Hans-Otto Walther

Justus-Liebig-Universität Gießen | JLU · Mathematisches Institut

Let h>0, open, and continuously differentiable. If f satisfies two mild additional smoothness conditions then the setis a C1-submanifold of codimension n in , the maximal solutions xφ of the initial value problemsdefine a continuous semiflow F on X, and all operators F(t,·) are continuously differentiable. Their derivatives D2F(t,φ) are given in th...

The first part of this paper is a general approach towards chaotic dynamics for a continuous map (Formula presented.) which employs the fixed point index and continuation. The second part deals with the differential equation (Formula presented.)with state-dependent delay. For a suitable parameter (Formula presented.) close to (Formula presented.) w...

We consider a periodic function \(p:{\mathbb {R}}\rightarrow {\mathbb {R}}\) of minimal period 4 which satisfies a family of delay differential equations $$\begin{aligned} x'(t)=g(x(t-d_{\Delta }(x_t))),\quad \Delta \in {\mathbb {R}}, \end{aligned}$$ (0.1) with a continuously differentiable function \(g:{\mathbb {R}}\rightarrow {\mathbb {R}}\) and...

We construct a delay functional dU with values in (0,r) and find a positive number h<r such that the negative feedback equationx′(t)=−x(t−dU(xt,r)), with the segment xt,r:[−r,0]→R given by xt,r(s)=x(t+s), has a solution whose short segments xt,h=xt,r|[−h,0] are dense in an open subset of the space C1([−h,0],R). The domain U of dU is open in C1([−r,...

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 decrea...

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 $\sigma\pm\mu$, $\sigma<0<\mu$, with $0<\sigma+u$.

For autonomous delay differential equations x′(t) = f(xt) we construct a continuous semiflow of continuously differentiable solution operators x0 ↦ xt, t ≥ 0, on open subsets of the Fréchet space C((−∞, 0], ℝn). For nonautonomous equations this yields a continuous process of differentiable solution operators. As an application we obtain processes w...

UDC 517.9 Differential equations with state-dependent delays specify a semiflow of continuously differentiable solution operators, in general, only on an associated submanifold of the Banach space C 1 ( [ - h ,0 ] , R n ) . We extend a recent result on the simplicity of these {\it solution manifolds} to systems in which the delay is given by the st...

Differential equations with state-dependent delays define a semiflow of continuously differen-tiable solution operators in general only on an associated submanifold of the Banach space C 1 ([−h, 0], R n). We extend a recent result on simplicity of these solution manifolds to systems where the delay is given by the state only implicitly in an extra...

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([...

For a differential equation with a state-dependent delay we show that the associated solution manifold X f of codimension 1 in the space C 1 ( [ − r , 0 ] , 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.

Let r > 0, n ∈ N, k ∈ N. Consider the delay differential equation x (t) = g(x(t − d1(Lxt)),. .. , x(t − d k (Lxt))) for g : (R n) k ⊃ V → R n continuously differentiable, L a continuous linear map from C([−r, 0], R n) into a finite-dimensional vector-space F , each d k : F ⊃ W → [0, r], k = 1,. .. , k, continuously differentiable, and xt(s) = x(t +...

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...

For differential equations with state-dependent delays the associated initial value problem is well-posed, with differentiable solution operators, on submanifolds of the space \begin{document}$ C^1_n=C^1([-r,0],\mathbb{R}^n) $\end{document}, under mild smoothness assumptions. We study these solution manifolds and find that for a large class of equa...

For autonomous delay differential equations x'(t)=f(xt){x'(t)=f(x_t)} we construct a continuous semiflow of continuously differentiable solution operators x0xt{x_0 \to x_t}, t0{t \le 0}, on open subsets of the Frechet space C((-,0],Rn){C((-\infty, 0], R^n)}. For nonautonomous equations this yields a continuous process of differentiable solution ope...

We construct examples of nonlinear maps on function spaces which are continuously differentiable in the sense of Michal and Bastiani but not in the sense of Fréchet. The search for such examples is motivated by studies of delay differential equations with the delay variable and not necessarily bounded.

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\s...

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...

We construct a delay functional d on an open subset of the space $$C^1_r=C^1([-r,0],\mathbb {R})$$ C r 1 = C 1 ( [ - r , 0 ] , R ) and find $$h\in (0,r)$$ h ∈ ( 0 , r ) so that the equation $$\begin{aligned} x'(t)=-x(t-d(x_t)) \end{aligned}$$ x ′ ( t ) = - x ( t - d ( x t ) ) defines a continuous semiflow of continuously differentiable solution ope...

Differential equations with state-dependent delays which generalize the scalar example(0)x′(t)=g(x(t),x(t−d(xt))) where g:R2→R and d:C([−r,0],R)→[0,r] are continuously differentiable, and with xt:[−r,0]→R given by xt(s)=x(t+s), define semiflows of differentiable solution operators on an associated submanifold of the state space C1=C1([−r,0],Rn). Wh...

We construct a delay functional \(d_Y\) on an infinite-dimensional subset \(Y\subset C^1([-r,0],\mathbb {R})\), \(r>1\), so that the delay differential equation $$\begin{aligned}x'(t)=-\alpha \,x(t-d_Y(x_t)),\quad \alpha >0,\end{aligned}$$has a solution whose short segments \(x_t|[-1,0]\) are dense in \(C^1([-1,0],\mathbb {R})\). This implies compl...

In infinite-dimensional spaces there are non-equivalent notions of continuous differentiability which can be used to derive the familiar results of calculus up to the Implicit Function Theorem and beyond. For autonomous differential equations with variable delay, not necessarily bounded, the search for a state space in which solutions are unique an...

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 Fr\'echet space $C((-\infty,0],\mathbb{R}^n)$. For nonautonomous equations this yields a continuous process of differentiable solution operators. As an appl...

We introduce and discuss Fr\'echet differentiability for maps between Fr\'echet 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 Fr\'echet space $C^1((-\infty,0],\mathbb{R}^n)$, and establish local invariant...

We construct examples of nonlinear maps on function spaces which are continuously differentiable in the sense of Michal and Bastiani but not in the sense of Fre´chet. The search for such examples is motivated by studies of delay differential equations with the delay variable and not necessarily bounded.

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 onse...

Consider the delay differential equation $x'(t)=f(x_t)$ with the history $x_t:(-\infty,0]\to\mathbb{R}^n$ of $x$ at 'time' $t$ defined by $x_t(s)=x(t+s)$. In order not to lose any possible entire solution of any example we work in the Fréchet space $C^1((-\infty,0],\mathbb{R}^n)$, with the topology of uniform convergence of maps and their derivativ...

We construct a semiflow of continuously differentiable solution operators for delay differential equations x′(t) = f(xt) with f defined on an open subset of the Fréchet space C¹ = C¹((-∞, 0],ℝⁿ). This space has the advantage that it contains all histories xt = x(t+ ·), t ε ℝ, of every possible entire solution of the delay differential equation, in...

For neutral delay differential equations of the form $$\begin{aligned} \dot{x}(t)=g(\partial x_t,x_t), \end{aligned}$$ with \(g\) defined on an open subset of the space \(C([-h,0],\mathbb {R}^n)\times C^1([-h,0],\mathbb {R}^n)\) , we extend an earlier principle of linearized stability. The present result applies to a wider class of neutral differen...

For \(\frac{\pi }{2}<\alpha <\frac{5\pi }{2}\) the zero solution of the linear equation $$\begin{aligned} x'(t)=-\alpha x(t-1) \end{aligned}$$is hyperbolic with two-dimensional unstable space. For \(\alpha \) close to \(\frac{5\pi }{2}\) we replace the constant delay \(1\) by a state-dependent delay $$\begin{aligned} d_U:C\supset U\rightarrow (0,2)...

For π2<α<5π2 the linear differential equationx′(t)=−αx(t−1) with constant time lag is unstable and hyperbolic. For α sufficiently close to 5π2 we construct a state-dependent delay dU(ϕ)∈(0,2) on an open set in the space C1([−2,0]), with dU(ϕ)=1 for ϕ close to 0, so that the nonlinear equationx′(t)=−αx(t−dU(xt)) has a one-parameter-family of entire...

We consider nonautonomous retarded functional differential equations under hypotheses which are designed for the application to equations with variable time lags, which may be unbounded, and construct an evolution system of solution operators which are continuously differentiable. These operators are defined on manifolds of continuously differentia...

There exist a delay functional d on an open set U⊂C([-2,0], ℝ) and a parameter α>0 so that the equation x ' (t)=-945x(t-d(x t )) has a solution which is homoclinic to zero, the zero equilibrium is hyperbolic with 2-dimensional unstable manifold, and the stable manifold and unstable manifold intersect transversely along the homoclinic flowline. We p...

We introduce delay differential equations, give some motivation by applications, review basic facts about initial value problems from wellposedness to the variation-of-constants formula in the sun-star-framework, and discuss two topics in greater detail: (a) The dynamics generated by autonomous scalar equations with a single, constant time lag, fro...

In 1806 Poisson published one of the first papers on functional differential equations. Among others he studied an example with a state-dependent delay, which is motivated by a geometric problem. This example is not covered by recent results on initial value problems for differential equations with state-dependent delay. We show that the example ge...

We show that under mild hypotheses neutral functional differential equations where delays may be state-dependent generate continuous semiflows, a larger one on a thin set in a Banach space of C 1-functions and a smaller one, with better smoothness properties, on a closed subset in a Banach manifold of C 2-functions. The hypotheses are satisfied for...

The life of Dov Tamari is described, including a brief introduction to his mathematical work.

For differential delay equations of the general form x′(t)=g(xt)x′(t)=g(xt) which include equations with unbounded finite state-dependent delays we construct semiflows of continuously differentiable solution operators on suitable Banach manifolds and provide local stable and unstable manifolds at equilibria. Examples occur in feedback systems with...

The note describes the role of the Mackey-Glass equation and of a similar equation due to Lasota in the study of nonlinear delay differential equations between 1977 and 2012, as far as rigorous mathematical results are concerned, and from the very personal point of view of the author's involvement.

We prove a principle of linearized stability for semiflows generated by neutral functional differential equations of the form $$ x'(t)=g(\partial\,x_t,x_t). $$ The state space is a closed subset in a manifold of C 2-functions. Applications include equations with state-dependent delay, as for example $$ x'(t)=A(x'(t+d(x(t))))+f(x(t+r(x(t)))) $$...

We prove a principle of linearized stability for semiflows generated by neutral functional differential equations of the form x′(t) = g(∂ x t , x t ). The state space is a closed subset in a manifold of C 2-functions. Applications include equations with state-dependent delay, as for example x′(t) = a x′(t + d(x(t))) + f (x(t + r(x(t)))) with \({a\i...

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 c...

Consider the functional differential equation (FDE) ẋ(t) = f(x t) with f defined on an open subset of the space C1 = C1([-h, 0], Rn). Under mild smoothness assumptions, which are designed for the application to differential equations with state-dependent delays, the FDE generates a semiflow on a submanifold of C 1 with continuously differentiable t...

Systems of the form $$\begin{array}{ll}x'(t) = g(r(t),x_t)\\ \quad\,\, 0 = \Delta(r(t),x_t) \end{array}$$generalize differential equations with delays r(t) < 0 which are given implicitly by the history x t of the state. We show that the associated initial value problem generates a semiflow with differentiable solution operators on a Banach manifold...

We study a differential equation for delayed negative feedback which models a situation where the delay depends on the present state and becomes effective in the future. The main result is existence of a periodic solution in case the equilibrium is linearly unstable. The proof employs the ejective fixed point principle on a compact convex set K0⊂C(...

Consider the functional differential equation (FDE) ẋ(t) = f(x t) with f defined on an open subset of the space C1 = C1([-h, 0], ℝn). Under mild smoothness assumptions, which are designed for the application to differential equations with state-dependent delays, the FDE generates a semiflow on a submanifold of C 1 with continuously differentiable t...

Automatic soft landing is modeled by a differential equation with state-dependent delay. It is shown that in the model soft landing occurs for an open set of initial data, which is determined by means of a smooth invariant manifold.

We consider a one-parameter family of delay differential equations which has been proposed as a model for a prize and prove that at a critical parameter where the linearization at equilibrium has a double zero eigenvalue periodic solutions bifurcate off with periods descending from infinity.

For periodic solutions to the autonomous delay differential equation x¢(t) = -mx(t) + f(x(t-1))x^{\prime}(t) =-\mu x(t) + f(x(t-1)) with rational periods we derive a characteristic equation for the Floquet multipliers. This generalizes a result from an earlier paper where only periods larger than 2 were considered. As an application we obtain a cri...

We consider a one-parameter family of delay differential equations which has been proposed as a model for a prize and prove that at a critical parameter where the linearization at equilibrium has a double zero eigenvalue periodic solutions bifurcate off with periods descending from infinity.

Proofs for most of the basic results presented in this part are found in the monograph [57]. See also [105].

An unusual bifurcation to time-periodic oscillations of a class of delay differential equations is investigated. As we approach the bifurcation point, both the amplitude and the frequency of the oscillations go to zero. The class of delay differential equations is a nonlinear extension of a nonevasive control method and is motivated by a recent stu...

It is shown that periodic solutions of a delay differential equation approach a square wave if a parameter becomes large. The equation models short-term prize fluctuations. The proof relies on the fact that the branches of the unstable manifold at equilibrium tend to the periodic orbit.

We consider periodic solutions of nonlinear functional differential equations with rational periods less than 2. We study the spectral properties of monodromy operators and state a hyperbolicity criterion for such solutions.

We consider periodic solutions to the equation x ' (t)=-μx(t)+f(x(t-1)),(1) where μ>0 and f:ℝ→ℝ is an odd continuously differentiable function. In the case of rational periods greater than 2, a criterion for hyperbolicity was obtained in H.O. Walther and A.L. Skubachevskii [Trans. Mosc. Math. Soc. 2003, 1–44 (2003); translation from Tr. Mosk. Mat....

Differential equations with state-dependent delay can often be written as ${\dot x}$ (t)=f(x_t) with a continuously differentiable map f from an open subset of the space C1=C1([-h,0], {ℝ}^n), {h>0}, into {ℝ}^n. In a previous paper we proved that under two mild additional conditions the set $X = \{ \phi \in U:\left( 0 \right) = f(\phi )\} $ is...

A delay differential equation is presented which models how the behavior of traders influences the short time price movements of an asset. Sensitivity to price changes is measured by a parameter a. There is a single equilibrium solution, which is non-hyperbolic for all a>0. We prove that for a< 1 the equilibrium is asymptotically stable, and that f...

A delay differential equation is presented which models how the behavior of traders influences the short time price movements of an asset. Sensitivity to price changes is measured by a parameter a. There is a single equilibrium solution, which is non-hyperbolic for all a>0. We prove that for a< 1 the equilibrium is asymptotically stable, and that f...

It is shown that an autonomous delay dierential system for a damped spring with a delayed restoring force has a periodic solution whose orbit is exponentially stable with asymptotic phase.

We study a system of equations which is based on Newton''s law and models automatic position control by echo. Rewritten as a delay differential equation with state-dependent delay the model defines a semiflow on a submanifold in the space of continuously differentiable maps [-h, 0] 2. All time-t maps and the restriction of the semiflow given by t >...

We consider an autonomous system of a differential and a functional equation for one-dimensional motion of an object which attempts to regulate its distance from a given point by means of reflected signals. In a suitable, compact state space the forward initial value problem is well-posed. For certain configurations of the parameters involved we pr...

For equations (i) over dot(t) = -mux(t) + f(x(t - 1)) with continuous odd nonlinearities close to a step function f(a)(xi) = -a sign xi, a > 0, we find sets of initial data to which solutions return. For Lipschitz nonlinearities the associated return maps become Lipschitz continuous. Monotonicity of f close to 0 permits sharp estimates of the Lipsc...

The delay differential equation, [(x)\dot]\dot x (t)=–x(t)+f(x(t–1)), with >0 and a real function f satisfying f(0)=0 and f>0 models a system governed by delayed positive feedback and instantaneous damping. Recently the geometric, topological, and dynamical properties of a three-dimensional compact invariant set were described in the phase space C...

Let a C 1 -function f:ℝ→ℝ be given which satisfies f(0)=0, f ' (ξ)<0 for all ξ∈ℝ, and supf<∞ or -∞<inff. Let C=C([-1,0],ℝ). For an open-dense set of initial data the phase curves [0,∞)→C given by the solutions [-1,∞)→ℝ to the negative feedback equation x ' (t)=-μx(t)+f(x(t-1)),withμ>0, are absorbed into the positively invariant set S⊂C of data ϕ≠0...

We construct a smooth function g* : IR ℝ IR with such that the equation has a slowly oscillating periodic solution y, and a slowly oscillating solution z* whose phase curve is homoclinic with respect to the orbit o of y in the space C = C0([1,0],IR). For an associated Poincaré map we obtain a transversal homoclinic loop. The proof of transversality...

We study the equation x’(t) = g(x(t-1)) (g) for smooth functions g: ℝ → R satisfying ξg(ξ) < 0 for ξ ≠ 0, and the equation x’(t) = b(t)x(t-1) (b) with a periodic coefficient b: ℝ → (-∞, 0). Equation (b) generalizes variational equations along periodic solutions y of equation (g) in case g’(ξ) < 0 for all ξ ∈ y(ℝ). We investigate the largest Floquet...

We prove that certain slowly oscillating periodic solutions of differential delay equations x ˙(t)=-μx(t)+f(x(t-1)) are unstable and hyperbolic, with precisely one simple Floquet multiplier outside the unit circle.

In an earlier paper we generalized the notion of a hyperbolic set and proved that the Shadowing Lemma remains valid, for C1-maps which need not be invertible. Here we establish the existence of (generalized) hyperbolic structures along transversal homoclinic trajectories of C1-maps. The hyperbolic structure and shadowing are then used to give a new...