Matthew Kvalheim     Research

Research


I like to think about interesting mathematical problems arising from those in the physical universe.

My research interests are nonlinear systems in general, and in particular dynamics and control, geometry and topology, and stochastic processes.

Specific topics related to my work: the feedback stabilization problem in control theory, Koopman operator theory, data-driven algorithms, closed one-forms in dynamics, Morse/Conley theory, stochastic processes and topology, invariant manifolds, bifurcations, and geometric mechanics.


Published or accepted journal papers:

  • Necessary conditions for feedback stabilization and safety, with Daniel E. Koditschek, Journal of Geometric Mechanics, Accepted (2022). (article, arXiv)
  • Existence and uniqueness of global Koopman eigenfunctions for stable fixed points and periodic orbits, with Shai Revzen, Physica D, 425 (2021), pp. 132959. (article, arXiv)
  • Families of periodic orbits: closed 1-forms and global continuability, with Anthony M. Bloch, Journal of Differential Equations, 285 (2021), pp. 211-257. (article, arXiv)
  • Conley’s fundamental theorem for a class of hybrid systems, with Paul Gustafson and Daniel E. Koditschek, SIAM Journal on Applied Dynamical systems, 20.2 (2021), pp. 784-825. (article, arXiv)
  • Gait modeling and optimization for the perturbed Stokes regime, with Brian Bittner and Shai Revzen, Nonlinear Dynamics, 97.4 (2019), pp. 2249-2270. (article, arXiv)
  • Global linearization and fiber bundle structure of invariant manifolds, with Jaap Eldering and Shai Revzen, Nonlinearity, 31.9 (2018), pp. 4202-4245. (article, arXiv)

Submitted journal papers and selected other preprints:

  • Obstructions to asymptotic stabilization, (2022). (arxiv)
  • A generalization of the Hopf degree theorem, (2022). (arxiv)
  • Estimating phase from observed trajectories using the temporal 1-form, with Simon Wilshin, Clayton Scott, and Shai Revzen, (2022). (arxiv)
  • Phase response curves and the role of coordinates, with Simon Wilshin and Shai Revzen, (2021). (arXiv)
  • A pasting lemma for Lipschitz functions, with Paul Gustafson and Samuel A. Burden, (2021). (arXiv)
  • Poincaré-Hopf theorem for hybrid systems, (2021). (arXiv)
  • Flux in tilted potential systems: negative resistance and persistence, with Yuliy Baryshnikov, (2021). (arXiv)

Peer-reviewed conference papers:

  • Generic properties of Koopman eigenfunctions for stable fixed points and periodic orbits, with David Hong and Shai Revzen, IFAC-PapersOnline, 54.9 (2021), pp. 267-272. (article, arXiv)
  • Data-driven models of legged locomotion, with Shai Revzen, SPIE Defense + Security, International Society for Optics and Photonics (2015). (article)

Book chapters:

  • Templates and anchors, with Shai Revzen, Bioinspired legged locomotion, Ch. 3.2, Butterworth-Heinemann, Elsevier, Oxford (2017). (link)
  • Locomotion as an oscillator, with Shai Revzen, Bioinspired legged locomotion, Ch. 3.5, Butterworth-Heinemann, Elsevier, Oxford (2017). (link)

PhD thesis:

Aspects of invariant manifold theory and applications, 207 pages, December 2018. (link) Written under the supervision of Professor Shai Revzen.


Selected talks:

  • M. Kvalheim. “Large deviations, persistent homology, and Brownian conductors with negative resistance.” Invited talk, Cornell University Probability Seminar (2022). Online. (slides minus movies)
  • M. Kvalheim. “When can hybrid systems operate safely?” Invited talk, University of Michigan Control Seminar (2022). Online. (video, slides)
  • M. Kvalheim. “Flux in small noise dynamics: persistence and negative resistance.” Invited talk, Brown University, Division of Applied Mathematics (2022). Online.
  • M. Kvalheim. “Flux in small noise dynamics: negative resistance and persistence.” Invited talk, University of Illinois Urbana-Champaign Mathematical Biology Seminar (2021). Online.
  • S. Burden and M. Kvalheim. “When and how are hybrid dynamical systems conjugate to their classical quotients?” Invited talk, SIAM Conference on Control and its Applications (2021). Online.
  • D. Koditschek, P. Gustafson, and M. Kvalheim. “Toward a physically grounded type theory for robot task composition.” Invited talk, ICRA workshop on Compositional Robotics: Mathematics and Tools (2021). Online.
  • M. Kvalheim and Y. Baryshnikov. “Small noise dynamics, persistence, and negative resistance.” Invited talk, SIAM Conference on Applications of Dynamical Systems (2021). Online.
  • M. Kvalheim. “Towards a Conley theory for hybrid dynamical systems.” Invited talk, TRIPODS/DATA-INSPIRE Workshop on Dynamics, Topology, and Robotic Control (2021). Online.
  • M. Kvalheim. “Hierarchical Composition via Collapse of Dimension in Dynamical Systems.” Invited talk, ETH Zürich Autonomy Talks (2021). Online.
  • A. M. Bloch and M. Kvalheim. “Geometry and dynamics of circulant systems.” Invited talk, AMS sectional meeting, special session on Geometric Dynamics (2020). Online.
  • M. Kvalheim, P. Gustafson, D. Koditschek. “Toward a task planning theory for robot hybrid dynamics.” Invited talk, BIRS-CMO workshop on Topological Complexity and Motion Planning (2020). Online.
  • M. Kvalheim, S. Wilshin, C. Scott, and S. Revzen. “Isochrons from short, noisy data. Invited talk, Dynamics Days Europe (2020). Online.
  • S. Revzen, G. Council, and M. Kvalheim. “Is legged locomotion almost smooth?” Dynamic Walking (2019). Calgary, AB, Canada.
  • A. M. Bloch and M. Kvalheim. “Dynamics of circulant systems of ODE’s.” Invited talk, SIAM Conference on Applications of Dynamical Systems (2019). Snowbird, UT.
  • M. Kvalheim and S. Revzen. “Reduced-order models for locomotion in the perturbed Stokes regime.” Invited talk, SIAM Conference on Applications of Dynamical Systems (2019). Snowbird, UT.
  • S. Revzen and M. Kvalheim. “Hybrid oscillators: phase and amplitude in a class of non-smooth systems.” Invited talk, SIAM Conference on Applications of Dynamical Systems (2019). Snowbird, UT.
  • M. Kvalheim. “Oscillators, asymptotic phase, and reduction of dynamical systems.” Invited talk, University of Michigan student Applied and Interdisciplinary Mathematics seminar (2018). Ann Arbor, MI.
  • M. Kvalheim and S. Revzen. “Testing an extended ‘posture principle’.” SICB (2018). San Francisco, CA.
  • M. Kvalheim. “Introduction to homology: towards topological data analysis.” Invited talk, University of Michigan Student Signal Processing Seminar (2015). Ann Arbor, MI.

Selected posters:

  • M. Kvalheim and S. Revzen. “Principal Koopman eigenfunctions for nonlinear and nonsmooth systems” UCLA IPAM workshop on ‘Operator theoretic methods in dynamic data analysis and control’ (2019). Los Angeles, CA.
  • M. Kvalheim, J. Eldering, and S. Revzen. “Global linearization and fiber bundle structure of invariant manifolds.” AIMS (2018). Taipei, Taiwan.
  • M. Kvalheim and S. Revzen. “Templates and Anchors: a review of notions of model reduction.” Dynamic Walking (2018). Pensacola, FL.
  • M. Kvalheim and S. Revzen. “Asymptotic phase, model reduction, and control of Templates & Anchors.” OSU MBI workshop on ‘Sensorimotor control of animals and robots’ (2017). Columbus, OH.
  • M. Kvalheim and S. Revzen. “A dynamical systems perspective on Templates & Anchors: some general methods for anchoring templates.” IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Robotics-inspired Biology Workshop (2017). Vancouver, BC.