arXiv subjects

Luu Hoang Duc

Publications and source records attributed to Luu Hoang Duc.

18 recordsLinked to original sources

arXiv2026

Taming discrete rough paths via strong Lyapunov functions

Based on the newly introduced concept of {\it strong Lyapunov functions} for rough differential equations \cite{ducjost25}, we study a tamed numerical scheme to approximate the solutions of the continuous system. We derive explicit estimates of solution norms of the tamed system which look similar to those of the continuous system. As a result, we prove the convergence of the tamed scheme in the $L^1$ sense. For systems with the negative gradient condition, we prove the existence of a numerical pullback attractor for the generated random dynamical system from the tamed numerical scheme which is integrable and upper semi-continuous w.r.t. the scheme step size.

math.NA↗

arXiv2025

Strong Lyapunov functions for rough systems

We extend the Lyapunov function technique, a fundamental tool for investigating asymptotic stability and existence of attractors for ordinary differential equations, by introducing the notion of a {\it strong Lyapunov function} for an autonomous drift under stochastic perturbation driven by general Hölder-continuous multiplicative noise, not necessarily Brownian. The mathematical setting within which our method proceeds consists of rough path calculus and the framework of random dynamical systems. We conclude that if such a function exists for the drift then the perturbed system admits a global random pullback attractor that is upper semi-continuous w.r.t. the noise intensity coefficient and the dyadic approximation of the noise. Moreover, in case the drift is globally Lipschitz continuous, then there exists a numerical attractor for the discretization which is upper semi-continuous w.r.t. the noise intensity and converges to the continuous attractor as the step size tends to zero. Several applications, including dissipative systems, the pendulum, the FitzHugh-Nagumo neuro-system and the Lorenz system, demonstrate the power of our approach. We also prove that strong Lyapunov functions can be approximated in practice by Lyapunov neural networks.

math.DS↗

arXiv2025

Stability criteria for rough systems

We propose a quantitative direct method to prove the local stability of a stationary solution for a rough differential equation and its regular discretization scheme. Using Doss-Sussmann technique and stopping time analysis, we provide stability criteria for a stationary solution of the continuous system to be exponentially stable, provided the diffusion term is bounded and its derivatives exhibit small growth. The same conclusions hold for the regular discretization scheme with a sufficiently small step size, but one needs to apply the sewing lemma and stopping times for the discrete time set. Our stability criteria are based on the linearization of the drift and require only information about the bound and growth rates of the diffusion, making them data-driven criteria.

math.DS↗

arXiv2020

Controlled differential equations as rough integrals

We study controlled differential equations with unbounded drift terms, where the driving paths is $ν$ - Hölder continuous for $ν\in (\frac{1}{3},\frac{1}{2})$, so that the rough integral are interpreted in the Gubinelli sense \cite{gubinelli} for controlled rough paths. Similar to the rough differential equations in the sense of Lyons \cite{lyons98} or of Friz-Victoir \cite{friz}, we prove the existence and uniqueness theorem for the solution in the sense of Gubinelli, the continuity on the initial value, and the solution norm estimates.

math.PR↗

arXiv2020

Pullback attractors for stochastic Young differential delay equations

We study the asymptotic dynamics of stochastic Young differential delay equations under the regular assumptions on Lipschitz continuity of the coefficient functions. Our main results show that, if there is a linear part in the drift term which has no delay factor and has eigenvalues of negative real parts, then the generated random dynamical system possesses a random pullback attractor provided that the Lipschitz coefficients of the remaining parts are small.

math.DS↗

arXiv2020

Asymptotic stability of controlled differential equations. Part II: rough integrals

We continue the approach in Part I \cite{duchong19} to study stationary states of controlled differential equations driven by rough paths, using the framework of random dynamical systems and random attractors. Part II deals with driving paths of finite $ν$ - Hölder norms with $ν\in (\frac{1}{3},\frac{1}{2})$ so that the integrals are interpreted in the Gubinelli sense for controlled rough paths. We prove sufficient conditions for the attractor to be a singleton, thus the pathwise convergence is in both pullback and forward senses.

math.PR↗

arXiv2020

Asymptotic stability of controlled differential equations. Part I: Young integrals

We provide a unified analytic approach to study stationary states of controlled differential equations driven by rough paths, using the framework of random dynamical systems and random attractors. Part I deals with driving paths of finite $p$-variations with $1 \leq p <2$ so that the integrals are interpreted in the Young sense. Our method helps to generalize recent results \cite{GAKLBSch2010}, \cite{ducGANSch18}, \cite{duchongcong18} 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.

math.PR↗

arXiv2019

Asymptotic stability for stochastic dissipative systems with a Hölder noise

We prove the exponential stability of the zero solution of a stochastic differential equation with a Hölder noise, under the strong dissipativity assumption. As a result, we also prove that there exists a random pullback attractor for a stochastic system under a multiplicative fractional Brownian noise.

math.PR↗

arXiv2019

Stochastic FitzHugh-Nagumo neuron model in excitable regime embeds a leaky integrate-and-fire model

In this paper, we provide a complete mathematical construction for a stochastic leaky-integrate-and-fire model (LIF) mimicking the interspike interval (ISI) statistics of a stochastic FitzHugh-Nagumo neuron model (FHN) in the excitable regime, where the unique fixed point is stable. Under specific types of noises, we prove that there exists a global random attractor for the stochastic FHN system. The linearization method is then applied to estimate the firing time and to derive the associated radial equation representing a LIF equation. This result confirms the previous prediction in [Ditlevsen, S. and Greenwood, P. (2013). The Morris-Lecar neuron model embeds a leaky integrate-and-fire model. Journal of Mathematical Biology, 67(2):239-259] for the Morris-Lecar neuron model in the bistability regime consisting of a stable fixed point and a stable limit cycle.

math.DS↗

arXiv2019

Lyapunov spectrum of nonautonomous linear Young differential equations

We show that a linear Young differential equation generates a topological two-parameter flow, thus the notions of Lyapunov exponents and Lyapunov spectrum are well-defined. The spectrum can be computed using the discretized flow and is independent of the driving path for triangular systems which are regular in the sense of Lyapunov. In the stochastic setting, the system generates a stochastic two-parameter flow which satisfies the integrability condition, hence the Lyapunov exponents are random variables of finite moments. Finally, we prove a Millionshchikov theorem stating that almost all, in a sense of an invariant measure, linear nonautonomous Young differential equations are Lyapunov regular.

math.DS↗

arXiv2019

Stability theory for Gaussian rough differential equations. Part I

We propose a quantitative direct method of proving the stability result for Gaussian rough differential equations in the sense of Gubinelli \cite{gubinelli}. Under the strongly dissipative assumption of the drift coefficient function, we prove that the trivial solution of the system under small noise is exponentially stable.

math.PR↗

arXiv2019

Stability theory for Gaussian rough differential equations. Part II

math.PR↗

arXiv2018

Young differential delay equations driven by Hölder continuous paths

In this paper we prove the existence and uniqueness of the solution of Young differential delay equations under weaker conditions than it is known in the literature. We also prove the continuity and differentiability of the solution with respect to the initial function and give an estimate for the growth of the solution. The proofs use techniques of stopping times, Shauder-Tychonoff fixed point theorem and a Gronwall-type lemma.

math.DS↗

arXiv2017

Nonautonomous Young differential equations revisited

In this paper we prove that under weak conditions a nonautonomous Young differential equation possesses a unique solution which depends continuously on initial conditions. The proofs use estimates in p-variation norms, greedy time techniques, and Gronwall-type lemma with the help of Shauder theorem of fixed points.

math.PR↗

arXiv2017

Exponential stability of stochastic evolution equations driven by small fractional Brownian motion with Hurst parameter in $(1/2,1)$

This paper addresses the exponential stability of the trivial solution of some types of evolution equations driven by Hölder continuous functions with Hölder index greater than $1/2$. The results can be applied to the case of equations whose noisy inputs are given by a fractional Brownian motion $B^H$ with covariance operator $Q$, provided that $H\in (1/2,1)$ and ${\rm tr}(Q)$ is sufficiently small.

math.AP↗

arXiv2014

Asymptotic Behavior of Linear Almost Periodic Differential Equations

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 inhomogeneous equation.

math.DS↗

arXiv2014

A new concept of local metric entropy for finite-time nonautonomous dynamical systems

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 dynamical structures such as Lagrangian coherent structures.

math.DS↗

arXiv2014

Generation of random dynamical systems from fractional stochastic delay differential equations

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 Hölder space which is separable.

math.DS↗