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After relating the notion of $\omega$-recurrence in skew products to the range of values taken by partial ergodic sums and Lyapunov exponents, ergodic $\mathbb{Z}$-valued cocycles over an irrational rotation are presented in detail. First,…

We introduce the concept of mixed random-quasiperiodic linear cocycles. We characterize the ergodicity of the base dynamics and establish a large deviations type estimate for certain types of observables. For the fiber dynamics we prove the…

In this paper we show that if a linear cocycle is robustly periodical stable then it is uniformly stable.

This article is concerned with stability analysis and stabilization of randomly switched nonlinear systems. These systems may be regarded as piecewise deterministic stochastic systems: the discrete switches are triggered by a stochastic…

We study the regularity of the Lyapunov exponent for quasi-periodic cocycles $(T_\omega, A)$ where $T_\omega$ is an irrational rotation $x\to x+ 2\pi\omega$ on $\SS^1$ and $A\in {\cal C}^l(\SS^1, SL(2,\mathbb{R}))$, $0\le l\le \infty$. For…

We show that the probability distribution function that best fits the distribution of return times between two consecutive visits of a chaotic trajectory to finite size regions in phase space deviates from the exponential statistics by a…

This paper deals with random dynamical systems of polynomial automorphisms (complex generalized H\'{e}non maps and their conjugate maps) of $\Bbb{C}^{2}.$ We show that a generic random dynamical system of polynomial automorphisms has ``mean…

We study the regularity of Lyapunov exponents for random linear cocycles taking values in $\Mat_m(\R)$ and driven by i.i.d. processes. Under three natural conditions - finite exponential moments, a spectral gap between the top two Lyapunov…

Classes of polynomial differential equations of degree n are considered. An explicit upper bound on the size of the coefficients are given which implies that each equation in the class has exactly n complex periodic solutions. In most of…

We investigate global bounded solutions of higher regularity to boundary value problems for a general linear nonautonomous first order 1D hyperbolic system in a strip. We establish the existence of such solutions under the assumption of…

This article aims to investigate sufficient conditions for the stability of stochastic differential equations with a random structure, particularly in contexts involving the presence of concentration points. The proof of asymptotic…

We study semi-dynamical systems associated to delay differential equations. We give a simple criteria to obtain weak and strong persistence and provide sufficient conditions to guarantee uniform persistence. Moreover, we show the existence…

In this paper, we study periodic linear systems on periodic time scales which include not only discrete and continuous dynamical systems but also systems with a mixture of discrete and continuous parts (e.g. hybrid dynamical systems). We…

Based on our studies done on two-dimensional autonomous systems, forced non-autonomous systems and time-delayed systems, we propose a unified methodology - that uses renormalization group theory - for finding out existence of periodic…

In this paper we establish uniform large deviations estimates of exponential type and H\"older continuity of the Lyapunov exponents for random non-invertible cocycles with constant rank.

In this paper, we study the existence of random periodic solutions for nonlinear stochastic differential equations with additive white noise. We extend the input-to-state characteristic operator of the system to the non-autonomous…

We prove the existence of periodic solutions in a class of nonlinear partial differential equations, including the nonlinear Schroedinger equation, the nonlinear wave equation, and the nonlinear beam equation, in higher dimension. Our…

This paper aims at providing rigorous numerical computation procedure for finite-time singularities in dynamical systems. Combination of time-scale desingularization as well as Lyapunov functions validation on stable manifolds of invariant…

We study random dynamical systems on the real line, considering each dynamical system together with the one generated by the inverse maps. We show that there is a duality between forward and inverse behaviour for such systems, splitting…

We establish concentration inequalities for random dynamical systems (RDSs), assuming that the observables of interest are separately Lipschitz. Under a weak average contraction condition, we obtain deviation bounds for several random…