scholarly journals Generation of One-Sided Random Dynamical Systems by Stochastic Differential Equations

1997 ◽  
Vol 2 (0) ◽  
Author(s):  
Gerald Kager ◽  
Michael Scheutzow
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Random Dynamical Systems

Random dynamical systems generated by coalescing stochastic flows on ℝ

2018 ◽  
Vol 18 (04) ◽  
pp. 1850031 ◽  
Author(s):  
Georgii V. Riabov
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
State Space ◽  
Random Dynamical Systems ◽  
Stochastic Flows

Existence of random dynamical systems for a class of coalescing stochastic flows on [Formula: see text] is proved. A new state space for coalescing flows is built. As particular cases coalescing flows of solutions to stochastic differential equations and coalescing Harris flows are considered.

scholarly journals Almost periodicity and periodicity for nonautonomous random dynamical systems

2020 ◽  
pp. 2150034
Author(s):  
Paul Raynaud de Fitte
Keyword(s):  
Partial Differential Equations ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Hilbert Spaces ◽  
Stochastic Partial Differential Equations ◽  
Random Processes ◽  
Random Dynamical Systems ◽  
Almost Periodicity ◽  
Almost Periodic

We present a notion of almost periodicity which can be applied to random dynamical systems as well as almost periodic stochastic differential equations in Hilbert spaces (abstract stochastic partial differential equations). This concept allows for improvements of known results of almost periodicity in distribution, for general random processes and for solutions to stochastic differential equations.

scholarly journals Asymptotic Dynamics of Young Differential Equations

2021 ◽  
Author(s):  
Luu Hoang Duc ◽  
Phan Thanh Hong
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Random Dynamical Systems ◽  
Analytic Approach ◽  
Pullback Attractors ◽  
Random Attractors ◽  
Asymptotic Dynamics

AbstractWe provide a unified analytic approach to study the asymptotic dynamics of Young differential equations, using the framework of random dynamical systems and random attractors. Our method helps to generalize recent results (Duc et al. in J Differ Equ 264:1119–1145, 2018, SIAM J Control Optim 57(4):3046–3071, 2019; Garrido-Atienza et al. in Int J Bifurc Chaos 20(9):2761–2782, 2010) on the existence of the global pullback attractors for the generated random dynamical systems. We also prove sufficient conditions for the attractor to be a singleton, thus the pathwise convergence is in both pullback and forward senses.

scholarly journals On Smoother Attributions using Neural Stochastic Differential Equations

2021 ◽  
Author(s):  
Sumit Jha ◽  
Rickard Ewetz ◽  
Alvaro Velasquez ◽  
Susmit Jha
Keyword(s):  
Neural Networks ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Systems Theory ◽  
Dynamical Systems Theory ◽  
Small Perturbations

Several methods have recently been developed for computing attributions of a neural network's prediction over the input features. However, these existing approaches for computing attributions are noisy and not robust to small perturbations of the input. This paper uses the recently identified connection between dynamical systems and residual neural networks to show that the attributions computed over neural stochastic differential equations (SDEs) are less noisy, visually sharper, and quantitatively more robust. Using dynamical systems theory, we theoretically analyze the robustness of these attributions. We also experimentally demonstrate the efficacy of our approach in providing smoother, visually sharper and quantitatively robust attributions by computing attributions for ImageNet images using ResNet-50, WideResNet-101 models and ResNeXt-101 models.

THE NUMERICAL SOLUTION OF NONLINEAR STOCHASTIC DYNAMICAL SYSTEMS: A BRIEF INTRODUCTION

1991 ◽  
Vol 01 (02) ◽  
pp. 277-286 ◽  
Author(s):  
P. E. KLOEDEN ◽  
E. PLATEN ◽  
H. SCHURZ
Keyword(s):  
Numerical Analysis ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Numerical Solution ◽  
Stochastic Differential Equations ◽  
Taylor Expansion ◽  
Rapid Development ◽  
Stochastic Calculus ◽  
Stochastic Dynamical Systems ◽  
The Subject

The numerical analysis of stochastic differential equations, currently undergoing rapid development, differs significantly from its deterministic counterpart due to the peculiarities of stochastic calculus. This article presents a brief, pedagogical introduction to the subject from the perspective of stochastic dynamical systems. The key tool is the stochastic Taylor expansion. Strong, pathwise approximations are distinguished from weak, functional approximations, and their role in stability with Lyapunov exponents and stiffness is discussed.

Sign in / Sign up

Export Citation Format

Share Document