Peter E. Kloeden
YOU?
Author Swipe
View article: Strong order-one convergence of the Euler method for random ordinary differential equations driven by semi-martingale noises
Strong order-one convergence of the Euler method for random ordinary differential equations driven by semi-martingale noises Open
It is well known that the Euler method for a random ordinary differential equation d X t /dt = f ( t, X t , Y t ) driven by a stochastic process { Y t } t∈I , on a time interval I , with θ -Hölder sample paths is of strong order θ with res…
View article: Fokker-Planck equation for stochastic heat equations
Fokker-Planck equation for stochastic heat equations Open
This work is devoted to the study of the Fokker--Planck equation for a stochastic heat equation with an additive $Q$-Wiener noise and non-homogeneous boundary conditions. We explicitly construct the probability density function and establi…
View article: The Exponential of the Lattice Laplacian Operator and the Mean-Square Attractor of A Stochastic Lattice System
The Exponential of the Lattice Laplacian Operator and the Mean-Square Attractor of A Stochastic Lattice System Open
The exponential $$e^{\Lambda t}$$ of the lattice Laplacian operator $$\Lambda $$ was introduced in the 1970 s, but, unlike its counterpart for parabolic partial differential equations, has not yet been used to investigate lattice d…
View article: theta-scheme for solving Caputo fractional differential equations
theta-scheme for solving Caputo fractional differential equations Open
We formulate a \(\theta\)-numerical scheme for solving Caputo fractional differential equations (FDEs) of order \(\alpha\in(0,1)\), with vector fields satisfying a standard Lipschitz continuity condition in the state variable and a Holder …
View article: Dynamics of a random Hopfield neural lattice model with adaptive synapses and delayed Hebbian learning
Dynamics of a random Hopfield neural lattice model with adaptive synapses and delayed Hebbian learning Open
UDC 517.9 A Dong–Hopfield neural lattice model with random external forcing and delayed response to the evolution of interconnection weights is developed and studied. The interconnection weights evolve according to the Hebbian learning rul…
View article: Mean-square stability analysis of stochastic delay evolution equations driven by fractional Brownian motion with Hurst index $ H\in(0,1) $
Mean-square stability analysis of stochastic delay evolution equations driven by fractional Brownian motion with Hurst index $ H\in(0,1) $ Open
In this paper, we consider stochastic evolution equations with finite delay driven by fractional Brownian motion (fBm) with Hurst index $ H\in(0,1) $. First, the global existence and uniqueness of mild solutions are established by using a …
View article: Random attractors of a stochastic Hopfield neural network model with delays
Random attractors of a stochastic Hopfield neural network model with delays Open
The global asymptotic behavior of a stochastic Hopfield neural network model (HNNM) with delays is explored by studying the existence and structure of random attractors. It is first proved that the trajectory field of the stochastic delaye…
View article: Numerical dynamics of integrodifference equations: Forward dynamics and pullback attractors
Numerical dynamics of integrodifference equations: Forward dynamics and pullback attractors Open
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 s…
View article: Robustness of exponential attractors for infinite dimensional dynamical systems with small delay and application to 2D nonlocal diffusion delay lattice systems
Robustness of exponential attractors for infinite dimensional dynamical systems with small delay and application to 2D nonlocal diffusion delay lattice systems Open
We first present some sufficient conditions for the construction of a robust family of exponential attractors for infinite dimensional dynamical systems with small time delay perturbation. In particular, we prove that this family of expone…
View article: Numerical Dynamics of Integrodifference Equations: Forward Dynamics and Pullback Attractors
Numerical Dynamics of Integrodifference Equations: Forward Dynamics and Pullback Attractors Open
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 s…
View article: Exponential attractors for two-dimensional nonlocal diffusion lattice systems with delay
Exponential attractors for two-dimensional nonlocal diffusion lattice systems with delay Open
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 dimens…
View article: A nonautonomous chemostat model for the growth of gut microbiome with varying nutrient
A nonautonomous chemostat model for the growth of gut microbiome with varying nutrient Open
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…
View article: Pullback and forward dynamics of nonautonomous Laplacian lattice systems on weighted spaces
Pullback and forward dynamics of nonautonomous Laplacian lattice systems on weighted spaces Open
A nonautonomous lattice system with discrete Laplacian operator is revisited in the weighted space of infinite sequences . First the existence of a pullback attractor in is established by utilizing the dense inclusion of . Moreover, the p…
View article: Attractors of Caputo fractional differential equations with triangular vector fields
Attractors of Caputo fractional differential equations with triangular vector fields Open
It is shown that the attractor of an autonomous Caputo fractional differential equation of order $α\in(0,1)$ in $\mathbb{R}^d$ whose vector field has a certain triangular structure and satisfies a smooth condition and dissipativity conditi…
View article: Robustness of a dynamical systems model with a plastic self-organising vector field to noisy input signals
Robustness of a dynamical systems model with a plastic self-organising vector field to noisy input signals Open
We investigate the robustness with respect to random stimuli of a dynamical system with a plastic self-organising vector field, previously proposed as a conceptual model of a cognitive system and inspired by the self-organised plasticity o…
View article: Coupled nonautonomous inclusion systems with spatially variable exponents
Coupled nonautonomous inclusion systems with spatially variable exponents Open
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…
View article: Pullback attractors for stochastic recurrent neural networks with discrete and distributed delays
Pullback attractors for stochastic recurrent neural networks with discrete and distributed delays Open
In this paper, we investigate a class of stochastic recurrent neural networks with discrete and distributed delays for both biological and mathematical interests. We do not assume any Lipschitz condition on the nonlinear term, just a conti…
View article: Forward and Pullback Dynamics of Nonautonomous Integrodifference Equations: Basic Constructions
Forward and Pullback Dynamics of Nonautonomous Integrodifference Equations: Basic Constructions Open
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 a…
View article: Asymptotic behaviour of a neural field lattice model with delays
Asymptotic behaviour of a neural field lattice model with delays Open
The asymptotic behaviour of an autonomous neural field lattice system with delays is investigated. It is based on the Amari model, but with the Heaviside function in the interaction term replaced by a sigmoidal function. First, the lattice…
View article: Sigmoidal approximations of a delay neural lattice model with Heaviside functions
Sigmoidal approximations of a delay neural lattice model with Heaviside functions Open
The approximation of Heaviside coefficient functions in delay neural lattice models with delays by sigmoidal functions is investigated. The solutions of the delay sigmoidal models are shown to converge to a solution of the delay differenti…
View article: Strong <inline-formula><tex-math id="M1">$ (L^2,L^\gamma\cap H_0^1) $</tex-math></inline-formula>-continuity in initial data of nonlinear reaction-diffusion equation in any space dimension
Strong -continuity in initial data of nonlinear reaction-diffusion equation in any space dimension Open
In this paper we study the continuity in initial data of a classical reaction-diffusion equation with arbitrary order nonlinearity and in any space dimension . It is proved that the weak solutions can be -continuous in initial data for ar…
View article: Attractors of Hopfield-type lattice models with increasing neuronal input
Attractors of Hopfield-type lattice models with increasing neuronal input Open
Two Hopfield-type neural lattice models are considered, one with local -neighborhood nonlinear interconnections among neurons and the other with global nonlinear interconnections among neurons. It is shown that both systems possess global …
View article: Asymptotic behavior of coupled inclusions with variable exponents
Asymptotic behavior of coupled inclusions with variable exponents Open
This work concerns the study of asymptotic behavior of the solutions of a nonautonomous coupled inclusion system with variable exponents. We prove the existence of a pullback attractor and that the system of inclusions is asymptotically au…
View article: Semi-dynamical systems generated by autonomous Caputo fractional differential equations
Semi-dynamical systems generated by autonomous Caputo fractional differential equations Open
An autonomous Caputo fractional differential equation of order $α\in(0,1)$ 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 con…
View article: Strong $(L^2,L^\gamma\cap H_0^1)$-continuity in initial data of nonlinear reaction-diffusion equation in any space dimension
Strong $(L^2,L^\gamma\cap H_0^1)$-continuity in initial data of nonlinear reaction-diffusion equation in any space dimension Open
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…
View article: Strong $(L^2,L^γ\cap H_0^1)$-continuity in initial data of nonlinear reaction-diffusion equation in any space dimension
Strong $(L^2,L^γ\cap H_0^1)$-continuity in initial data of nonlinear reaction-diffusion equation in any space dimension Open
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^γ\cap H_0^…