29 citations · 52 across the 16 of their papers we have counts for
26 papers
The Maxwell Conjecture is False
Philip Arathoon, Gavin Ball, Matthew D. Kvalheim
We exhibit a configuration of five point charges in Euclidean space whose electrostatic potential admits at least 24 critical points all of which are non-degenerate. Maxwell's conj…
Stabilizability of first-order dynamics in second-order systems
Matthew D. Kvalheim
We study whether second-order systems can be made to behave like prescribed first-order dynamical systems through feedback control. More precisely, we study whether prescribed vect…
Autoencoding Dynamics: Topological Limitations and Capabilities
Matthew D. Kvalheim, Eduardo D. Sontag
Given a "data manifold" and "latent space" , an autoencoder is a pair of continuous maps consisting of an "encoder" $E\colon \mathbb{R}^n\t…
Differential topology of the spaces of asymptotically stable vector fields and Lyapunov functions
Matthew D. Kvalheim
We study the topology of the space of all smooth asymptotically stable vector fields on , as well as the space of all proper smooth Lyapunov functions for such vector…
Global linearization of asymptotically stable systems without hyperbolicity
Matthew D. Kvalheim, Eduardo D. Sontag
We give a proof of an extension of the Hartman-Grobman theorem to nonhyperbolic but asymptotically stable equilibria of vector fields. Moreover, the linearizing topological conjuga…
Relationships Between Necessary Conditions for Feedback Stabilizability
Matthew D. Kvalheim
The author's extensions of Brockett's and Coron's necessary conditions for stabilizability are shown to be independent in the fiber bundle picture of control, but the latter is sho…
Why should autoencoders work?
Matthew D. Kvalheim, Eduardo D. Sontag
Deep neural network autoencoders are routinely used computationally for model reduction. They allow recognizing the intrinsic dimension of data that lie in a -dimensional subset…
Koopman Embedding and Super-Linearization Counterexamples with Isolated Equilibria
Philip Arathoon, Matthew D. Kvalheim
A frequently repeated claim in the "applied Koopman operator theory'' literature is that a dynamical system with multiple isolated equilibria cannot be linearized in the sense of a…
Linearizability of flows by embeddings
Matthew D. Kvalheim, Philip Arathoon
We consider the problem of determining the class of continuous-time dynamical systems that can be globally linearized in the sense of admitting an embedding into a linear system on…
A Compositional Approach to Certifying the Almost Global Asymptotic Stability of Cascade Systems
Jake Welde, Matthew D. Kvalheim, Vijay Kumar
In this work, we give sufficient conditions for the almost global asymptotic stability of a cascade in which the subsystems are only almost globally asymptotically stable. The resu…
The Role of Symmetry in Constructing Geometric Flat Outputs for Free-Flying Robotic Systems
Jake Welde, Matthew D. Kvalheim, Vijay Kumar
Mechanical systems naturally evolve on principal bundles describing their inherent symmetries. The ensuing factorization of the configuration manifold into a symmetry group and an…
Obstructions to asymptotic stabilization
Matthew D. Kvalheim
Necessary conditions for asymptotic stability and stabilizability of subsets for dynamical and control systems are obtained. The main necessary condition is homotopical and is in t…
Estimating Phase from Observed Trajectories Using the Temporal 1-Form
Simon Wilshin, Matthew D. Kvalheim, Clayton Scott +1
Oscillators are ubiquitous in nature, and usually associated with the existence of an asymptotic phase that governs the long-term dynamics of the oscillator. % We show that asympto…
A generalization of the Hopf degree theorem
Matthew D. Kvalheim
The Hopf theorem states that homotopy classes of continuous maps from a closed connected oriented smooth -manifold to the -sphere are classified by their degree. Such a m…
Phase Response Curves and the Role of Coordinates
Simon Wilshin, Matthew D. Kvalheim, Shai Revzen
The "Phase Response Curve" (PRC) is a common tool used to analyze phase resetting in the natural sciences. We make the observation that the PRC with respect to a coordinate $y\in\m…
A pasting lemma for Lipschitz functions
Matthew D. Kvalheim, Paul Gustafson, Samuel A. Burden
We give a necessary and sufficient condition ensuring that any function which is separately Lipschitz on two fixed compact sets is Lipschitz on their union.
Poincaré-Hopf theorem for hybrid systems
Matthew D. Kvalheim
A generalization of the Poincaré-Hopf index theorem applicable to hybrid dynamical systems is obtained. For the hybrid systems considered, guard sets are not assumed to be smooth;…
Flux in tilted potential systems: negative resistance and persistence
Yuliy Baryshnikov, Matthew D. Kvalheim
Many real-world systems are well-modeled by Brownian particles subject to gradient dynamics plus noise arising, e.g., from the thermal fluctuations of a heat bath. Of central impor…
Necessary conditions for feedback stabilization and safety
Matthew D. Kvalheim, Daniel E. Koditschek
Brockett's necessary condition yields a test to determine whether a system can be made to stabilize about some operating point via continuous, purely state-dependent feedback. For…
Generic Properties of Koopman Eigenfunctions for Stable Fixed Points and Periodic Orbits
Matthew D. Kvalheim, David Hong, Shai Revzen
Our recent work established existence and uniqueness results for globally defined linearizing semiconjugacies for flows having a gl…
Conley's fundamental theorem for a class of hybrid systems
Matthew D. Kvalheim, Paul Gustafson, Daniel E. Koditschek
We establish versions of Conley's (i) fundamental theorem and (ii) decomposition theorem for a broad class of hybrid dynamical systems. The hybrid version of (i) asserts that a glo…
Existence and uniqueness of global Koopman eigenfunctions for stable fixed points and periodic orbits
Matthew D. Kvalheim, Shai Revzen
We consider dynamical systems having an attracting hyperbolic fixed point or periodic orbit and prove existence and uniqueness results for (actually $C^{k,α}_{\text{loc…
Gait modeling and optimization for the perturbed Stokes regime
Matthew D. Kvalheim, Brian Bittner, Shai Revzen
Many forms of locomotion, both natural and artificial, are dominated by viscous friction in the sense that without power expenditure they quickly come to a standstill. From geometr…
Families of periodic orbits: closed 1-forms and global continuability
Matthew D. Kvalheim, Anthony M. Bloch
We investigate global continuation of periodic orbits of a differential equation depending on a parameter, assuming that a closed 1-form satisfying certain properties exists. We be…