Berger, P Turaev, D

We show that any area-preserving Cr-diffeomorphism of a two-dimensional surface displaying an elliptic fixed point can be Cr-perturbed to one exhibiting a chaotic island whose metric entropy is positive, for every 1≤r≤∞. This proves a conjecture of Herman stating that the identity map of the disk can be C∞-perturbed to a conservative diffeomorphism...

Al-Hdaibat, B Govaerts, Willy van Kekem, DL Kuznetsov, YuA

It is known that in the Bogdanov–Takens map there exists a zone of transversal homoclinic intersections bounded by two curves of homoclinic tangencies. In this paper, we derive an improved asymptotic formula for the homoclinic parameter values of the BT map. We compare two methods to approximate the Bogdanov–Takens map by the time-1 flow of a vecto...

Li, Dongchen Turaev, Dmitry V.

We prove that a pair of heterodimensional cycles can be born at the bifurcations of a pair of Shilnikov loops (homoclinic loops to a saddle-focus equilibrium) having a one-dimensional unstable manifold in a volume-hyperbolic flow with a $\mathbb{Z}_2$ symmetry. We also show that these heterodimensional cycles can belong to a chain-transitive attrac...

Ovsyannikov, II Turaev, D

We give an analytic (free of computer assistance) proof of the existence of a classical Lorenz attractor for an open set of parameter values of the Lorenz model in the form of Yudovich–Morioka–Shimizu. The proof is based on detection of a homoclinic butterfly with a zero saddle value and rigorous verification of one of the Shilnikov criteria for th...

Knobloch, J Lamb, JSW Webster, KN

We consider non-elementary T-points in reversible systems in R2n+1. We assume that the leading eigenvalues are real. We prove the existence of shift dynamics in the unfolding of this T-point. Furthermore, we study local bifurcations of symmetric periodic orbits occurring in the process of dissolution of the chaotic dynamics.

Grines, VZ Pochinka, OV Van Strien, S

This paper is a step towards the complete topological classification of Ω-stable diffeomorphisms on an orientable closed surface, aiming to give necessary and sufficient conditions for two such diffeomorphisms to be topologically conjugate without assuming that the diffeomorphisms are necessarily close to each other. In this paper we will establish...

Asaoka, Masayuki Shinohara, Katsutoshi Turaev, Dmitry
Published in
Mathematische Annalen

We consider semigroup actions on the unit interval generated by strictly increasing Cr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^r$$\end{document}-maps. We assum...

Cotter, C. J. Eldering, J. Holm, D. D. Jacobs, H. O. Meier, D. M.
Published in
Journal of Nonlinear Science

We review the role of dual pairs in mechanics and use them to derive particle-like solutions to regularized incompressible fluid systems. In our case we have a dual pair resulting from the action of diffeomorphisms on point particles (essentially by moving the points). We then augment our dual pair by considering the action of diffeomorphisms on Ta...

Turaev, Dmitry
Published in
Communications in Mathematical Physics

Given an n-dimensional Cr-diffeomorphism g, its renormalized iteration is an iteration of g, restricted to a certain n-dimensional ball and taken in some Cr-coordinates in which the ball acquires radius 1. We show that for any r ≥ 1 the renormalized iterations of Cr-close to identity maps of an n-dimensional unit ball Bn (n ≥ 2) form a residual set...

Tal, Fabio Armando

CNPq