动力系统
Non-uniform cellular automata (NUCA) are an extension of cellular automata with multiple local rules in different cells. We show that if the distribution of local rules is uniformly recurrent, or recurrent in the one-dimensional case, the…
Non-uniform cellular automata (NUCA) are an extension of cellular automata (CA), which transform cells according to multiple different local rules. A NUCA is defined by a configuration of local rules called a local rule distribution. We…
Accurate knowledge of the state variables in a dynamical system is critical for effective control, diagnosis, and supervision, especially when direct measurements of all states are infeasible. This paper presents a novel approach to…
The study of the dynamics of a continuous observable and non-controllable three-dimensional symmetric piecewise linear system with three zones can be reduced to the study of the existence of limit cycles for the piecewise differential…
We prove that there exists a group which is not finitely generated, but admits a minimal sofic shift. This answers a question of Doucha, Melleray and Tsankov. The group is of the form $(F_4 \times F_2) \rtimes F_{\infty}$. The construction…
We develop a contraction-based framework to establish the existence and exponential stability of periodic solutions in planar nonsmooth dynamical systems governed by Filippov differential inclusions. The method integrates a time- and…
We discuss the isomorphism problem for ergodic actions of locally compact groups. In particular we show that the conjugacy relation is not Borel for ergodic measure preserving actions of indicable groups.
Given a topologically transitive subshift of finite type, the $u$-dimension of the $\alpha$-level set of the quotient of Birkhoff sums is a well-studied topic. In this paper, we consider the uniform $\alpha$-level set, which is a subset of…
We consider autonomous Hamiltonian systems and present an algorithm to compute at the same time partially hyperbolic invariant tori (whiskered tori), as well as high-order expansions of their stable and unstable manifolds. Such whiskered…
In this paper, we study the Hausdorff dimension of self-similar measures and sets on the real line, where the generating iterated function system consists of some maps that share the same fixed point. In particular, we will show that out of…
We investigate the bifurcation structure of equilibria in a class of non-autonomous ordinary differential equations governed by a season length parameter, $\tau$, which determines the alternation between growth and decline dynamics. This…
The study of neural computation aims to understand the function of a neural system as an information processing machine. Neural systems are undoubtedly complex, necessitating principled and automated tools to abstract away details to…
Constrained mechanical systems occur in many applications, such as modeling of robots and other multibody systems. In this case, the motion is governed by a system of differential-algebraic equations (DAE), often with large and sparse…
Adapted invariant measures, such as the natural area measure (Liouville), have a central place in the development of ergodic theory for billiards. These measures ensure local Pesin charts can be constructed almost everywhere even in the…
Building on work of Doyle and Hyde on polynomial maps in one variable, we produce for each odd integer $d \geq 2$ a H\'enon map of degree $d$ defined over $\mathbb{Q}$ with at least $(d-4)^2$ integral periodic points. This provides a…
We prove a result on equilibrium measures for potentials with summable variation on arbitrary subshifts over a countable amenable group. For finite configurations $v$ and $w$, if $v$ is always replaceable by $w$, we obtain a bound on the…
For a class of potentials $\psi$ satisfying a condition depending on the roof function of a suspension (semi)flow, we show an EKP inequality, which can be interpreted as a H\"older continuity property in the weak${^*}$ norm of measures,…
We study locally constant skew-product maps over full shifts of finite symbols with arbitrary compact metric spaces as fiber spaces. We introduce a new criterion to determine the density of leaves of the strong unstable (and strong stable)…
To investigate the degree $d$ connectedness locus, Thurston studied \emph{$\sigma_d$-invariant laminations}, where $\sigma_d$ is the $d$-tupling map on the unit circle, and built a topological model for the space of quadratic polynomials…
In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms. As an expository contribution we demonstrate how to re-formulate a wide variety of challenges from deep neural networks, (stochastic)…