动力系统

Under some non-invertibility and irreducibility condition, for nilmanifold Anosov maps with one-dimensional stable bundle, we get the equivalence among the existence of invariant unstable bundle, the existence of topological conjugacy to…

Data-driven methods for the identification of the governing equations of dynamical systems or the computation of reduced surrogate models play an increasingly important role in many application areas such as physics, chemistry, biology, and…

Let $\Lambda_1$, $\Lambda_2$ be two discrete orbits under the linear action of a lattice $\Gamma<\mathrm{SL}_2(\mathbb{R})$ on the Euclidean plane. We prove a Siegel$-$Veech-type integral formula for the averages $$…

Natural fluctuations have played a crucial role in affecting the dynamics of pervasive diseases such as the coronavirus. Examining the effects of irregular unsettling disturbances on epidemic models is important for understanding these…

This article focuses on recent investigations on equilibria of the Frenkel-Kontorova models subjected to potentials generated by quasi-crystals. We present a specific one-dimensional model with an explicit potential driven by the Fibonacci…

We study the limiting distribution of the rational points under a horizontal translation along a sequence of expanding closed horocycles on the modular surface. Using spectral methods we confirm equidistribution of these sample points for…

We prove effective bounds on the rate in the quadratic growth asymptotics for the orbit of a non-uniform lattice of SL(2,R), acting linearly on the plane. This gives an error bound in the count of saddle connection holonomies, for some…

In this paper we apply techniques from nonstandard analysis to study expansive dynamical systems. Among other results, we provide a necessary and sufficient condition for an expansive homeomorphism on a compact metric space to admit…

This paper provides two results for the omega limit sets of a dynamical system. We show that omega limit sets can be estimated by using functions that satisfy different (and in many cases less demanding) assumptions than the usual…

We prove a generalized non-stationary version of the fiber contraction mapping theorem. It was originally used in [HirschPugh70] to prove that the stable foliation of a $C^2$ Anosov diffeomorphism of a surface is $C^1$. Our generalized…

The main goal of this paper is to give a complete fractal analysis of piecewise smooth (PWS) slow-fast Li\'{e}nard equations. For the analysis, we use the notion of Minkowski dimension of one-dimensional orbits generated by slow relation…

In a polydiagonal subspace of the Euclidean space, certain components of the vectors are equal (synchrony) or opposite (anti-synchrony). Polydiagonal subspaces invariant under a matrix have many applications in graph theory and dynamical…

This work proposes a general framework for capturing noise-driven transitions in spatially extended non-equilibrium systems and explains the emergence of coherent patterns beyond the instability onset. The framework relies on stochastic…

Recent work has pioneered the use of system-theoretic passivity to study equilibrium stability for the dynamics of noncooperative strategic interactions in large populations of learning agents. In this and related works, the stability…

Constructing sparse, effective reduced-order models (ROMs) for high-dimensional dynamical data is an active area of research in applied sciences. In this work, we study an efficient approach to identifying such sparse ROMs using an…

For any exact twist map $f$ and any cohomology class $c\in\mathbb{R}$, let $u_c$ be any associated discrete weak K.A.M. solution, and we introduce an inherent Lipschitz dynamics $\Sigma_+$ given by the discrete forward Lax-Oleinik…

Recurrence is a fundamental characteristic of dynamical systems with complicated behavior. Understanding the inner structure of recurrence is challenging, especially if the system has many degrees of freedom and is subject to noise. We…

We consider generic 1-parameter unfoldings of parabolic vector fields. It is known that the box dimension of orbits of their time-one maps is discontinuous at the bifurcation value. Here, we expand asymptotically the Lebesgue measure of the…

We study the topological dynamics of H\'enon maps. For a parameter set generalizing the Benedicks-Carleson parameters (the Wang-Young parameter set) we obtain the following: The pruning front conjecture (due to Cvitanovi\'c); A kneading…

We study Birkhoff sums as distributions. We obtain regularity results on such distributions for various dynamical systems with hyperbolicity, as hyperbolic linear maps on the torus and piecewise expanding maps on the interval. We also give…