26 citations · 61 across the 24 of their papers we have counts for
26 papers · 1 filter
On a tropicalization of planar polynomial ODEs with finitely many structurally stable phase portraits
K. U. Kristiansen, A. H. Sarantaris
Recently, concepts from the emerging field of tropical geometry have been used to identify different scaling regimes in chemical reaction networks where dimension reduction may tak…
Revisiting the Kepler problem with linear drag using the blowup method and normal form theory
Kristian Uldall Kristiansen
In this paper, we revisit the Kepler problem with linear drag. With dissipation, the energy and the angular momentum are both decreasing, but in \cite{margheri2017a} it was shown t…
The dud canard: Existence of strong canard cycles in
Kristian Uldall Kristiansen
In this paper, we provide a rigorous description of the birth of canard limit cycles in slow-fast systems in through the folded saddle-node of type II and the singula…
A dynamical systems approach to WKB-methods: The simple turning point
K. Uldall Kristiansen, P. Szmolyan
In this paper, we revisit the classical linear turning point problem for the second order differential equation with for . Writte…
Geometric singular perturbation analysis of the multiple-timescale Hodgkin-Huxley equations
Panagiotis Kaklamanos, Nikola Popović, Kristian Uldall Kristiansen
We present a novel and global three-dimensional reduction of a non-dimensionalised version of the four-dimensional Hodgkin-Huxley equations [J. Rubin and M. Wechselberger, Giant sq…
Mixed-mode oscillations in coupled FitzHugh-Nagumo oscillators: blowup analysis of cusped singularities
Kristian Uldall Kristiansen, Morten Gram Pedersen
In this paper, we use geometric singular perturbation theory and blowup, as our main technical tool, to study the mixed-mode oscillations (MMOs) that occur in two coupled FitzHugh-…
Blowup analysis of a hysteresis model based upon singular perturbations
Kristian Uldall Kristiansen
In this paper, we provide a geometric analysis of a new hysteresis model that is based upon singular perturbations. Here hysteresis refers to a type of regularization of piecewise…
The Geometry of the Painlevé paradox
Noah Cheesman, S. John Hogan, Kristian Uldall Kristiansen
Painlevé showed that there can be inconsistency and indeterminacy in solutions to the equations of motion of a 2D rigid body moving on a sufficiently rough surface. The study of Pa…
Regularisation of isolated codimension-2 discontinuity sets
Noah Cheesman, Kristian Uldall Kristiansen, S. J. Hogan
Work on standard piecewise-smooth (PWS) dynamical systems, with codimension-1 discontinuity sets, relies on the Filippov framework, which does not always readily generalise to syst…
Singularly Perturbed Boundary-Equilibrium Bifurcations
Samuel Jelbart, Kristian Uldall Kristiansen, Martin Wechselberger
Boundary equilibria bifurcation (BEB) arises in piecewise-smooth systems when an equilibrium collides with a discontinuity set under parameter variation. Singularly perturbed BEB r…
A stiction oscillator under slowly varying forcing: Uncovering small scale phenomena using blowup
Kristian Uldall Kristiansen
In this paper, we consider a mass-spring-friction oscillator with the friction modelled by a regularized stiction model in the limit where the ratio of the natural spring frequency…
Singularly Perturbed Boundary-Focus Bifurcations
Samuel Jelbart, Kristian Uldall Kristiansen, Martin Wechselberger
We consider smooth systems limiting as to piecewise-smooth (PWS) systems with a boundary-focus (BF) bifurcation. After deriving a suitable local normal form, we study the…
On the pitchfork bifurcation of the folded node and other unbounded time-reversible connection problems in
Kristian Uldall Kristiansen
In this paper, we revisit the folded node and the bifurcations of secondary canards at resonances . In particular, we prove for the first time that pitchfork bifurc…
Regularization and geometry of piecewise smooth systems with intersecting discontinuity sets
P. Kaklamanos, K. Uldall Kristiansen
In this work, we study the dynamics of piecewise smooth systems on a codimension-2 transverse intersection of two codimension-1 discontinuity sets. The Filippov convention can be e…
The regularized visible fold revisited
Kristian Uldall Kristiansen
The planar visible fold is a simple singularity in piecewise smooth systems. In this paper, we consider singularly perturbed systems that limit to this piecewise smooth bifurcation…
A new type of relaxation oscillation in a model with rate-and-state friction
K. Uldall Kristiansen
In this paper we prove the existence of a new type of relaxation oscillation occurring in a one-block Burridge-Knopoff model with Ruina rate-and-state friction law. In the relevant…
Canards in stiction: on solutions of a friction oscillator by regularization
Elena Bossolini, Morten Brøns, Kristian Uldall Kristiansen
We study the solutions of a friction oscillator subject to stiction. This discontinuous model is non-Filippov, and the concept of Filippov solution cannot be used. Furthermore some…
On the regularization of impact without collision: the Painlevé paradox and compliance
S. J. Hogan, K. Uldall Kristiansen
We consider the problem of a rigid body, subject to a unilateral constraint, in the presence of Coulomb friction. We regularize the problem by assuming compliance (with both stiffn…
Resolution of the piecewise smooth visible-invisible two-fold singularity in using regularization and blowup
Kristian Uldall Kristiansen, S. John Hogan
Two-fold singularities in a piecewise smooth (PWS) dynamical system in have long been the subject of intensive investigation. The interest stems from the fact that t…
On the approximation of the canard explosion point in epsilon-free systems
Morten Brøns, Kristian Uldall Kristiansen
A canard explosion is the dramatic change of period and amplitude of a limit cycle of a system of non-linear ODEs in a very narrow interval of the bifurcation parameter. It occurs…
Regularizations of two-fold bifurcations in planar piecewise smooth systems using blow up
K. Uldall Kristiansen, S. J. Hogan
We use blowup to study the regularization of codimension one two-fold singularities in planar piecewise smooth (PWS) dynamical systems. We focus on singular canards, pseudo-equlibr…
On the use of blow up to study regularizations of singularities of piecewise smooth dynamical systems in
K. Uldall Kristiansen, S. J. Hogan
In this paper we use the blow up method of Dumortier and Roussarie \cite{dumortier_1991,dumortier_1993,dumortier_1996}, in the formulation due to Krupa and Szmolyan \cite{krupa_ext…
Computation of saddle type slow manifolds using iterative methods
Kristian Uldall Kristiansen
This paper presents an alternative approach for the computation of trajectory segments on slow manifolds of saddle type. This approach is based on iterative methods rather than col…
Periodic orbits near a bifurcating slow manifold
Kristian Uldall Kristiansen
This paper studies a class of -degree-of-freedom Hamiltonian systems with a slowly varying phase that unfolds a Hamiltonian pitchfork bifurcation. The main result of the…