# Peter KloedenAuburn University | AU · Department of Mathematics & Statistics

Peter Kloeden

BA, PhD, DSc

## About

562

Publications

64,718

Reads

**How we measure 'reads'**

A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more

23,168

Citations

Citations since 2016

Introduction

Until recently Peter Kloeden worked at the School of Mathematics and Statistics, Huazhong University of Science and Technology. He is now an Affiliated Professor at Auburn University and and a visiting researcher at the University of Tuebingen. He lives in Tuebingen. Peter does research in Analysis, Applied Mathematics and Probability Theory. One current project is 'Higher order numerical approximation methods for stochastic partial differential equations (SPDEs)'.

Additional affiliations

December 2014 - November 2017

September 1997 - September 2014

January 1997 - August 1997

## Publications

Publications (562)

This paper examines the stationary distribution of the stochastic Lotka-Volterra model with infinite delay. Since the solutions of stochastic functional or delay differential equations depend on their history, they are non-Markov, which implies that traditional techniques based on the Markov property, cannot be applicable. This paper uses the varia...

This paper examines the stationary distribution of the stochastic Lotka-Volterra model with infinite delay.
Since the solutions of stochastic functional or delay differential equations depend on their history, they are
non-Markov, which implies that traditional techniques based on the Markov property, cannot be applicable.
This paper uses the varia...

We provide a tool for diagnosing preference heterogeneity (e.g., present bias) in a resource game. This game constitutes of two present-biased players, i.e., resource harvesters. Each resource harvester has three decision-making mechanisms: naive, precommitted and sophisticated paradigms. It is proven that (i) the harvester’s decision to extract re...

In theoretical ecology, models describing the spatial dispersal and the temporal evolution of species having non-overlapping generations are often based on integrodifference equations. For various such applications the environment has an aperiodic influence on the models leading to nonautonomous integrodifference equations. In order to capture thei...

In order to determine the dynamics of nonautonomous equations both their forward and pullback behavior need to be understood. For this reason we provide sufficient criteria for the existence of such attracting invariant sets in a general setting of nonautonomous difference equations in metric spaces. In addition it is shown that both forward and pu...

It is shown that the attractor of an autonomous Caputo fractional differential equation of order α∈(0,1) in Rd whose vector field has a certain triangular structure and satisfies a smooth condition and dissipativity condition is essentially the same as that of the ordinary differential equation with the same vector field. As an application, we esta...

Due to the restriction of duality structure, existing discounting methods in Lévy models have not met the requirements from psychologists and economists. To address this weakness, our paper invents a stochastic heterogeneous quasi-hyperbolic (SHQH) clock. The SHQH clock has the following strengths: (i) being invisible in the duality structure; (ii)...

A dynamical system with a plastic self-organising velocity vector field was introduced in Janson and Marsden (Sci Rep 7:17007, 2017) as a mathematical prototype of new explainable intelligent systems. Although inspired by the brain plasticity, it does not model or explain any specific brain mechanisms or processes, but instead expresses a hypothesi...

In theoretical ecology, models describing the spatial dispersal and the temporal evolution of species having non-overlapping generations are often based on integrodifference equations. For various such applications the environment has an aperiodic influence on the models leading to nonautonomous integrodifference equations. In order to capture thei...

In this paper, we approximate a nonautonomous neural network with infinite delay and a Heaviside signal function by neural networks with sigmoidal signal functions. We show that the solutions of the sigmoidal models converge to those of the Heaviside inclusion as the sigmoidal parameter vanishes. In addition, we prove the existence of pullback attr...

Lattice difference equations are essentially difference equations on a Hilbert space of bi-infinite sequences. They are motivated by the discretization of the spatial variable in integrodifference equations arising in theoretical ecology. It is shown here that under similar assumptions to those used for such integrodifference equations they have a...

In this paper, we study the long term dynamical behavior of a two-dimensional nonlocal diffusion lattice system with delay. First some sufficient conditions for the construction of an exponential attractor are presented for infinite dimensional autonomous dynamical systems with delay. Then, the existence of exponential attractors for the two-dimens...

A mathematical model describing the growth of gut microbiome inside and on the wall of the gut is developed based on the chemostat model with wall growth. Both the concentration and flow rate of the nutrient input are time-dependent, which results in a system of non-autonomous differential equations. First the stability of each meaningful equilibri...

In the first part of this paper, we establish the well-posedness for solutions of Caputo stochastic fractional differential equations (for short Caputo SFDE) of order α∈(12,1) in Lp spaces with p≥2 whose coefficients satisfy a standard Lipschitz condition. More precisely, we first show a result on the existence and uniqueness of solutions, next we...

It is shown that the attractor of an autonomous Caputo fractional differential equation of order $\alpha\in(0,1)$ in $\mathbb{R}^d$ whose vector field has a certain triangular structure and satisfies a smooth condition and dissipativity condition is essentially the same as that of the ordinary differential equation with the same vector field. As an...

The aim of this work is to analyze the mean-square convergence rates of numerical schemes for random ordinary differential equations (RODEs). First, a relation between the global and local mean-square convergence order of one-step explicit approximations is established. Then, the global mean-square convergence rates are investigated for RODE-Taylor...

Kloeden, Peter E.Yang, MeihuaThe nature of time in a nonautonomous dynamicalDynamical system system isDynamical system, autonomous very different from that in autonomous systemsAutonomous, which depend on the time that has elapsed since starting rather on the actual time. This requires new concepts of invariant sets and attractors. Pullback and for...

First some sufficient conditions are presented for the existence and the construction of pullback exponential attractors for infinite dimensional non-autonomous dynamical systems with delays. This abstract result is then used to establish the existence of pullback exponential attractors for an infinite lattice model of non-autonomous recurrent neur...

This article develops a general framework for preference heterogeneity. This framework includes a discount function, a nonstandard Hamilton–Jacobi–Bellman equation (HJB) and a behavioral equation. When controlling parameters, our discount function and its HJB can reduce to those in Marín‐Solano and Patxot (2012) and among many others. In various fi...

In this paper, we study a local equi-attraction of pullback attractors for non-autonomous processes. By the local equi-attraction we mean that any local part of sections of a pullback attractor A={A(τ)}τ∈R are pullback attracting at the same rate, i.e., for any bounded (but arbitrarily large) interval I,limt→∞(supτ∈IdistX(U(t,τ−t,B),A(τ)))=0,∀B⊂X...

A family of nonautonomous coupled inclusions governed by $p(x)$-Laplacian operators with large diffusion is investigated. The existence of solutions and pullback attractors as well as the generation of a generalized process are established. It is shown that the asymptotic dynamics is determined by a two dimensional ordinary nonautonomous coupled in...

An autonomous Caputo fractional differential equation of order α ∈ (0,1) in a finite dimensional space whose vector field satisfies a global Lipschitz condition is shown to generate a semi-dynamical system in the function space \(\mathfrak {C}\) of continuous functions with the topology uniform convergence on compact subsets. This contrasts with a...

In this paper we study the long-time behavior of a 2D Navier–Stokes equation. It is shown that under small forcing intensity the global attractor of the equation is a singleton. When endowed with additive or multiplicative white noise no sufficient evidence was found that the random attractor keeps the singleton structure, but the estimate of the c...

A nonautonomous lattice system with discrete Laplacian operator is revisited in the weighted space of infinite sequences \begin{document}$ {{\ell_{\rho}^2}} $\end{document}. First the existence of a pullback attractor in \begin{document}$ {{\ell_{\rho}^2}} $\end{document} is established by utilizing the dense inclusion of \begin{document}$ \ell^2 \...

An infinite lattice model of a recurrent neural network with random connection strengths between neurons is developed and analyzed. To incorporate the presence of various type of delays in the neural networks, both discrete and distributed time varying delays are considered in the model. For the existence of random pullback attractors and periodic...

In [3] the solutions of a sigmoidal neural field lattice system based on an Amari-Hopfield neural field lattice model, in which the Heaviside function is replaced by a simplifying sigmoidal function characterized by a small parameter ε, were shown to converge to the solution of the Heaviside lattice system as ε→0, through an inflated lattice system...

The existence of numerical attractors for lattice dynamical systems is established, where the implicit Euler scheme is used for time discretisation. Infinite dimensional discrete lattice systems as well as their finite dimensional truncations are considered. It is shown that the finite dimensional numerical attractors converge upper semicontinuousl...

When solving for optimal strategies, a financial engineer needs to take into consideration of preferences heterogeneities, which involve not only present bias, but also future-focused preferences. We provide a reusable tool (i.e. algorithm) for explicitly solving optimal strategy in the presence of preferences variation over time, decision-makers a...

This article establishes optimal upper and lower error estimates for strong full-discrete numerical approximations of the stochastic heat equation driven by space-time white noise. Thereby, this work proves the optimality of the strong convergence rates for certain full-discrete approximations of stochastic Allen–Cahn equations with space-time whit...

A non-autonomous free boundary model for tumor growth is studied. The model consists of a nonlinear reaction diffusion equation describing the distribution of vital nutrients in the tumor and a nonlinear integro-differential equation describing the evolution of the tumor size. First the global existence and uniqueness of a transient solution is est...

This paper establishes a reusable framework, including a stochastic heterogeneous quasi-hyperbolic (SHQH) discount function, its non-standard Hamilton-Jacobi-Bellman equation (HJB) and its naive and precommitted solutions. To gurantee the broad generalities of the framework, we adopt a game theoretic approach in the sense of refraining from imposin...

In this paper, we first construct a Euler–Maruyama type scheme for Caputo stochastic fractional differential equations (for short Caputo SFDE) of order α∈(12,1) whose coefficients satisfy a standard Lipschitz and a linear growth bound conditions. The strong convergence rate of this scheme is established. In particular, it is α−12 when the coefficie...

Random attractors and their higher-order regularity properties are studied for stochastic reaction–diffusion equations on time-varying domains. Some new a priori estimates for the difference of solutions near the initial time and the continuous dependence in initial data in [Formula: see text] are proved. Then attraction of the random attractors in...

A free boundary model for tumor growth with time dependent nutritional supply and infinite time delays is studied. The governing system consists of a nonlinear reaction diffusion equation describing the distribution of vital nutrients in the tumor and a nonlinear integro-differential equation describing the evolution of the tumor size. First, the g...

This edited volume gathers selected, peer-reviewed contributions presented at the fourth International Conference on Differential & Difference Equations Applications (ICDDEA), which was held in Lisbon, Portugal, in July 2019.
First organized in 2011, the ICDDEA conferences bring together mathematicians from various countries in order to promote coo...

A sigmoidal neural field lattice system is developed based on an Amari-Hopfield neural field lattice model, in which the Heaviside function is replaced by a simplifying sigmoidal function characterized by a small parameter ε. First the existence and uniqueness of solutions of the resulting sigmoidal lattice system is studied. Then the solutions of...

In stochastic financial and biological models, the diffusion coefficients often involve the term x, or more general |x|r for r∈(0,1), which is non-Lipschitz. In this paper, we study the strong convergence of the truncated Euler–Maruyama (EM) approximation first proposed by Mao (2015) for one-dimensional stochastic differential equations (SDEs) with...

This chapter focuses on dynamical behavior of lattice models arising in the biological sciences, in particular, attractors for such systems. Three types of lattice dynamical systems are investigated; they are lattice reaction-diffusion systems, Hopfield neural lattice systems, and neural field lattice systems. For each system the existence of a glo...

A nonautonomous difference equation is aymptotically autonomous if its right-hand side becomes more and more like that of an autonomous difference equation as time increases. It can then be shown that the component sets of a pullback attractor of the nonautonomous converge to the attractor of the autonomous system. Various conditions ensuring this...

The existence and uniqueness of global solutions for a fractional functional differential equation is established. The asymptotic behaviour of a lattice system with a fractional substantial time derivative and variable time delays is investigated. The existence of a global attracting set is established. It is shown to be a singleton set under a cer...

An autonomous Caputo fractional differential equation of order $\alpha\in(0,1)$ in $\mathbb{R}^d$ whose vector field satisfies a global Lipschitz condition is shown to generate a semi-dynamical system in the function space $\mathfrak{C}$ of continuous functions $f:\R^+\rightarrow \R^d$ with the topology uniform convergence on compact subsets. This...

In this paper, we study the continuity in initial data of a classical reaction-diffusion equation with arbitrary $p>2$ order nonlinearity and in any space dimension $N\geq 1$. It is proved that the weak solutions can be $(L^2, L^\gamma\cap H_0^1)$-continuous in initial data for any $\gamma\geq 2$ (independent of the physical parameters of the syste...

In this paper we study pullback attractors of multi-valued dynamical systems that are asymptotically convergent. It is shown that, under certain conditions, the components of the pullback attractor of a dynamical system can converge in time to those of the pullback attractor of the limiting dynamical system. Particular examples are asymptotically a...

A Hopfield neural lattice model is developed as the infinite dimensional extension of the classical finite dimensional Hopfield model. In addition, random external inputs are considered to incorporate environmental noise. The resulting random lattice dynamical system is first formulated as a random ordinary differential equation on the space of squ...

Reaction-diffusion equations on time-variable domains are instrin-sically nonautonomous even if the coefficients in the equation do not depend explicitly on time. Thus the appropriate asymptotic concepts, such as attractors, are nonautonomous. Forward attracting sets based on omega-limit sets are considered in this paper. These are related to the V...

Motivated by the study of lattice dynamical systems, i.e., infinite dimensional systems of ordinary differential equations, with nonlinear and state dependent diffusion, a new sequence space with variable exponents is introduced. In particular, given an exponent sequence \({\boldsymbol p} = (p_i)_{i \in \mathbb {Z}}\), a discrete Musielak-Orlicz sp...

A mathematical model for Zika virus dynamics under randomly varying environmental conditions is developed, in which the birth and loss rates for mosquitoes, and environmental influence are modeled as random processes. The resulting system of random ordinary differential equations are studied by the theory of random dynamical systems and dynamical a...

In [Janson & Marsden 2017] a dynamical system with a plastic self-organising velocity vector field was introduced, which was inspired by the architectural plasticity of the brain and proposed as a possible conceptual model of a cognitive system. Here we provide a more rigorous mathematical formulation of this problem, make several simplifying assum...

Let (Ω,ℱ,ℙ) be a probability space and let X be a separable Banach space. It is shown a subset V of Lp((Ω,ℱ,ℙ);X), where 1≤p<∞, is relatively compact in Lp((Ω,ℱ,ℙ);X) if and only if it is uniformly Lp-integrable and uniformly tight. The additional condition of scalarly relatively compact required in the literature is shown to hold by a probabilisti...

Optimal upper and lower error estimates for strong full-discrete numerical approximations of the stochastic heat equation driven by space-time white noise are obtained. In particular, we establish the optimality of strong convergence rates for full-discrete approximations of stochastic Allen-Cahn equations with space-time white noise which have rec...

Motivated by the importance of discrete structures of neuron networks, a neural field lattice system arising from the discretization of neural field models in the form of integro-differential equations is studied. The neural field lattice system is first formulated as a differential inclusion on a weighted space of infinite sequences, due to the sw...

Random ODEs are ordinary differential equations that include a stochastic process in their vector field. They can be analysed pathwise using deterministic calculus. Since the sample paths of the driving stochastic process is at most Hölder continuous, they lack the smoothness in their time variable to justify the convergence analysis of classical n...

Forward attractors, especially invariant ones, of non-autonomous and random dynamical systems have not been as well studied as pullback attractors. This is mainly due to the different role that time plays in each case. Pullback attractors involve convergence at each instant of current time as the initial data is pulled back to distant past, whereas...

The well-posedness and asymptotic dynamics of second-order-in-time stochastic evolution equations with state-dependent delay is investigated. This class covers several important stochastic PDE models arising in the theory of nonlinear plates with additive noise. We first prove well-posedness in a certain space of functions which are C¹ in time. The...

The pullback attractor of a non-autonomous random dynamical system is a time-indexed family of random sets, typically having the form {At(⋅)}t∈R with each At(⋅) a random set. This paper is concerned with the nature of such time-dependence. It is shown that the upper semi-continuity of the mapping t↦At(ω) for each ω fixed has an equivalence relation...

In stochastic financial and biological models, the diffusion coefficients often involve the term (Formula presented.), or more general |x|r, r ∈ (0, 1). These coefficients do not satisfy the local Lipschitz condition, which implies that the existence and uniqueness of the solution cannot be obtained by the standard conditions. This article establis...

Random ordinary differential equations (RODEs) are ordinary differential equations (ODEs) which have a stochastic process in their vector field functions. They have been used in a wide range of applications such as biology, medicine and engineering and play an important role in the theory of random dynamical systems. RODEs can be investigated pathw...

This book is intended to make recent results on the derivation of higher order numerical schemes for random ordinary differential equations (RODEs) available to a broader readership, and to familiarize readers with RODEs themselves as well as the closely associated theory of random dynamical systems. In addition, it demonstrates how RODEs are being...

Using a temporally weighted norm we first establish a result on the global existence and uniqueness of solutions for Caputo fractional stochastic differential equations of order $\alpha\in(\frac{1}{2},1)$ whose coefficients satisfy a standard Lipschitz condition. For this class of systems we then show that the asymptotic distance between two distin...

A general theorem on the local and global existence of solutions is
established for an impulsive fractional delay differential equation
with Caputo fractional substantial derivative in a separable Hilbert
space under the assumption that the nonlinear term is weakly continuous.
The uniqueness of solutions is also considered under an additional
Lipsc...

The scientific literature contains a number of numerical approximation results for stochastic partial differential equations (SPDEs) with superlinearly growing nonlinearities but, to the best of our knowledge, none of them prove strong or weak convergence rates for full-discrete numerical approximations of space-time white noise driven SPDEs with s...

Taylor expansions are derived for RODEs with affine noise and general Affine-RODE-Taylor approximations are formulated.

An immune system virus model formulated in terms of RODEs is investigated and shown to generate a random dynamical system, which has a random attractor. Simulations using various numerical schemes developed in the previous chapters are presented.

RODE-Taylor schemes are presented for general RODEs with Hölder continuous noise, and in particular essential RODE-Taylor schemes are considered. These schemes are also applied to RODEs with affine noise and compared with the heuristically derived schemes presented in the first chapter.

Itô stochastic ordinary differential equations (SODEs) and the basic ideas of Itô stochastic calculus are reviewed. The relationship between RODEs and SODEs is considered, in particular the Doss-Sussmann transformation.

Comparative simulations of the numerical schemes derived in the previous chapters are made in the context of RODEs occurring in the biological sciences. These include models of tumor inhibition, toggle switching and seashell pattern formation. In particular, relative computational times are compared for different schemes.

Strong Itô-Taylor schemes are adapted to RODEs with Itô noise to derive one-step numerical schemes for RODEs. These are then used to derive derivative-free explicit and implicit schemes and are applied to RODEs with affine noise. Multi-step schemes for RODEs are also derived.

Taylor approximations for ODEs and SODEs are reviewed. In particular, general Itô-Taylor approximations for SODEs are reformulated in terms of hierarchical sets of multi-indices and their pathwise convergence is considered.

A chemostat is associated with a laboratory device which consists of three interconnected vessel and is used to grow microorganisms in a cultured environment (see Fig. 16.1). In its basic form, the outlet of the first vessel is the inlet for the second vessel and the outlet of the second vessel is the inlet for the third. The first vessel is called...

Taylor-like expansions are derived for general RODEs with Hölder continuous noise in terms of classical Taylor expansions of the vector field, followed from which general RODE-Taylor approximations are formulated. Essential RODE-Taylor approximations with minimal number of terms are determined.

One-step numerical schemes for ODEs and Itô SODEs are recalled.

Methods for the simulations and approximation of stochastic integrals are given for Wiener processes, Ornstein-Uhlenbeck processes, fractional Brownian motions and Poisson processes. A method for calculating a finer approximation of the same sample path of a Winer process based on the Lévy construction is presented.

Continuous Markov chains in a random environment are modeled as a linear RODE on a probability simplex. The positivity of the solutions of such linear RODEs is established and the resulting linear random dynamical system is shown to be strongly contracting and to possess a random attractor consisting of singleton sets.

Random dynamical systems and random attractors are introduced through examples. The special case of contractive cocycle mappings is considered for latter use.

The numerical stability of implicit averaged and implicit multi-step schemes for nonlinear RODEs is determined.

Existence and uniqueness theorems are given for RODEs under classical and Carathéodory assumptions. In the latter case the measurability of solutions is also established. Conditions ensuring the positivity of solutions are stated and RODEs with canonical noise are formulated.