Kristian Uldall Kristiansen

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 2 n , n ∈ N , at the turning point, i.e. x ′ = − x 2 n ( 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 → 0 . The associated Dulac map is smooth w.r.t. ( ϵ , ϵ log ⁡ ϵ − 1 ) . On the other hand for the cases n ⩾ 2 , we show that the entry–exit relation requires additional control parameters. Our approach follows the one used by De Maesschalck and Schecter (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 relevant graphics in the Dumortier-Roussarie-Rousseau program. Here we also demonstrate that the singular transformation of y due to De Maesschalck and Schecter (JDE 2016) has practical significance in numerical computations.

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.

In this paper, we show that the coefficients of the formal series expansions of center manifolds of planar analytic saddle‐nodes grow like (after rescaling ) as . Here, the quantity is the formal analytic invariant associated with the saddle‐node (following the work of Martinet and Ramis). This growth property of , which is optimal, was recently (2024) described for a restricted class of nonlinearities by the present author in collaboration with 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. Finally, we apply the result to a family of Riccati equations and obtain a partial classification of the analytic center manifolds.

We prove the existence of a family of traveling wave solutions in a variant of the 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 they provide a rigorous and geometrically informative counterpart to formal asymptotic results that have been obtained for similar problems using high activation energy asymptotics. We also go beyond the existing results by (i) proving smoothness of the minimum wave speed function [Formula: see text], where [Formula: see text] is the small parameter, and (ii) providing an asymptotic series for a flat slow manifold which plays a role in the construction of traveling wave solutions for nonminimal wave speeds [Formula: see text]. The analysis is complicated by the presence of an exponential nonlinearity which leads to two different scaling regimes as [Formula: see text], which we refer to herein as the convective-diffusive and diffusive-reactive zones. The main idea of the proof is to use the geometric blow-up method to identify and characterize a [Formula: see text]-family of heteroclinic orbits which traverse both of these regimes, and correspond to traveling 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 of dynamical systems with singular exponential nonlinearities.

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 centre 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 ∞ -smooth self-similar solutions of the compressible Euler equations.

The goal of this paper is to study the number of sliding limit cycles of regularized piecewise linear visible–invisible twofolds 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.

In this paper, we revisit the damped Kepler problem within a general family of nonlinear damping forces with magnitude [Formula: see text], depending on three parameters, [Formula: see text], and [Formula: see text], 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 [Formula: see text] is an important quantity, with the special case [Formula: see text] separating circularization ([Formula: see text]) where the eccentricity converges to zero, i.e., [Formula: see text] as [Formula: see text], from cases ([Formula: see text]) where [Formula: see text] as [Formula: see text], both on open sets of initial conditions. We find that circularization for [Formula: see text] occurs due to asymptotic stability of a zero-Hopf equilibrium point (i.e., the eigenvalues are [Formula: see text]) 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. The case [Formula: see text] relates to a certain scaling symmetry (that is also present in the undamped Kepler problem) and in this case the system can be reduced to a planar system. We find that the eccentricity limits to [Formula: see text] on an open subset of initial conditions for ([Formula: see text] and [Formula: see text]). We also describe different properties of the solutions, including finite time blowup and the limit of the eccentricity vector. Interestingly, we find that circularization [Formula: see text] implies finite time blowup of solutions. We believe that our approach can be used to describe unbounded solutions.