arXiv subjects

Christian Kuehn

Publications and source records attributed to Christian Kuehn.

At least 163 records · Page 9Linked to original sources

arXiv2018

Discretized Fast-Slow Systems near Transcritical Singularities

We extend slow manifolds near a transcritical singularity in a fast-slow system given by the explicit Euler discretization of the corresponding continuous-time normal form. The analysis uses the blow-up method and direct trajectory-based estimates. We prove that the qualitative behaviour is preserved by a time-discretization with sufficiently small step size. This step size is fully quantified relative to the time scale separation. Our proof also yields the continuous-time results as a special case and provides more detailed calculations in the classical (or scaling) chart.

math.DS↗

arXiv2018

Network topology near criticality in adaptive epidemics

We study structural changes of adaptive networks in the co-evolutionary susceptible-infected-susceptible (SIS) network model along its phase transition. We clarify to what extent these changes can be used as early-warning signs for the transition at the critical infection rate $λ_c$ at which the network collapses and the system disintegrates. We analyze the interplay between topology and node-state dynamics near criticality. Several network measures exhibit clear maxima or minima close to the critical threshold that could potentially serve as early-warning signs. These measures include the $SI$ link density, triplet densities, clustering, assortativity and the eigenvalue gap. For the $SI$ link density and triplet densities the maximum is found to originate from the co-existence of two power laws. Other network quantities, such as the degree, the branching ratio, or the harmonic mean distance, show scaling with a singularity at $λ=0$ and not at $λ_c$, which means that they are incapable of detecting the transition.

physics.soc-ph↗

arXiv2018

Pattern formation in the doubly-nonlocal Fisher-KPP equation

We study the existence, bifurcations, and stability of stationary solutions for the doubly-nonlocal Fisher-KPP equation. We prove using Lyapunov-Schmidt reduction that under suitable conditions on the parameters, a bifurcation from the non-trivial homogeneous state can occur. The kernel of the linearized operator at the bifurcation is two-dimensional and periodic stationary patterns are generated. Then we prove that these patterns are, again under suitable conditions, locally asymptotically stable. We also compare our results to previous work on the nonlocal Fisher-KPP equation containing a local diffusion term and a nonlocal reaction term. If the diffusion is approximated by a nonlocal kernel, we show that our results are consistent and reduce to the local ones in the local singular diffusion limit. Furthermore, we prove that there are parameter regimes, where no bifurcations can occur for the doubly-nonlocal Fisher-KPP equation. The results demonstrate that intricate different parameter regimes are possible. In summary, our results provide a very detailed classification of the multi-parameter dependence of the stationary solutions for the doubly-nonlocal Fisher-KPP equation.

math.AP↗

arXiv2018

Analysis and Predictability for Tipping Points with Leading-Order Nonlinear Terms

Tipping points have been actively studied in various applications as well as from a mathematical viewpoint. A main technique to theoretically understand early-warning signs for tipping points is to use the framework of fast-slow stochastic differential equations. A key assumption in many arguments for the existence of variance and auto-correlation growth before a tipping point is to use a linearization argument, i.e., the leading-order term governing the deterministic (or drift) part of stochastic differential equation is linear. This assumption guarantees a local approximation via an Ornstein-Uhlenbeck process in the normally hyperbolic regime before, but sufficiently bounded away from, a bifurcation. In this paper, we generalize the situation to leading-order nonlinear terms for the setting of one fast variable. We work in the quasi-steady regime and prove that the fast variable has a well-defined stationary distribution and we calculate the scaling law for the variance as a bifurcation-induced tipping point is approached. We cross-validate the scaling law numerically. Furthermore, we provide a computational study for the predictability using early-warning signs for leading-order nonlinear terms based upon receiver-operator characteristic curves.

nlin.PS↗

arXiv2018

A Note on Kernel Methods for Multiscale Systems with Critical Transitions

We study the maximum mean discrepancy (MMD) in the context of critical transitions modelled by fast-slow stochastic dynamical systems. We establish a new link between the dynamical theory of critical transitions with the statistical aspects of the MMD. In particular, we show that a formal approximation of the MMD near fast subsystem bifurcation points can be computed to leading-order. In particular, this leading order approximation shows that the MMD depends intricately on the fast-slow systems parameters and one can only expect to extract warning signs under rather stringent conditions. However, the MMD turns out to be an excellent binary classifier to detect the change point induced by the critical transition. We cross-validate our results by numerical simulations for a van der Pol-type model.

nlin.PS↗

arXiv2018

Early-warning signals for bifurcations in random dynamical systems with bounded noise

We consider discrete-time one-dimensional random dynamical systems with bounded noise, which generate an associated set-valued dynamical system. We provide necessary and sufficient conditions for a discontinuous bifurcation of a minimal invariant set of the set-valued dynamical system in terms of the derivatives of the so-called extremal maps. We propose an algorithm for reconstructing the derivatives of the extremal maps from a time series that is generated by iterations of the original random dynamical system. We demonstrate that the derivative reconstructed for different parameters can be used as an early-warning signal to detect an upcoming bifurcation, and apply the algorithm to the bifurcation analysis of the stochastic return map of the Koper model, which is a three-dimensional multiple time scale ordinary differential equation used as prototypical model for the formation of mixed-mode oscillation patterns. We apply our algorithm to data generated by this map to detect an upcoming transition.

math.DS↗

arXiv2018

Scaling Laws and Warning Signs for Bifurcations of SPDEs

Critical transitions (or tipping points) are drastic sudden changes observed in many dynamical systems. Large classes of critical transitions are associated to systems, which drift slowly towards a bifurcation point. In the context of stochastic ordinary differential equations (SODEs), there are results on growth of variance and autocorrelation before a transition, which can be used as possible warning signs in applications. A similar theory has recently been developed in the simplest setting also for stochastic partial differential equations (SPDEs) for self-adjoint operators in the drift term. This setting leads to real discrete spectrum and growth of the covariance operator via a certain scaling law. In this paper, we develop this theory substantially further. We cover the cases of complex eigenvalues, degenerate eigenvalues as well as continuous spectrum. This provides a fairly comprehensive theory for most practical applications of warning signs for SPDE bifurcations.

math.DS↗

arXiv2018

Smoluchowski's discrete coagulation equation with forcing

In this article we study an extension of Smoluchowski's discrete coagulation equation, where particle in- and output takes place. This model is frequently used to describe aggregation processes in combination with sedimentation of clusters. More precisely, we show that the evolution equation is well-posed for a large class of coagulation kernels and output rates. Additionally, in the long-time limit we prove that solutions converge to a unique equilibrium with exponential rate under a suitable smallness condition on the coefficients.

math-ph↗

arXiv2017

Stochastic mixed-mode oscillations in a three-species predator-prey model

The effect of demographic stochasticity, in the form of Gaussian white noise, in a predator-prey model with one fast and two slow variables is studied. We derive the stochastic differential equations (SDEs) from a discrete model. For suitable parameter values, the deterministic drift part of the model admits a folded node singularity and exhibits a singular Hopf bifurcation. We focus on the parameter regime near the Hopf bifurcation, where small amplitude oscillations exist as stable dynamics in the absence of noise. In this regime, the stochastic model admits noise-driven mixed-mode oscillations (MMOs), which capture the intermediate dynamics between two cycles of population outbreaks. We perform numerical simulations to calculate the distribution of the random number of small oscillations between successive spikes for varying noise intensities and distance to the Hopf bifurcation. We also study the effect of noise on a suitable Poincaré map. Finally, we prove that the stochastic model can be transformed into a normal form near the folded node, which can be linked to recent results on the interplay between deterministic and stochastic small amplitude oscillations. The normal form can also be used to study the parameter influence on the noise level near folded singularities.

math.DS↗

arXiv2017

Validity of amplitude equations for non-local non-linearities

Amplitude equations are used to describe the onset of instability in wide classes of partial differential equations (PDEs). One goal of the field is to determine simple universal/generic PDEs, to which many other classes of equations can be reduced, at least on a sufficiently long approximating time scale. In this work, we study the case, when the reaction terms are non-local. In particular, we consider quadratic and cubic convolution-type non-linearities. As a benchmark problem, we use the Swift-Hohenberg equation. The resulting amplitude equation is a Ginzburg-Landau PDE, where the coefficients can be calculated from the kernels. Our proof relies on separating critical and non-critical modes in Fourier space in combination with suitable kernel bounds.

math.AP↗

arXiv2017

Generalized Play Hysteresis Operators in Limits of Fast-Slow Systems

Hysteresis operators appear in many applications such as elasto-plasticity and micromagnetics, and can be used for a wider class of systems, where rate-independent memory plays a role. A natural approximation for systems of evolution equations with hysteresis operators are fast-slow dynamical systems, which - in their used approximation form - do not involve any memory effects. Hence, viewing differential equations with hysteresis operators in the non-linearity as a limit of approximating fast-slow dynamics involves subtle limit procedures. In this paper, we give a proof of Netushil's "observation" that broad classes of planar fast-slow systems with a two-dimensional critical manifold are expected to yield generalized play operators in the singular limit. We provide two proofs of this "observation" based upon the fast-slow systems paradigm of decomposition into subsystems. One proof strategy employs suitable convergence in function spaces, while the second approach considers a geometric strategy via local linearization and patching adapted originally from problems in stochastic analysis. We also provide an illustration of our results in the context of oscillations in forced planar non-autonomous fast-slow systems. The study of this example also strongly suggests that new canard-type mechanisms can occur for two-dimensional critical manifolds in planar systems.

math.DS↗

arXiv2017

Model Spaces of Regularity Structures for Space-Fractional SPDEs

We study model spaces, in the sense of Hairer, for stochastic partial differential equations involving the fractional Laplacian. We prove that the fractional Laplacian is a singular kernel suitable to apply the theory of regularity structures. Our main contribution is to study the dependence of the model space for a regularity structure on the three-parameter problem involving the spatial dimension, the polynomial order of the nonlinearity, and the exponent of the fractional Laplacian. The goal is to investigate the growth of the model space under parameter variation. In particular, we prove several results in the approaching subcriticality limit leading to universal growth exponents of the regularity structure. A key role is played by the viewpoint that model spaces can be identified with families of rooted trees. Our proofs are based upon a geometrical construction similar to Newton polygons for classical Taylor series and various combinatorial arguments. We also present several explicit examples listing all elements with negative homogeneity by implementing a new symbolic software package to work with regularity structures. We use this package to illustrate our analytical results and to obtain new conjectures regarding coarse-grained network measures for model spaces.

math.PR↗

arXiv2017

Geometry and numerical continuation of multiscale orbits in a nonconvex variational problem

We investigate a singularly perturbed, non-convex variational problem arising in materials science with a combination of geometrical and numerical methods. Our starting point is a work by Stefan Müller, where it is proven that the solutions of the variational problem are periodic and exhibit a complicated multi-scale structure. In order to get more insight into the rich solution structure, we transform the corresponding Euler-Lagrange equation into a Hamiltonian system of first order ODEs and then use geometric singular perturbation theory to study its periodic solutions. Based on the geometric analysis we construct an initial periodic orbit to start numerical continuation of periodic orbits with respect to the key parameters. This allows us to observe the influence of the parameters on the behavior of the orbits and to study their interplay in the minimization process. Our results confirm previous analytical results such as the asymptotics of the period of minimizers predicted by Müller. Furthermore, we find several new structures in the entire space of admissible periodic orbits.

math.DS↗

arXiv2016

Tracking Particles in Flows near Invariant Manifolds via Balance Functions

Particles moving inside a fluid near, and interacting with, invariant manifolds is a common phenomenon in a wide variety of applications. One elementary question is whether we can determine once a particle has entered a neighbourhood of an invariant manifold, when it leaves again. Here we approach this problem mathematically by introducing balance functions, which relate the entry and exit points of a particle by an integral variational formula. We define, study, and compare different natural choices for balance functions and conclude that an efficient compromise is to employ normal infinitesimal Lyapunov exponents. We apply our results to two different model flows: a regularized solid-body rotational flow and the asymmetric Kuhlmann--Muldoon model developed in the context of liquid bridges. Furthermore, we employ full numerical simulations of the Navier-Stokes equations of a two-way coupled particle in a shear--stress-driven cavity to test balance functions for a particle moving near an invariant wall. In conclusion, our theoretically-developed framework seems to be applicable to models as well as data to understand particle motion near invariant manifolds.

math.DS↗

arXiv2016

A meeting point of entropy and bifurcations in cross-diffusion herding

A cross-diffusion system modeling the information herding of individuals is analyzed in a bounded domain with no-flux boundary conditions. The variables are the species' density and an influence function which modifies the information state of the individuals. The cross-diffusion term may stabilize or destabilize the system. Furthermore, it allows for a formal gradient-flow or entropy structure. Exploiting this structure, the global-in-time existence of weak solutions and the exponential decay to the constant steady state is proved in certain parameter regimes. This approach does not extend to all parameters. We investigate local bifurcations from homogeneous steady states analytically to determine whether this defines the validity boundary. This analysis shows that generically there is a gap in the parameter regime between the entropy approach validity and the first local bifurcation. Next, we use numerical continuation methods to track the bifurcating non-homogeneous steady states globally and to determine non-trivial stationary solutions related to herding behaviour. In summary, we find that the main boundaries in the parameter regime are given by the first local bifurcation point, the degeneracy of the diffusion matrix and a certain entropy decay validity condition. We study several parameter limits analytically as well as numerically, with a focus on the role of changing a linear damping parameter as well as a parameter controlling the cross-diffusion. We suggest that our paradigm of comparing bifurcation-generated obstructions to the parameter validity of global-functional methods could also be of relevance for many other models beyond the one studied here.

math.AP↗

arXiv2016

A dynamical systems approach for the contact-line singularity in thin-film flows

We are interested in a complete characterization of the contact-line singularity of thin-film flows for zero and nonzero contact angles. By treating the model problem of source-type self-similar solutions, we demonstrate that this singularity can be understood by the study of invariant manifolds of a suitable dynamical system. In particular, we prove regularity results for singular expansions near the contact line for a wide class of mobility exponents and for zero and nonzero dynamic contact angles. Key points are the reduction to center manifolds and identifying resonance conditions at equilibrium points. The results are extended to radially-symmetric source-type solutions in higher dimensions. Furthermore, we give dynamical systems proofs for the existence and uniqueness of self-similar droplet solutions in the nonzero dynamic contact-angle case.

math.DS↗

arXiv2015

Uncertainty Transformation via Hopf Bifurcation in Fast-Slow Systems

Propagation of uncertainty in dynamical systems is a significant challenge. Here we focus on random multiscale ordinary differential equation models. In particular, we study Hopf bifurcation in the fast subsystem for random initial conditions. We show that a random initial condition distribution can be transformed during the passage near a delayed/dynamic Hopf bifurcation: (I) to certain classes of symmetric copies, (II) to an almost deterministic output, (III) to a mixture distribution with differing moments, and (IV) to a very restricted class of general distributions. We prove under which conditions the cases (I)-(IV) occur in certain classes vector fields.

math.DS↗

arXiv2015

Predictability of Critical Transitions

Critical transitions in multistable systems have been discussed as models for a variety of phenomena ranging from the extinctions of species to socio-economic changes and climate transitions between ice-ages and warm-ages. From bifurcation theory we can expect certain critical transitions to be preceded by a decreased recovery from external perturbations. The consequences of this critical slowing down have been observed as an increase in variance and autocorrelation prior to the transition. However especially in the presence of noise it is not clear, whether these changes in observation variables are statistically relevant such that they could be used as indicators for critical transitions. In this contribution we investigate the predictability of critical transitions in conceptual models. We study the quadratic integrate-and-fire model and the van der Pol model, under the influence of external noise. We focus especially on the statistical analysis of the success of predictions and the overall predictability of the system. The performance of different indicator variables turns out to be dependent on the specific model under study and the conditions of accessing it. Furthermore, we study the influence of the magnitude of transitions on the predictive performance.

physics.data-an↗