24 citations · 28 across the 16 of their papers we have counts for
19 papers
On the slow passage through a Hopf in a class of real-analytic systems: Exponential asymptotics and maximal delay
Johan Møller Christensen, Kristian Uldall Kristiansen
In this paper, we consider the slow passage through a Hopf in in a certain class of real-analytic systems. Classically, the (simplified) Shishkova problem: $ε\frac{dz…
A geometric approach to inner problems associated with exponentially small splitting phenomena in local bifurcations
Kristian Uldall Kristiansen
In this paper, we study analytic nonlinear partial differential equations for of the form $x^2 \mathbf y'_x(1+\mathcal O(x)) + \mathbf y'_…
A geometric approach to exponentially small splitting: The generic zero-Hopf bifurcation of co-dimension two
Kristian Uldall Kristiansen
In this paper, we consider the unfolding of the real-analytic and generic zero-Hopf bifurcation of co-dimension two. It is well-known that in an open set of parameter space the spl…
A geometric approach to exponentially small splitting: Zero-Hopf bifurcations of arbitrary co-dimension
Kristian Uldall Kristiansen
In this paper, we present a geometric approach to exponentially small splitting in zero-Hopf bifurcations of arbitrary co-dimension. In further details, we consider a family of pro…
On entry-exit formulas for degenerate turning point problems in planar slow-fast systems
Renato Huzak, Kristian Uldall Kristiansen
In this paper, we study degenerate entry-exit problems associated with planar slow-fast systems having an invariant line with a turning point at . The dege…
Switching, Multiple Time-Scales and Geometric Blow-Up in a Low-Dimensional Gene Regulatory Network
Samuel Jelbart, Kristian Uldall Kristiansen, Peter Szmolyan
ODE-based models for gene regulatory networks (GRNs) can often be formulated as smooth singular perturbation problems with multiple small parameters, some of which are related to t…
A dynamical systems approach to WKB-methods: The eigenvalue problem for a single well potential
Kristian Uldall Kristiansen, Peter Szmolyan
In this paper, we revisit the eigenvalue problem of the one-dimensional Schr{ö}dinger equation for smooth single well potentials. In particular, we provide a new interpretation of…
On -summable normal forms of vector fields with one zero eigenvalue
Peter De Maesschalck, Kristian Uldall Kristiansen
In this paper, we study normal forms of analytic saddle-nodes in with any Poincaré rank . The approach and the results generalize those of Bonckae…
Improved Gevrey-1 estimates of formal series expansions of center manifolds
Kristian Uldall Kristiansen
In this paper, we show that the coefficients of the formal series expansions of center manifolds of planar analytic saddle-no…
Travelling Waves and Exponential Nonlinearities in the Zeldovich-Frank-Kamenetskii Equation
Samuel Jelbart, Kristian Uldall Kristiansen, Peter Szmolyan
We prove the existence of a family of travelling wave solutions in a variant of the , a reaction-diffusion equation which model…
Analytic weak-stable manifolds in unfoldings of saddle-nodes
Kristian Uldall Kristiansen, Peter Szmolyan
Any attracting, hyperbolic and proper node of a two-dimensional analytic vector-field has a unique strong-stable manifold. This manifold is analytic. The corresponding weak-stable…
Circularization in the damped Kepler problem
Kristian Uldall Kristiansen, Rafael Ortega
In this paper, we revisit the damped Kepler problem within a general family of nonlinear damping forces with magnitude , depending on three…
Sliding cycles of the regularized piecewise linear two-fold
Renato Huzak, Kristian Uldall Kristiansen
The goal of this paper is to study the number of sliding limit cycles of a regularized piecewise linear two-fold using the notion of slow divergence integral. We focus on li…
Slow divergence integral in regularized piecewise smooth systems
Renato Huzak, Kristian Uldall Kristiansen, Goran Radunović
In this paper we define the notion of slow divergence integral along sliding segments in regularized planar piecewise smooth systems. The boundary of such segments may contain dive…
The number of limit cycles for regularized piecewise polynomial systems is unbounded
R. Huzak, K. Uldall Kristiansen
In this paper, we extend the slow divergence-integral from slow-fast systems, due to De Maesschalck, Dumortier and Roussarie, to smooth systems that limit onto piecewise smooth one…
Bifurcations of mixed-mode oscillations in three-timescale systems: an extended prototypical example
Panagiotis Kaklamanos, Nikola Popović, Kristian Uldall Kristiansen
We study a class of multi-parameter three-dimensional systems of ordinary differential equations that exhibit dynamics on three distinct timescales. We apply geometric singular per…
Singularly Perturbed Oscillators with Exponential Nonlinearities
Samuel Jelbart, Kristian Uldall Kristiansen, Peter Szmolyan +1
Singular exponential nonlinearities of the form with small occur in many different applications. These terms have essential singularities for leading t…
Relaxation oscillations in substrate-depletion oscillators close to the nonsmooth limit
Kristian Uldall Kristiansen, Peter Szmolyan
In this paper, we describe a novel type of relaxation oscillations occurring in a model of substrate-depletion oscillators. Using geometric singular perturbation theory, with blow-…
Blowup for flat slow manifolds
Kristian Uldall Kristiansen
In this paper we present a method for extending the blowup method, in the formulation of Krupa and Szmolyan, to flat slow manifolds that lose hyperbolicity beyond any algebraic ord…