Stefan Siegmund

TU Dresden | TUD · Department of Mathematics

For nonautonomous linear difference equations with bounded coefficients on \(\mathbb {N}\) which have a bounded inverse, we introduce two different notions of spectra and discuss their relation to the well-known exponential dichotomy spectrum. The first new spectral notion is called Bohl spectrum and is based on an extended notion of the concept of...

Bohl dichotomy is a notion of hyperbolicity for linear nonautonomous difference equations that is weaker than the classical concept of exponential dichotomy. In the class of systems with bounded invertible coefficient matrices which have bounded inverses, we study the relation between the set $\mathrm{BD}$ of systems with Bohl dichotomy and the set...

For nonautonomous linear difference equations with bounded coefficients on $\mathbb{N}$ which have a bounded inverse, we introduce two different notions of spectra and discuss their relation to the well-known exponential dichotomy spectrum. The first new spectral notion is called Bohl spectrum and is based on an extended notion of the concept of Bo...

Using Maxwell's mental imagery of a tube of fluid motion of an imaginary fluid, we derive his equations $\operatorname{curl} \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$, $\operatorname{curl} \mathbf{H} = \frac{\partial \mathbf{D}}{\partial t} + \mathbf{j}$, $\operatorname{div} \mathbf{D} = \varrho$, $\operatorname{div} \mathbf{B} = 0$, wh...

A damped harmonic oscillator ẍ(t)+a1ẋ(t)+a0x(t)=0 with a0,a1>0 is known to be exponentially stable. We extend this result to time-varying positive coefficients a0(t), a1(t), t≥0, which are bounded from above and below and satisfy supt≥0a0(t)<(inft≥0a1(t))2 and we thus further extend the sufficient condition supt≥0a0(t)≤14(inft≥0a1(t))2 by Levin (19...

We consider a version of the pole placement problem for tempered one-sided linear discrete-time time-varying linear systems. We prove a sufficient condition for assignability of the nonuniform dichotomy spectrum by linear feedback. The main result is that the nonuniform dichotomy spectrum is assignable if the system is completely controllable and c...

We study fractional differential equations of Riemann–Liouville and Caputo type in Hilbert spaces. Using exponentially weighted spaces of functions defined on R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\...

We consider a local version of the assignment problem for the dichotomy spectrum of linear continuous time-varying systems defined on the half-line. Our aim is to show that uniform complete controllability is a sufficient condition to place the dichotomy spectrum of the closed-loop system in an arbitrary position within some Hausdorff neighborhood...

This work comprehensively describes and extends the results on asymptotic properties of linear discrete time-varying fractional order systems with Caputo and Riemann-Liouville forward and backward difference operators. In our considerations we take into account various definitions from the literature of fractional difference operators and we compar...

For nonautonomous linear difference equations, we introduce the notion of the so-called nonuniform dichotomy spectrum and prove a spectral theorem. As an application of the spectral theorem, we prove a reducibility result.

In this paper, we establish some criteria for boundedness, stability properties, and separation of solutions of autonomous nonlinear nabla Riemann‐Liouville scalar fractional difference equations. To derive these results, we prove the variation of constants formula for nabla Riemann‐Liouville fractional difference equations.

In this paper, we study the asymptotic behavior of solutions to a scalar fractional delay differential equations around the equilibrium points. More precise, we provide conditions on the coefficients under which a linear fractional delay equation is asymptotically stable and show that the asymptotic stability of the trivial solution is preserved un...

We formulate fractional difference equations of Riemann–Liouville and Caputo type in a functional analytical framework. Main results are existence of solutions on Hilbert space-valued weighted sequence spaces and a condition for stability of linear fractional difference equations. Using a functional calculus, we relate the fractional sum to fractio...

We study fractional differential equations of Riemann-Liouville and Caputo type in Hilbert spaces. Using exponentially weighted spaces of functions defined on $\mathbb{R}$, we define fractional operators by means of a functional calculus using the Fourier transform. Main tools are extrapolation- and interpolation spaces. Main results are the existe...

We consider general difference equations \(u_{n+1} = F(u)_n\) for \(n \in \mathbb {Z}\) on exponentially weighted \(\ell _2\) spaces of two-sided Hilbert space-valued sequences u and discuss initial value problems. As an application of the Hilbert space approach, we characterize exponential stability of linear equations and prove a stable manifold...

This paper is devoted to the numerical analysis of the abstract semilinear fractional problem \(D^\alpha u(t) = Au(t) + f(u(t)), u(0)=u^0,\) in a Banach space E. We are developing a general approach to establish a semidiscrete approximation of stable manifolds. The phase space in the neighborhood of the hyperbolic equilibrium can be split in such a...

We formulate fractional difference equations of Riemann-Liouville and Caputo type in a functional analytical framework. Main results are existence of solutions on Hilbert space-valued weighted sequence spaces and a condition for stability of linear fractional difference equations. Using a functional calculus, we relate the fractional sum to fractio...

In this paper, we establish variation of constant formulas for both Caputo and RiemannLiouville fractional difference equations. The main technique is the Z -transform. As an application, we prove a lower bound on the separation between two different solutions of a class of nonlinear scalar fractional difference equations.

The paper presents a comprehensive numerical study of mathematical models used to describe complex biological systems in the framework of integrated pest management. Our study considers two specific ecosystems that describe the application of control mechanisms based on pesticides and natural enemies, implemented in an impulsive and periodic manner...

We consider general difference equations $u_{n+1} = F(u)_n$ for $n \in \mathbb{Z}$ on exponentially weighted $\ell_2$ spaces of two-sided Hilbert space valued sequences $u$ and discuss initial value problems. As an application of the Hilbert space approach, we characterize exponential stability of linear equations and prove a stable manifold theore...

The continued scaling of CMOS semiconductor technology, together with the corresponding reduction of operating voltages, pose serious challenges for SRAM arrays regarding susceptibility to parameter variations, reduction of stability margins and reliability of read and write operations. One approach, alternative to various noise margin definitions,...

The paper is concerned with the development and numerical analysis of mathematical models used to describe complex biological systems in the framework of Integrated Pest Management (IPM). Established in the late 1950s, IPM is a pest management paradigm that involves the combination of different pest control methods in ways that complement one anoth...

In this paper, a discrete-time predator–prey model with Crowley–Martin functional response is investigated based on the center manifold theorem and bifurcation theory. It is shown that the system undergoes flip bifurcation and Neimark–Sacker bifurcation. An explicit approximate expression of the invariant curve, caused by Neimark–Sacker bifurcation...

In this paper, we investigate some aspects of the qualitative theory for multi-order fractional differential equation systems. First, we obtain a fundamental result on the existence and uniqueness for multi-order fractional differential equation systems. Next, a representation of solutions of homogeneous linear multi-order fractional differential e...

Strong gales can cause damages to buildings by lifting their roofs. The work by Siegmund (2015) suggests a method for a dynamic redistribution and redirection of wind force for securing buildings from damage as occasioned by hurricanes and the like. It is based on a semi-permeable textile membrane to counteract the aerodynamical uplift. The apparat...

In recent decades, Dengue fever and its deadly complications, such as Dengue hemorrhagic fever, have become one of the major mosquito-transmitted diseases, with an estimate of 390 million cases occurring annually in over 100 tropical and subtropical countries, most of which belonging to the developing world. Empirical evidence indicates that the mo...

We study local activity and its contrary, local passivity, for linear systems and show that generically an eigenvalue of the system matrix with positive real part implies local activity. If all state variables are port variables we prove that the system is locally active if and only if the system matrix is not dissipative. Local activity was sugges...

In this article we establish a bridge between dynamical systems, including topological and measurable dynamical systems as well as continuous skew product flows and nonautonomous dynamical systems; and coalgebras in categories having all finite products. We introduce a straightforward unifying definition of abstract dynamical system on finite produ...

In biochemical networks transient dynamics plays a fundamental role, since the activation of signalling pathways is determined by thresholds encountered during the transition from an initial state (e.g. an initial concentration of a certain protein) to a steady-state. These thresholds can be defined in terms of the inflection points of the stimulus...

The central theme of complex systems research is to understand the emergent macroscopic properties of a system from the interplay of its microscopic constituents. The emergence of macroscopic properties is often intimately related to the structure of the microscopic interactions. Here, we present an analytical approach for deriving necessary condit...

Our aim in this paper is to establish stable manifolds near hyperbolic equilibria of fractional differential equations in arbitrary finite dimensional spaces.

We study local activity and its opposite, local passivity, for linear systems and show that generically an eigenvalue of the system matrix with positive real part implies local activity. If all state variables are port variables we prove that the system is locally active if and only if the system matrix is not dissipative. Local activity was sugges...

The present paper is concerned with strong stability of solutions of non-autonomous equations of the form \(\dot{u}(t) = A(t)u(t)\), where A(t) is an unbounded operator in a Banach space depending almost periodically on t. A general condition on strong stability is given in terms of Perron conditions on the solvability of the associated inhomogeneo...

The mesochronic velocity is the average of the velocity field along trajectories generated by the same velocity field over a time interval of finite duration. In this paper we classify initial conditions of trajectories evolving in incompressible vector fields according to the character of motion of material around the trajectory. In particular, we...

In this paper, we present a family of three-point with eight-order convergence methods for finding the simple roots of nonlinear equations by suitable approximations and weight function based on Maheshwari’s method. Per iteration this method requires three evaluations of the function and one evaluation of its first derivative. These class of method...

This work concerns the modeling and dynamical study of a chemo-fluidic oscillator with the ability of coupling chemical and fluidic domains. The coupling is made possible by means of stimuli-responsive (also referred to as smart) hydrogels, which are able to change their volume under small variations of special thermodynamic parameters, in a revers...

The mesochronic velocity is the average of the velocity field along trajectories generated by the same velocity field over a time interval of finite duration. In this paper we classify initial conditions of trajectories evolving in incompressible vector fields according to the character of motion of material around the trajectory. In particular, we...

Cooperative behaviour is often accompanied by the incentives to defect, i.e., to reap the benefits of others' efforts without own contribution. We provide evidence that cooperation and defection can coexist under very broad conditions in the framework of evolutionary games on graphs under deterministic imitation dynamics. Namely, we show that for a...

We generalize the Picard–Lindelöf theorem on the unique solvability of initial value problems ẋ = f (t, x), x(t0) = x0, by replacing the sufficient classical Lipschitz condition of f with respect to x with a more general Lipschitz condition along hyperspaces of the (t, x)-space. A comparison with known results is provided and the generality of the...

A chemofluidic oscillator circuit that employs a hydrogel-based chemofluidic transistor for chemical-fluidic coupling is presented. It shows a period between 200 and 1000 s and alcohol concentrations oscillating between 2 wt% and 10 wt%. Because of the direct interaction with chemistry, chemofluidic transistors have the potential to facilitate labs...

In article 1600005, Andreas Richter and co-workers present a chemofluidic oscillator circuit that employs a hydrogel-based valve, with the valves state dependent on the chemical composition of the liquid. Operating with constant fluidic sources the circuit generates oscillations of flow rates, pressures and chemical concentrations with excellent po...

Inhomogeneous percolation, for its closer relationship with real-life, can be more useful and reasonable than homogeneous percolation to illustrate the critical phenomena and dynamical behaviour of complex networks. However, due to its intricacy, the theoretical framework of inhomogeneous percolation is far from being complete and many challenging...

We prove the theorem of linearized asymptotic stability for fractional differential equations. More precisely, we show that an equilibrium of a nonlinear Caputo fractional differential equation is asymptotically stable if its linearization at the equilibrium is asymptotically stable. As a consequence we extend Lyapunov's first method to fractional...

In this paper we consider the existence of periodic solutions of one-parameter and two-parameter families of second order singular differential equations. © 2015 Juliusz Schauder Centre for Nonlinear Studies Nicolaus Copernicus University.

In this paper we investigate a differential game in which countably many dynamical objects pursue a single one. All the players perform simple motions. The duration of the game is fixed. The controls of a group of pursuers are subject to integral constraints, and the controls of the other pursuers and the evader are subject to geometric constraints...

In this paper we investigate a differential game in which countably many dynamical objects pursue a single one. All the players perform simple motions. The duration of the game is fixed. The controls of a group of pursuers are subject to integral constraints, and the controls of the other pursuers and the evader are subject to geometric constraints...

In this paper, we present an extension of a theorem due to Katznelson and Tzafriri to non-autonomous linear differential equations with almost periodic coefficients of the form x′(t) = A(t)x(t). To this end, we consider the evolution semigroup associated with the equation in a small invariant function space consisting of almost periodic functions w...

The multiplicative ergodic theorem by Oseledets on Lyapunov spec-trum and Oseledets subspaces is extended to linear random difference equations with random delay. In contrast to the general multiplica-tive ergodic theorem by Lian and Lu, we can prove that a random dynamical system generated by a difference equation with random de-lay cannot have in...

We prove a partial linearization theorem for planar nonautonomous differential equations with one center-like direction and one hyperbolic direction. Gap conditions are formulated in terms of the dichotomy spectral intervals. The result is applied to the Duffing-van der Pol oscillator under a nonautonomous parametric perturbation.

Evolutionary games on graphs play an important role in the study of evo-lution of cooperation in applied biology. We provide so far nonexistent rigorous math-ematical concepts from a dynamical systems and graph theoretical point of view. We prove results on attractors for different utility functions and update orders. For complete graphs we charact...

In this paper, we present a family of three-point with eight-order convergence methods for finding the simple roots of nonlinear equations by suitable approximations and weight function based on Maheshwari method. Per iteration this method requires three evaluations of the function and one evaluation of its first derivative. This class of methods h...

In this paper, we present an iterative three-point method with memory based on the family of King's methods to solve nonlinear equations. This proposed method has eighth order convergence and costs only four function evaluations per iteration which supports the Kung-Traub conjecture on the optimal order of convergence. An acceleration of the conver...

We construct two optimal Newton–Secant like iterative methods for solving nonlinear equations. The proposed classes have convergence order four and eight and cost only three and four function evaluations per iteration, respectively. These methods support the Kung and Traub conjecture and possess a high computational efficiency. The new methods are...

We introduce a new class of optimal iterative methods without memory for approximating a simple root of a given nonlinear equation. The proposed class uses four function evaluations and one first derivative evaluation per iteration and it is therefore optimal in the sense of Kung and Traub's conjecture. We present the construction, convergence anal...

In this paper we investigate a differential game in which countably many dynamical objects pursue a single one. All the players perform simple motions. The duration of the game is fixed. The controls of a group of pursuers are subject to geometric constraints and the controls of the other pursuers and the evader are subject to integral constraints....

We introduce a concept of adjoint equation and Lyapunov regularity of a stochastic differential algebraic equation (SDAE) of index 1. The notion of adjoint SDAE is introduced in a similar way as in the deterministic di�erential algebraic equation (DAE) case. We prove a multiplicative ergodic theorem (MET) for the adjoint SDAE and the adjoint Lyapun...

This paper is devoted to the numerical analysis of the abstract semilinear parabolic problem u (t) = Au(t) + f (u(t)), u(0) = u 0 , in a Banach space E. We are developing a general approach to establish a discrete dichotomy in a very general setting and prove shadowing theorems that compare solutions of the continuous problem with those of discrete...

This paper presents a spatiotemporal dynamic model based on the interaction between multiple shear bands in the plastic flow of metallic glasses during compressive deformation. Various sizes of sliding events burst in the plastic deformation as the generation of different scales of shear branches occurred; microscopic creep events and delocalized s...

The classical Lipschitz-type criteria guarantee unique solvability of the scalar initial value problem $\dot x=f(t,x)$, $x(t_0)=x_0,$ by putting restrictions on $|f(t,x)-f(t,y)|$ in dependence of $|x-y|$. Geometrically it means that the field differences are estimated in the direction of the $x$-axis. In 1989, Stettner and the second author could e...

For nonautonomous linear differential equations with nonuniform hyperbolicity, we introduce a definition for nonuniform dichotomy spectrum, which can be seen as a generalization of Sacker-Sell spectrum. We prove a spectral theorem and use the spectral theorem to prove a reducibility result.

In networked control systems, the Try-Once-Discard (TOD) protocol is of high interest because its properties can be characterized by Lyapunov functions. This feature makes it practical to incorporate TOD into Lyapunov-based design of linear and nonlinear control systems, yielding a self-contained theory for system stabilization. In previous work, c...

We establish a new class of three-point methods for the computation of simple zeros of a scalar function. Based on the two-point optimal method by Ostrowski (1966), we construct a family of order eight methods which use three evaluations of f and one of f' and therefore have an efficiency index equal to and are optimal in the sense of the Kung and...

We introduce a new concept of finite-time entropy which is a local version of the classical concept of metric entropy. Based on that, a finite-time version of Pesin's entropy formula and also an explicit formula of finite-time entropy for $2$-D systems are derived. We also discuss about how to apply the finite-time entropy field to detect special d...

In this paper, we establish a local stable manifold theorem near a hyperbolic equilibrium point for planar fractional differential equations. The construction of this stable manifold is based on the associated Lyapunov–Perron operator. An example is provided to illustrate the result.

This paper is devoted to the numerical analysis of the abstract semilinear parabolic problem u (t) = Au(t) + f (u(t)), u(0) = u 0 , in a Banach space E. We are developing a general approach to establish a discrete dichotomy in a very general setting and prove shadowing theorems that compare solutions of the continuous problem with those of discrete...

In this paper we propose and explore a general notion of chaos in the abstract context of continuous actions of topological semigroups and show that any chaotic action on a Hausdorff uniform space is sensitive to initial conditions .

We provide sufficient criteria for the stability of positive linear switched systems on ordered Banach spaces. The switched systems can be generated by finitely many bounded operators in infinite-dimensional spaces with a general class of order-inducing cones. In the discrete-time case, we assume an appropriate interior point of the cone, whereas i...

Hyperbolicity of linear systems of difference and differential equations is a robust property. We provide a quantity to measure the maximal size of perturba-tions under which hyperbolicity is preserved. This so-called hyperbolicity radius is calculated by two methods, using the transfer operator and the input-output operator.

In this paper, we investigate a type of neutral difference equation with variable delay. By applications of Krasnosell skii's fixed point theorem and some new techniques, various sufficient conditions for the existence of one or twin positive periodic solutions are established.

In this paper we build on the spectral theory for linear fractional differential equations and prove that the fractional Lyapunov spectrum of solutions starting from a unit sphere is the union of a compact interval in R<0 and at most d distinct fractional Lyapunov exponents.

In this paper we propose and explore a general notion of chaos in the abstract context of continuous actions of topological semigroups and show that any chaotic action on a Hausdorff uniform space is sensitive to initial conditions.

In this note we prove that a fractional stochastic delay differential equation which satisfies natural regularity conditions generates a continuous random dynamical system on a subspace of a H\"older space which is separable.

In this paper we provide a geometric characterization of those locally compact Hausdorff topological groups which admit a faithful strongly chaotic continuous action on some Hausdorff space.

We prove a general result on the existence of periodic trajectories of systems of difference equations with finite state space which are phase-locked on certain components which correspond to cycles in the coupling structure. A main tool is the new notion of order-induced graph which is similar in spirit to a Lyapunov function. To develop a coheren...

We establish the link between linear Lyapunov functions for positive switched systems and corresponding Collatz-Wielandt sets. This leads to an algorithm to compute a linear Lyapunov function whenever a Lyapunov function exists.

The Multiplicative Ergodic Theorem by Oseledets on Lyapunov spectrum and Oseledets subspaces is extended to linear random differential equations with random delay, using a recent result by Lian and Lu. Random differential equations with bounded delay are discussed as an example.

The present paper is concerned with strong stability of solutions of non-autonomous equations of the form $\dot u(t)=A(t)u(t)$, where $A(t)$ is an unbounded operator in a Banach space depending almost periodically on $t$. A general condition on strong stability is given in terms of Perron conditions on the solvability of the associated inhomogeneou...

The central theme of complex systems research is understanding the emergent macroscopic properties of a system from the interplay of its microscopic constituents. Here, we ask what conditions a complex network of microscopic dynamical units has to meet to permit stationary macroscopic dynamics, such as stable equilibria or phase-locked states. We p...

A linear system $\dot x = Ax$, $A \in \mathbb{R}^{n \times n}$, $x \in \mathbb{R}^n$, with $\mathrm{rk} A = n-1$, has a one-dimensional center manifold $E^c = \{v \in \mathbb{R}^n : Av=0\}$. If a differential equation $\dot x = f(x)$ has a one-dimensional center manifold $W^c$ at an equilibrium $x^*$ then $E^c$ is tangential to $W^c$ with $A = Df(x...

We introduce a notion of attractivity for delay equations which are defined on bounded time intervals. Our main result shows that linear delay equations are finite-time attractive, provided that the delay is only in the coupling terms between different components, and the system is diagonally dominant. We apply this result to a nonlinear Lotka-Volt...

A hyperbolicity notion for linear differential equations $dotx = A(t)x$, $tin[t_-,t_+]$ , is defined which unifies different existing notions like finite-time Lyapunov exponents (Haller, 2001, [13], Shadden et al., 2005, [24]), uniform or M-hyperbolicity (Haller, 2001, [13], Berger et al., 2009, [6]) and $(t_-,(t_+-t_-))$-dichotomy (Rasmussen, 2010...