My research has two main ingredients: Multi-timescale dynamical systems theory and control theory. I am further interested in their applications to, for example, engineering, biochemistry, neuroscience, etc.

Many complex phenomena have intrinsically many different timescales. A usual system’s approach to understand them is to separate the processes at each timescale, hoping that information of each subprocess will provide enough insight on the overall dynamics. Sometimes this approach works well, especially when the timescale separation is, in some sense, global. There are instances, however, where such an approach is not suitable. In this latter situation, the timescale separation does not persist across the phase-space and trajectories that initially evolve slowly can suddenly jump to another state. The goal of my research is to understand systems with the aforementioned behaviour and to control them. Most of the following articles show work in that direction.

Preprints, work in progress and submitted works:
  • A survey on the blow-up method for fast-slow systems [arXiv],
    HJK and Christian Kuehn.
  • Geometric desingularization of a biochemical oscillator,
    Hadi Tafgvafarad, HJK, Peter Szmolyan and Ming Cao.
    To be submitted
Journals (peer-reviewed):

*CDC-XX = The contents of this paper were also selected by CDC XX Program Committee for presentation at the Conference

  1. Stabilization of a class of slow-fast control systems at non-hyperbolic points [Preprint]
    HJK, Jacquelien M.A. Scherpen, and D. del Puerto-Flores
    Automatica, vol. 99, pp. 13-21, 2019.
  2. Improving the region of attraction of a non-hyperbolic point in slow-fast systems with one fast direction [Preprint]
    HJK and Jacquelien M.A. Scherpen
    IEEE Control Systems Letters, vol. 2, no. 2, pp. 403-408, 2018. *CDC-57
  3. Parameter-robustness analysis for a biochemical oscillator model describing the social-behavior transition phase of myxobacteria
    Hadi Tafgvafarad, HJK, and Ming Cao
    Proceedings of the Royal Society A, 2018.
  4. Model order reduction and composite control for a class of slow-fast systems around a non-hyperbolic point [Preprint]
    HJK and Jacquelien M.A. Scherpen
    The IEEE Control Systems Letters, 2017. *CDC-56
  5. Limit sets within curves where trajectories converge to
    Pouria Ramazi, HJK and Ming Cao
    Applied Mathematics Letters, 2017.
  6. Vibrational Stabilization by Reshaping Arnold Tongues: A Numerical Approach
    Joaquín Collado, HJK
    Applied Mathematics, 2016.
  7. Analysis of a slow fast system near a cusp singularity [arXiv]
    HJK, Henk W. Broer and R. Roussarie
    Journal of Differential Equations, 2016.
  8. Formal normal form of Ak slow-fast systems [arXiv]
    Comptes Rendus Mathematique, 2015.
  9. Bifurcations of a non-gravitational interaction problem
    X. Liu and HJK
    AMC, 2015.
  10. Polynomial normal forms of Constrained Differential Equations with three parameters [arXiv]
    HJK and Henk W. Broer
    Journal of Differential Equations, 2014.
Conference proceedings (peer-reviewed):
  1. Nonlinear adaptive stabilization of a class of planar slow-fast systems at a non-hyperbolic point
    HJK, Jacquelien M.A. Scherpen and D. del Puerto-Flores
    ACC 2017.
  2. Stabilization of a planar slow-fast system at a non hyperbolic point
    HJK and Jacquelien M.A. Scherpen
    MTNS 2016.
  3. Model order reduction of a flexible-joint robot: a port-Hamiltonian approach
    HJK, M. Munoz-Arias and Jacquelien M.A. Scherpen
    NOLCOS 2016.