete approximations of continuous flows. However, in stiff systems,…
We consider a non-autonomous dynamical system formed by coupling two piecewise-smooth systems in $\RR^2$ through a non-autonomous periodic perturbation. We study the dynamics around one of the heteroclinic orbits of one of the…
Accurately modeling and verifying the correct operation of systems interacting in dynamic environments is challenging. By leveraging parametric uncertainty within the model description, one can relax the requirement to describe exactly the…
This paper develops a new approach to small time local attainability of smooth manifolds of any dimension, possibly with boundary and to prove H\"older continuity of the minimum time function. We give explicit pointwise conditions of any…