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Hans-Otto Walther

Publications and source records attributed to Hans-Otto Walther.

15 recordsLinked to original sources
arXiv2026

A direct approach to simplicity of solution manifolds

Differential equations with state-dependent delays define a semiflow of continuously differentiable solution operators in general only on the associated {\it solution manifold} $X\subset C^1([-h,0],\mathbb{R}^n)$. For systems with discrete state-dependent delays we construct a diffeomorphism on a neighbourhood of $X$ which takes $X$ to an open subset of the subspace given by $ϕ'(0)=0$. This is in line with earlier work on the nature of solution manifolds. The present approach, however, is new and dismisses all hypotheses beyond smoothness which have been instrumental so far. Compared to a recent case study it is more direct in the sense that theory of {\it algebraic delay systems} is avoided.

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arXiv2025

Dynamics of a state-dependent delay-differential equation

We present a detailed study of a scalar differential equation with threshold state-dependent delayed feedback. This equation arises as a simplification of a gene regulatory model. There are two monotone nonlinearities in the model: one describes the dependence of delay on state, and the other is the feedback nonlinearity. Both increasing and decreasing nonlinearities are considered. Our analysis is exhaustive both analytically and numerically as we examine the bifurcations of the system for various combinations of increasing and decreasing nonlinearities. We identify rich bifurcation patterns including Bautin, Bogdanov-Takens, cusp, fold, homoclinic, and Hopf bifurcations whose existence depend on the derivative signs of nonlinearities. Our analysis confirms many of these patterns in the limit where the nonlinearities are switch-like and change their value abruptly at a threshold. Perhaps one of the most surprising findings is the existence of a Hopf bifurcation to a periodic solution when the nonlinearity is monotone increasing and the time delay is a decreasing function of the state variable.

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arXiv2025

On Shilnikov's scenario with a homoclinic orbit in 3D

The paper provides a detailed proof that complicated motion exists in Shilnikov's scenario of a smooth vectorfield $V$ on $mathbb{R}^3$ with $V(0)=0$ so that the equation $x'=V(x)$ has a homoclinic solution $h$ with $\lim_{|t|\to\infty}h(t)=0$, and $DV(0)$ has eigenvalues $u>0$ and $σ\pmμ$, $σ<0<μ$, with $0<σ+u$.

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arXiv2025

On Shilnikov's scenario in 3D: Topological chaos for vectorfields of class $C^1$

Shilnikov's scenario in 3D consists of a vectorfield $V$ so that the equation $$ x'(t)=V(x(t))\in\mathbb{R}^3 $$ with $V(0)=0$ has a solution homoclinic to the origin and the eigenvalues of $DV(0)$ are $u>0$ and $σ\pm iμ$, $σ<0<μ$, with $0<σ+u$. We give a detailed proof that close to the homoclinic loop complicated motion exists provided $V$ is just once continuously differentiable. The result requires working with flows instead of an ODE, which necessitates major modifications compared to the earlier approach for twice continuously differentiable vectorfields in arXiv:2406.18289 .

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arXiv2024

On solution manifolds of some differential equations with more general state-dependent delay

Differential equations with state-dependent delays define a semiflow of continuously differentiable solution operators in general only on the associated {\it solution manifold} in the Banach space $C^1_n=C^1([-h,0],\mathbb{R}^n)$. For a prototypic example we develop a new proof that its solution manifold is diffeomorphic to an open subset of the subspace given by $ϕ'(0)=0$, without recourse to a restrictive hypothesis about the form of delays which is instrumental in earlier work on the nature of solution manifolds. The new proof uses the framework of algebraic-delay systems.

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arXiv2023

On solution manifolds for algebraic-delay systems

Differential equations with state-dependent delays define a semiflow of continuously differentiable solution operators in general only on an associated submanifold of the Banach space $C^1([-h,0],\mathbb{R}^n)$. We extend a recent result on simplicity of these {\it solution manifolds} to systems where the delay is given by the state only implicitly in an extra equation. Such algebraic-delay systems arise from various applications. Key words: Delay differential equation, implicit state-dependent delay, solution manifold

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arXiv2022

Solution manifolds of differential systems with discrete state-dependent delays are almost graphs

We show that for a system $$ x'(t)=g(x(t-d_1(Lx_t)),\dots,x(t-d_k(Lx_t))) $$ of $n$ differential equations with $k$ discrete state-dependent delays the solution manifold, on which solution operators are differentiable, is nearly as simple as a graph over a closed subspace in $C^1([-r,0],\mathbb{R}^n)$. The map $L$ is continuous and linear from $C([-r,0],\mathbb{R}^n)$ onto a finite-dimensional vectorspace, and $g$ as well as the delay functions $d_κ$ are assumed to be continuously differentiable.

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arXiv2022

On the solution manifold of a differential equation with a delay which has a zero

For a differential equation with a state-dependent delay we show that the associated solution manifold $X_f$ of codimnsion 1 in the space $C^1([-r,0],\mathbb {R})$ is an almost graph over a hyperplane, which implies that $X_f$ is diffeomorphic to the hyperplane. For the case considered previous results only provide a covering by 2 almost graphs.

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arXiv2021

A finite atlas for solution manifolds of differential systems with discrete state-dependent delays

Let $r>0, n\in\mathbb{N}, {\bf k}\in\mathbb{N}$. Consider the delay differential equation $$ x'(t)=g(x(t-d_1(Lx_t)),\ldots,x(t-d_{\bf k}(Lx_t))) $$ for $g:(\mathbb{R}^n)^{\bf k}\supset V\to\mathbb{R}^n$ continuously differentiable, $L$ a continuous linear map from $C([-r,0],\mathbb{R}^n)$ into a finite-dimensional vectorspace $F$, each $d_k:F\supset W\to[0,r]$, $k=1,\ldots,{\bf k}$, continuously differentiable, and $x_t(s)=x(t+s)$. The solutions define a semiflow of continuously differentiable solution operators on the submanifold $X_f\subset C^1([-r,0],\mathbb{R}^n)$ which is given by the compatibility condition $ϕ'(0)=f(ϕ)$ with $$ f(ϕ)=g(ϕ(-d_1(Lϕ)),\ldots,ϕ(-d_{\bf k}(Lϕ))). $$ We prove that $X_f$ has a finite atlas of at most $2^{\bf k}$ manifold charts, whose domains are almost graphs over $X_0$. The size of the atlas depends solely on the zerosets of the delay functions $d_k$.

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arXiv2021

Operon Dynamics with State Dependent Transcription and/or Translation Delays

Transcription and translation retrieve and operationalize gene encoded information in cells. These processes are not instantaneous and incur significant delays. In this paper we study Goodwin models of both inducible and repressible operons with state-dependent delays. The paper provides justification and derivation of the model, detailed analysis of the appropriate setting of the corresponding dynamical system, and extensive numerical analysis of its dynamics. Comparison with constant delay models shows significant differences in dynamics that include existence of stable periodic orbits in inducible systems and multistability in repressible systems. A combination of parameter space exploration, numerics, analysis of steady state linearization and bifurcation theory indicates the likely presence of Shilnikov-type homoclinic bifurcations in the repressible operon model.

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arXiv2020

Autonomous linear neutral equations with bounded Borel functions as initial data

For equations with the neutral term strictly delayed we construct the fundamental solution, derive a variation-of-constants formula for inhomogeneous equations, and prove growth estimates. Only unavoidable measure and integration theory, up to the Riesz Representation Theorem, is used. A key notion is pointwise convergence of bounded sequences of continuous functions.

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arXiv2018

Fréchet differentiability in Fréchet spaces, and differential equations with unbounded variable delay

We introduce and discuss Fréchet differentiability for maps between Fréchet spaces. For delay differential equations $x'(t)=f(x_t)$ we construct a continuous semiflow of continuously differentiable solution operators $x_0\mapsto x_t$, $t\ge0$, on submanifolds of the Fréchet space $C^1((-\infty,0],\mathbb{R}^n)$, and establish local invariant manifolds at stationary points by means of transversality and embedding properties. The results apply to examples with unbounded but locally bounded delay.

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arXiv2018

Delay differential equations with differentiable solution operators on open domains in $C((-\infty,0],\mathbb{R}^n)$, and processes for Volterra integro-differential equations

For autonomous delay differential equations $x'(t)=f(x_t)$ we construct a continuous semiflow of continuously differentiable solution operators $x_0\mapsto x_t$, $t\ge0$, on open subsets of the Fréchet space $C((-\infty,0],\mathbb{R}^n)$. For nonautonomous equations this yields a continuous process of differentiable solution operators. As an application we obtain processes which incorporate all solutions of Volterra integro-differential equations $x'(t)=\int_0^tk(t,s)h(x(s))ds$

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arXiv2016

Response of an oscillatory differential delay equation to a single stimulus

Here we analytically examine the response of a limit cycle solution to a simple differential delay equation to a single pulse perturbation of the piecewise linear nonlinearity. We construct the unperturbed limit cycle analytically, and are able to completely characterize the perturbed response to a pulse of positive amplitude and duration with onset at different points in the limit cycle. We determine the perturbed minima and maxima and period of the limit cycle and show how the pulse modifies these from the unperturbed case.

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