Related papers: Sensitive dependence and dense perodic points
We introduce pointwise measure expansivity for bi-measurable maps. We show through examples that this notion is weaker than measure expansivity. In spite of this fact, we show that many results for measure expansive systems hold true for…
The sensitivity of a wave field's evolution to small perturbations is of fundamental interest. For chaotic systems, there are two distinct regimes of either exponential or Gaussian overlap decay in time. We develop a semiclassical approach…
We propose a method for learning dynamical systems from high-dimensional empirical data that combines variational autoencoders and (spatio-)temporal attention within a framework designed to enforce certain scientifically-motivated…
We give a unified proof of the existence of turbulence for some classes of continuous interval maps which include, among other things, maps with periodic points of odd periods > 1, some maps with dense chain recurrent points and densely…
We consider the dynamics of a periodic chain of N coupled overdamped particles under the influence of noise. Each particle is subjected to a bistable local potential, to a linear coupling with its nearest neighbours, and to an independent…
We provide an example of an SDE with degenerate additive noise where synchronization depends on the strength of noise and the number of directions in which the noise acts. Here, synchronization means that the weak random attractor consists…
Frequentist-style large-sample properties of Bayesian posterior distributions, such as consistency and convergence rates, are important considerations in nonparametric problems. In this paper we give an analysis of Bayesian asymptotics…
The dynamics of extended many-body systems are generically chaotic. Classically, a hallmark of chaos is the exponential sensitivity to initial conditions captured by positive Lyapunov exponents. Supplementing chaotic dynamics with…
The intermittency phenomenon is the occurrence of very high but rare peaks, which despite their rarity influence the asymptotic behaviour of the underlying system. Mathematically this can be characterised with the asymptotics of moments. In…
We give a definition of generalized indicators of sensitivity to initial conditions and orbit complexity (a measure of the information that is necessary to describe the orbit of a given point). The well known Ruelle-Pesin and Brin-Katok…
We consider the system of $N$ points on the segment of the real line with the nearest-neighbor Coulomb repulsive interaction and external force $F$. For the fixed points of such systems (fixed configurations) we study the asymptotics (in…
The fundamental model of a periodic structure is a periodic point set up to rigid motion or isometry. Our recent paper in SoCG 2021 defined isometry invariants (density functions), which are complete in general position and continuous under…
Chaos thresholds of the $z$-logistic maps $x_{t+1}=1-a|x_t|^z$ $(z>1; t=0,1,2,...)$ are numerically analysed at accumulation points of cycles 2, 3 and 5. We verify that the nonextensive $q$-generalization of a Pesin-like identity is…
This work investigates the intersection property of conditional independence. It states that for random variables $A,B,C$ and $X$ we have that $X$ independent of $A$ given $B,C$ and $X$ independent of $B$ given $A,C$ implies $X$ independent…
In experiments, the dynamical behavior of systems is reflected in time series. Due to the finiteness of the observational data set it is not possible to reconstruct the invariant measure up to arbitrary fine resolution and arbitrary high…
Intermittent dynamics is characterized by long periods of different types of dynamical characteristics, for instance almost periodic dynamics alternated by chaotic dynamics. Critical intermittency is intermittent dynamics that can occur in…
We show that the output of systems with time-varying delay can exhibit a new kind of chaotic behavior characterized by laminar phases, which are periodically interrupted by irregular bursts. Within each laminar phase the output intensity…
A random dense countable set is characterized (in distribution) by independence and stationarity. Two examples are `Brownian local minima' and `unordered infinite sample'. They are identically distributed; the former ad hoc proof of this…
Continuous-time systems with switch-like behaviour occur in chemical kinetics, gene regulatory networks and neural networks. Networks with hard switching, as a limiting case of smooth sigmoidal switching, retain the richest possible range…
Stochastic feedback systems give rise to a variety of notions of stability. The conditions for the stability of the median, mean, and variance stability conditions differ. These conditions can be stated explicitly for scalar discrete-time…