arXiv subjects

Kristian Uldall Kristiansen

Publications and source records attributed to Kristian Uldall Kristiansen.

At least 19 recordsLinked to original sources

arXiv2026

On the slow passage through a Hopf in a class of real-analytic systems: Exponential asymptotics and maximal delay

In this paper, we consider the slow passage through a Hopf in $\mathbb R^3$ in a certain class of real-analytic systems. Classically, the (simplified) Shishkova problem: $ε\frac{dz}{dμ} = λ(μ) z+\mathcal O(ε)$, $λ(μ) = μ-ı$, has been used to illustrate the delay associated with slow passage through a Hopf at $μ=0$. In this paper, we consider a general $λ(μ)$ with $λ(0)\in i\mathbb R\setminus \{0\}$, $\real[λ'(0)]\ne 0$, and add an equation for the complex conjugate. It is a basic fact that these systems are local normal forms for any real-analytic $(1,2)$ slow-fast system in $\mathbb R^3$ experiencing slow passage through a Hopf. In this paper, we study these systems under additional (global) assumptions on $λ$. In particular, we suppose that $λ$ has a simple zero away from the real axis. The remaining assumptions then relate to invariant manifolds of the so-called elliptic system $\dot μ= -ı\overline{λ(μ)}$ as well as properties of the level set $\operatorname{Re}[ı{λ(μ)}{λ(\overline μ)}]=0$. Under these assumptions, we then provide an asymptotic formula for the exponentially small splitting of attracting and repelling slow manifolds. As a corollary, we also obtain an asymptotic formula for the maximal delay. Our approach is geometric and is inspired by the work of Hayes et al (2016) (using blowup) and Neishtadt (1987,1988) (using so-called elliptic paths) but also by recent work of the second author on exponentially small splitting in unfoldings of the zero-Hopf bifurcation.

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arXiv2026

Cyclicity of sliding cycles in regularizations of piecewise linear two-folds

We study limit cycles produced by Sotomayor-Teixeira regularizations of piecewise linear vector fields with generic two-fold singularities. We focus on the visible-visible and visible-invisible cases where sliding cycles occur, treating cycles from both sides of the switching manifold in a unified way. In further details, by relating the cyclicity of sliding cycles and zeros of the slow divergence integral, we prove that the cyclicity of compact families of sliding cycles is bounded by two when such integral does not vanish identically. In contrast to previous works, we do not perform a case-by-base study, but instead relate zeros of slow divergence integrals to crossing limit cycles of a suitably defined auxiliary piecewise linear system. For visible folds, we provide necessary and sufficient conditions that assure the existence of a unique simple zero of the slow divergence integral, which implies the existence of two limit cycles. We also show that, when a hyperbolic singularity of the sliding vector field lies at the boundary of the sliding segment, then the cyclicity is bounded by one.

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arXiv2026

A geometric approach to inner problems associated with exponentially small splitting phenomena in local bifurcations

In this paper, we study analytic nonlinear partial differential equations for $\mathbf y=\mathbf y(x,θ)\in \mathbb C^n$ of the form $x^2 \mathbf y'_x(1+\mathcal O(x)) + \mathbf y'_θ+x \mathbf A \mathbf y = \mathcal O(x^2)$, $()'_z=\frac{\partial}{\partial z}$, $z=x,θ$, with $x\in \mathbb C$ and $θ\in \mathbb C /(2π\mathbb Z)$ denoting the independent variables. We show that solutions are $1$-sums (with respect to $x$) of Fourier series with coefficients of type Gevrey-$1$. The motivation for studying these equations is that so-called inner problems, associated with two-dimensional formal connections in unfoldings of local bifurcations, can be brought into this form (by looking for invariant manifolds in blowup coordinates). In the present paper, we give two examples: The zero-Hopf bifurcation and the resonant Hopf-Hopf bifurcation. Importantly, these inner problems are given by the unperturbed problem (i.e. at the bifurcation) and the invariant manifolds are expressed directly in phase space (through blowup coordinates associated with the local bifurcation). This contrasts the Lazutkin-based formulation of inner problems which is based upon a blowup of singularities (with respect to complex time) of approximate solutions; for local bifurcations this requires (artificial) coordinate transformations. We solve the PDE by extending the Banach-convolution-algebra-approach to Borel-Laplace by Bonckaert and De Maesschalck (2008) to account for analytic Fourier series. In further details, we apply the Borel transform with respect to $x$ (keeping $θ$ fixed) and solve the resulting equation in an appropriate Banach space of Fourier series with coefficients that have exponential growth in the Borel plane. The solutions of the PDE are then obtained through application of the Laplace transformation.

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arXiv2026

A geometric approach to exponentially small splitting: The generic zero-Hopf bifurcation of co-dimension two

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 splitting of one-dimensional stable and unstable manifolds is beyond all orders. This paper provides a new geometric dynamical-systems-oriented proof for the exponentially small splitting. As a novel aspect, we relate the exponentially small splitting to the lack of analyticity of center-like manifolds of generalized saddle-nodes. Moreover, the blowup method plays an important technical role as a systematic way to relate dynamics on different orders of magnitude. Finally, as our approach takes place in the (complexified) phase space, we do not rely on an explicit time-parametrization of the unperturbed heteroclinic connection. We therefore believe that our approach has general interest.

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arXiv2026

A geometric approach to exponentially small splitting: Zero-Hopf bifurcations of arbitrary co-dimension

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 problems that generalizes the third order Michelsen/Kuramoto-Sivashinsky-type equations $ε^{2(κ-1)} f'''+f'={Q}(f)$, where ${Q}$ is an arbitrary real polynomial with $κ=\operatorname{degree}Q\ge 2$ simple real roots. For $ε=0$, the system has $(κ-1)$-many heteroclinic connections and we describe the exponentially small splitting for each connection for all $0<ε\ll 1$ under a separate nondegeneracy condition. In particular, we find that the $j$th-splitting is of the form $ε^{-\frac{3κ}{2}}\exp\left({-ε^{1-κ}T^j}\right)(C^j+\mathcal O(ε))$, where $T^j>0$ can be calculated explicitly and be interpreted as the blowup time of special unbounded solutions of the $ε=0$-limiting system in imaginary time $f'=iQ(f)$. Our approach extends a similar geometric method developed by the present author for the generic zero-Hopf bifurcation of co-dimension two, which does not rely on explicit time-parametrizations of the unperturbed heteroclinic connections and their singularities in the complex plane. Instead, we work exclusively in the complexified phase space and relate the exponentially small splitting to the lack of analyticity of center-like invariant manifolds of associated generalized saddle-nodes.

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arXiv2025

On $k$-summable normal forms of vector fields with one zero eigenvalue

In this paper, we study normal forms of analytic saddle-nodes in $\mathbb C^{n+1}$ with any Poincaré rank $k\in \mathbb N$. The approach and the results generalize those of Bonckaert and De Maesschalck from 2008 that considered $k=1$. In particular, we introduce a Banach convolutional algebra that is tailored to study differential equations in the Borel plane of order $k$. One of the subtleties that we take care of in this paper, is that nontrivial Jordan blocks are allowed in the linear part of the vector field. We anticipate that our approach can stimulate new research and be used to study different normal forms in future work.

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arXiv2025

Switching, Multiple Time-Scales and Geometric Blow-Up in a Low-Dimensional Gene Regulatory Network

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 time-scale separation, whereas others are related to 'switching' (proximity to a non-smooth singular limit). This motivates the study of reduced models obtained after (i) quasi-steady state reduction (QSSR), which utilises the time-scale separation, and (ii) piecewise-smooth approximations, which reduce the nonlinearity of the model by viewing highly nonlinear sigmoidal terms as singular perturbations of step functions. We investigate the interplay between the reduction methods (i)-(ii), in the context of a 4-dimensional GRN which has been used as a low-dimensional representative of an important class of (generally high-dimensional) GRN models in the literature. We begin by identifying a region in the small parameter plane for which this problem can be formulated as a smooth singularly perturbed system on a blown-up space, uniformly in the switching parameter. This allows us to apply Fenichel's coordinate-free theorems and obtain a rigorous reduction to a 2-dimensional system, that is a perturbation of the QSSR. Finally, we show that the reduced system features a Hopf bifurcation which does not appear in the QSSR system, due to the influence of higher order terms. Taken together, our findings suggest that the relative size of the small parameters is important for the validity of QSS reductions and the determination of qualitative dynamics in GRN models more generally. Although the focus is on the 4-dimensional GRN, our approach is applicable to higher dimensions.

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arXiv2025

On entry-exit formulas for degenerate turning point problems in planar slow-fast systems

In this paper, we study degenerate entry-exit problems associated with planar slow-fast systems having an invariant line $\{(x,y)\,:\,y=0\}$ with a turning point at $x=0$. The degeneracy stems from the fact that the slow flow has a saddle-node of even order $2n$, $n\in \mathbb N$, at the turning point, i.e. $x' = -x^{2n}(1+o(1))$ for $ε=0$. We are motivated by the appearance of such turning point problems (for $n=1$) in the graphics $(I_2^1)$ and $(I_4^1)$, through a nilpotent saddle-node singularity at infinity, in the Dumortier-Roussarie-Rousseau program (for solving the finiteness part of Hilbert's 16th problem for quadratic polynomial systems). Our results show, under additional hypothesis, that in the case $n=1$ there is a well-defined entry-exit relation for $ε\rightarrow 0$. The associated Dulac map is smooth w.r.t. $(ε,ε\log ε^{-1})$. On the other hand for the cases $n\ge 2$, we show that the entry-exit relation requires additional control parameters. Our approach follows the one used by De Maesschalck, P. and Schecter, S. (JDE 2016) for a different type of degenerate entry-exit problem. In particular, we apply blow-up {after} having first performed a singular coordinate transformation of $y$. The degeneracy at $x=0$ requires an additional blow-up. We finally apply the result for $n=1$ to a normal form for the unfolding of the \NEW{relevant} graphics in the Dumortier-Roussarie-Rousseau program. Here we also demonstrate that the singular transformation of $y$ due to De Maesschalck, P. and Schecter, S. (JDE 2016) has practical significance in numerical computations.

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arXiv2025

A dynamical systems approach to WKB-methods: The eigenvalue problem for a single well potential

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 the Bohr-Sommerfeld quantization formula. A novel aspect of our results, which are based on recent work of the authors on the turning point problem based upon dynamical systems methods, is that we cover all eigenvalues $E\in [0,\mathcal O(1)]$ and show that the Bohr-Sommerfeld quantitization formula approximates all of these eigenvalues (in a sense that is made precise). At the same time, we provide rigorous smoothness statements of the eigenvalues as functions of $ε$. We find that whereas the small eigenvalues $E=\mathcal O(ε)$ are smooth functions of $ε$, the large ones $E=\mathcal O(1)$ are smooth functions of $nε\in[c_1,c_2],\,0<c_1<c_2<\infty$, and $0\le ε^{1/3}\ll 1$; here $n\in \mathbb N_0$ is the index of the eigenvalues.

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arXiv2025

Improved Gevrey-1 estimates of formal series expansions of center manifolds

In this paper, we show that the coefficients $ϕ_n$ of the formal series expansions $y=\sum_{n=1}^\infty ϕ_n x^n\in x\mathbb C[[x]]$ of center manifolds of planar analytic saddle-nodes grow like $Γ(n+a)$ (after rescaling $x$) as $n\rightarrow \infty$. Here the quantity $a$ is the formal analytic invariant associated with the saddle node (following the work of J. Martinet and J.-P. Ramis). This growth property of $ϕ_n$, which cannot be improved when the center manifold is nonanalytic, was recently (2024) described for a restricted class of nonlinearities by the present author in collaboration with P. Szmolyan. This joint work was in turn inspired by the work of Merle, Raphaël, Rodnianski, and Szeftel (2022), which described the growth of the coefficients for a system related to self-similar solutions of the compressible Euler. In the present paper, we combine the previous approaches with a Borel-Laplace approach. Specifically, we adapt the Banach norm of Bonckaert and De Maesschalck (2008) in order to capture the singularity in the complex plane.

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arXiv2024

Analytic weak-stable manifolds in unfoldings of saddle-nodes

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 manifolds are, on the other hand, not unique, but in the nonresonant case there is a unique weak-stable manifold that is analytic. As the system approaches a saddle-node (under parameter variation), a sequence of resonances (of increasing order) occur. In this paper, we give a detailed description of the analytic weak-stable manifolds during this process. In particular, we relate a ``flapping-mechanism'', corresponding to a dramatic change of the position of the analytic weak-stable manifold as the parameter passes through the infinitely many resonances, to the lack of analyticity of the center manifold at the saddle-node. Our work is motivated and inspired by the work of Merle, Raphaël, Rodnianski, and Szeftel, where this flapping mechanism is the crucial ingredient in the construction of $C^\infty$-smooth self-similar solutions of the compressible Euler equations.

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arXiv2024

Travelling Waves and Exponential Nonlinearities in the Zeldovich-Frank-Kamenetskii Equation

We prove the existence of a family of travelling wave solutions in a variant of the $\textit{Zeldovich-Frank-Kamenetskii (ZFK) equation}$, a reaction-diffusion equation which models the propagation of planar laminar premixed flames in combustion theory. Our results are valid in an asymptotic regime which corresponds to a reaction with high activation energy, and provide a rigorous and geometrically informative counterpart to formal asymptotic results that have been obtained for similar problems using $\textit{high activation energy asymptotics}$. We also go beyond the existing results by (i) proving smoothness of the minimum wave speed function $\overline c(ε)$, where $0< ε\ll 1$ is the small parameter, and (ii) providing an asymptotic series for a flat slow manifold which plays a role in the construction of travelling wave solutions for non-minimal wave speeds $c > \overline c(ε)$. The analysis is complicated by the presence of an exponential nonlinearity which leads to two different scaling regimes as $ε\to 0$, which we refer to herein as the $\textit{convective-diffusive}$ and $\textit{diffusive-reactive}$ zones. The main idea of the proof is to use the geometric blow-up method to identify and characterise a $(c,ε)$-family of heteroclinic orbits which traverse both of these regimes, and correspond to travelling waves in the original ZFK equation. More generally, our analysis contributes to a growing number of studies which demonstrate the utility of geometric blow-up approaches to the study dynamical systems with singular exponential nonlinearities.

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arXiv2023

Circularization in the damped Kepler problem

In this paper, we revisit the damped Kepler problem within a general family of nonlinear damping forces with magnitude $δ\vert u\vert^β\vert \dot u\vert^{α+1}$, depending on three parameters $δ>0,α\ge 0$ and $β\ge 0$, and address the general question of circularization whereby orbits tend to become more circular as they approach the sun. Our approach is based on dynamical systems theory, using blowup and desingularization as our main technical tools. We find that $γ=α+2β-3$ is an important quantity, with the special case $γ=0$ separating circularization ($-3<γ<0$) where the eccentricity converges to zero, i.e. $e(t)\rightarrow 0$ as $u(t)\rightarrow 0$, from cases ($γ>0$) where $e(t)\rightarrow 1$ as $u(t)\rightarrow 0$, both on open sets of initial conditions. We find that circularization for $-3<γ<0$ occurs due to asymptotic stability of a zero-Hopf equilibrium point (i.e., the eigenvalues are $\pm i ω,0$) of a three-dimensional reduced problem (which is analytic in the blowup coordinates). The attraction is therefore not hyperbolic and in particular not covered by standard dynamical systems theory. Instead we use recent results on normal forms of the zero-Hopf to locally bring the system into a form where the stability can be addressed directly. We believe that our approach can be used to describe unbounded solutions.

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arXiv2023

Blowup analysis of a hysteresis model based upon singular perturbations

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 smooth differential equations where the past of a trajectory, in a small neighborhood of the discontinuity set, determines the vector-field at present. In fact, in the limit where the neighborhood of the discontinuity {vanishes}, hysteresis converges in an appropriate sense to Filippov's sliding vector-field. Recently {(2022)}, however, {Bonet and Seara} showed that hysteresis, in contrast to regularization through smoothing, leads to chaos in the regularization of grazing bifurcations, even in two dimensions. The hysteresis model we analyze in the present paper -- which was developed by {Bonet et al in a paper from 2017} as an attempt to unify different regularizations of piecewise smooth systems -- involves two singular perturbation parameters and includes a combination of slow-fast and nonsmooth effects. The description of this model is therefore -- from the perspective of singular perturbation theory -- challenging, even in two dimensions. Using blowup as our main technical tool, we prove existence of an invariant cylinder carrying fast dynamics in the azimuthal direction and a slow drift in the axial direction. We find that the slow drift is given by Filippov's sliding {vector-field} to leading order. Moreover, in the case of grazing, we identify two important parameter regimes that relate the model to smoothing (through a saddle-node bifurcation of limit cycles) and hysteresis (through chaotic dynamics, due to a folded saddle and a novel return mechanism).

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arXiv2023

The dud canard: Existence of strong canard cycles in $\mathbb R^3$

In this paper, we provide a rigorous description of the birth of canard limit cycles in slow-fast systems in $\mathbb R^3$ through the folded saddle-node of type II and the singular Hopf bifurcation. In particular, we prove -- in the analytic case only -- that for all $0<ε\ll 1$ there is a family of periodic orbits, born in the (singular) Hopf bifurcation and extending to $\mathcal O(1)$ cycles that follow the strong canard of the folded saddle-node. Our results can be seen as an extension of the canard explosion in $\mathbb R^2$, but in contrast to the planar case, the family of periodic orbits in $\mathbb R^3$ is not explosive. For this reason, we have chosen to call the phenomena in $\mathbb R^3$, the ``dud canard''. The main difficulty of the proof lies in connecting the Hopf cycles with the canard cycles, since these are described in different scalings. As in $\mathbb R^2$, we use blowup to overcome this, but we also have to compensate for the lack of uniformity near the Hopf bifurcation, due to its singular nature; it is a zero-Hopf bifurcation in the limit $ε=0$. In the present paper, we do so by imposing analyticity of the vector-field. This allows us to prove existence of an invariant slow manifold, that is not normally hyperbolic.

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arXiv2023

Sliding cycles of the regularized piecewise linear $VI_3$ two-fold

The goal of this paper is to study the number of sliding limit cycles of a regularized piecewise linear $VI_3$ two-fold using the notion of slow divergence integral. We focus on limit cycles produced by canard cycles located in the half-plane with an invisible fold point. We prove that the integral has at most $1$ zero counting multiplicity (when it is not identically zero). This will imply that the canard cycles can produce at most $2$ limit cycles. Moreover, we detect regions in the parameter space with $2$ limit cycles.

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arXiv2023

Slow divergence integral in regularized piecewise smooth systems

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 diverse tangency points. We show that the slow divergence integral is invariant under smooth equivalences. This is a natural generalization of the notion of slow divergence integral along normally hyperbolic portions of curve of singularities in smooth planar slow-fast systems. We give an interesting application of the integral in a model with visible-invisible two-fold of type $VI_3$. It is related to a connection between so-called Minkowski dimension of bounded and monotone "entry-exit" sequences and the number of sliding limit cycles produced by so-called canard cycles.

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arXiv2023

Revisiting the Kepler problem with linear drag using the blowup method and normal form theory

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 that the eccentricity vector has a well-defined limit in the case of linear drag. This limiting eccentricity vector defines a conserved quantity, and in the present paper, we prove that the corresponding invariant sets are smooth manifolds. These results rely on normal form theory and a blowup transformation, which reveals that the invariant manifolds are (nonhyperbolic) stable sets of (limiting) periodic orbits. Moreover, we identify a separate invariant manifold which corresponds to a zero limiting eccentricity vector. This manifold is obtained as a generalized center manifold over the zero eigenspace of a zero-Hopf point. Finally, we present a detailed blowup analysis, which provides a geometric picture of the dynamics. %We will use this to shed light on a degenerate case of the limiting eccentricity vector. %The aforementioned invariant manifolds only make up a subset hereof. We believe that our approach and results will have general interest in problems with blowup dynamics.

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