Piotr Oprocha

Distributional chaos revisited

In their famous paper, Schweizer and Smital introduced the definition of a distributionally chaotic pair and proved that the existence of such a pair implies positive topological entropy for continuous mappings of a compact interval. Further, their approach was extended to the general compact metric space case. In this article we provide an example which shows that the definition of distributional chaos (and as a result Li-Yorke chaos) may be fulfilled by a dynamical system with (intuitively) regular dynamics embedded in ℝ 3 . Next, we state strengthened versions of distributional chaos which, as we show, are present in systems commonly considered to have complex dynamics. We also prove that any interval map with positive topological entropy contains two invariant subsets X,Y C I such that f|x has positive topological entropy and f|y displays distributional chaos of type 1, but not conversely.

On almost specification and average shadowing properties

In this paper we study relations between almost specification property, asymptotic average shadowing property and average shadowing property for dynamical systems on compact metric spaces. We show implications between these properties and relate them to other important notions such as shadowing, transitivity, invariant measures, etc. We provide examples that compactness is a necessary condition for these implications to hold. As a consequence of our methodology we also obtain a proof that limit shadowing in chain transitive systems implies shadowing.

A quantum harmonic oscillator and strong chaos

It is known that many physical systems which do not exhibit deterministic chaos when treated classically may exhibit such behaviour if treated from the quantum mechanics point of view. In this paper, we will show that an annihilation operator of the unforced quantum harmonic oscillator exhibits distributional chaos as introduced in B Schweizer and J Smital (1994 Trans. Am. Math. Soc. 344 737–54). Our approach strengthens previous results on chaos in this model and provides a very powerful tool to measure chaos in other (quantum or classical) models.

On various definitions of shadowing with average error in tracing

When computing a trajectory of a dynamical system, influence of noise can lead to large perturbations which can appear, however, with small probability. Then when calculating approximate trajectories, it makes sense to consider errors small on average, since controlling them in each iteration may be impossible. Demand to relate approximate trajectories with genuine orbits leads to various notions of shadowing (on average) which we consider in the paper. As the main tools in our studies we provide a few equivalent characterizations of the average shadowing property, which also partly apply to other notions of shadowing. We prove that almost specification on the whole space induces this property on the measure center which in turn implies the average shadowing property. Finally, we study connections among sensitivity, transitivity, equicontinuity and (average) shadowing.

Specification property and distributional chaos almost everywhere

Our main result shows that a continuous map f acting on a compact metric space (X, p) with a weaker form of specification property and with a pair of distal points is distributionally chaotic in a very strong sense. Strictly speaking, there is a distributionally scrambled set S dense in X which is the union of disjoint sets homeomorphic to Cantor sets so that, for any two distinct points u, v S, the upper distribution function is identically 1 and the lower distribution function is zero at some £ > 0. As a consequence, we describe a class of maps with a scrambled set of full Lebesgue measure in the case when X is the k-dimensional cube I k . If X = I, then we can even construct scrambled sets whose complements have zero Hausdorff dimension.

Endocannabinoid and Cannabinoid-Like Fatty Acid Amide Levels Correlate with Pain-Related Symptoms in Patients with IBS-D and IBS-C: A Pilot Study

Aims Irritable bowel syndrome (IBS) is a functional gastrointestinal (GI) disorder, associated with alterations of bowel function, abdominal pain and other symptoms related to the GI tract. Recently the endogenous cannabinoid system (ECS) was shown to be involved in the physiological and pathophysiological control of the GI function. The aim of this pilot study was to investigate whether IBS defining symptoms correlate with changes in endocannabinoids or cannabinoid like fatty acid levels in IBS patients. Methods AEA, 2-AG, OEA and PEA plasma levels were determined in diarrhoea-predominant (IBS-D) and constipation-predominant (IBS-C) patients and were compared to healthy subjects, following the establishment of correlations between biolipid contents and disease symptoms. FAAH mRNA levels were evaluated in colonic biopsies from IBS-D and IBS-C patients and matched controls. Results Patients with IBS-D had higher levels of 2AG and lower levels of OEA and PEA. In contrast, patients with IBS-C had higher levels of OEA. Multivariate analysis found that lower PEA levels are associated with cramping abdominal pain. FAAH mRNA levels were lower in patients with IBS-C. Conclusion IBS subtypes and their symptoms show distinct alterations of endocannabinoid and endocannabinoid-like fatty acid levels. These changes may partially result from reduced FAAH expression. The here reported changes support the notion that the ECS is involved in the pathophysiology of IBS and the development of IBS symptoms.

Weak mixing and chaos in nonautonomous discrete systems

Abstract The paper is devoted to a study of chaotic properties of nonautonomous discrete systems (NDS) defined by a sequence f ∞ = { f i } i = 0 ∞ of continuous maps acting on a compact metric space. We consider such properties as chaos in the sense of Li and Yorke, topological weak mixing and topological entropy, all defined in a way suitable for NDS. We compare these concepts with the case of a single map (discrete dynamical system, DS for short) and relate them to recent results in the topic. While previous research of various authors were focusing on analogues to the DS case, we show that in general the dynamics of NDSs is much richer and quite different than what is expected from the DS case. We also provide a few new tools that can be used for the successful investigation of their qualitative behavior.

Relations between distributional and Devaney chaos.

Recently, it was proven that chaos in the sense of Devaney and weak mixing both imply chaos in the sense of Li and Yorke. In this article we give explicit examples that any of these two implications do not hold for distributional chaos.

Invariant scrambled sets and distributional chaos

In their famous paper ‘Period three implies chaos’, Li and Yorke started a study of a very important phenomena in dynamical systems (known presently under the name Li–Yorke chaos). Recently, it was proved by Du that an interval map f is turbulent if and only if there is an invariant scrambled set for f. We extend this approach and prove that exactly the same characterization is valid for distributional chaos.

Distributional chaos via semiconjugacy

We develop a new method for proving the existence of distributional chaos. It is based on the special properties of semiconjugacy. As an application we prove that the equation is uniformly distributionally chaotic, provided ? (0, 0.796] and g and f are small enough.We also give an example which shows that distributional chaos (DC1) does not transfer via semiconjugacy.