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Dmitry Turaev

Publications and source records attributed to Dmitry Turaev.

41 records · Page 3Linked to original sources
arXiv2014

Exponential energy growth in adiabatically changing Hamiltonian Systems

Fermi acceleration is the process of energy transfer from massive objects in slow motion to light objects that move fast. The model for such process is a time-dependent Hamiltonian system. As the parameters of the system change with time, the energy is no longer conserved, which makes the acceleration possible. One of the main problems is how to generate a sustained and robust energy growth. We show that the non-ergodicity of any chaotic Hamiltonian system must universally lead to the exponential growth of energy at a slow periodic variation of parameters. We build a model for this process in terms of a Geometric Brownian Motion with a positive drift, and relate it to the entropy increase.

nlin.CD↗
arXiv2013

Showcase of Blue Sky Catastrophes

Let a system of differential equations possess a saddle-node periodic orbit such that every orbit in its unstable manifold is homoclinic, i.e. the unstable manifold is a subset of the (global) stable manifold. We study several bifurcation cases where the splitting of such a homoclinic connection causes the Blue Sky Catastrophe, including the onset of complex dynamics. The birth of an invariant torus or a Klein bottle is also described.

math.DS↗
arXiv2012

Energy growth for a nonlinear oscillator coupled to a monochromatic wave

A system consisting of a chaotic (billiard-like) oscillator coupled to a linear wave equation in the three-dimensional space is considered. It is shown that the chaotic behaviour of the oscillator can cause the transfer of energy from a monochromatic wave to the oscillator, whose energy can grow without bound.

math.DS↗
arXiv2010

Maps close to identity and universal maps in the Newhouse domain

Given an n-dimensional C^r-diffeomorphism g, its renormalized iteration is an iteration of g, restricted to a certain n-dimensional ball and taken in some C^r-coordinates in which the ball acquires radius 1. We show that for any r >/- 1 the renormalized iterations of C^r -close to identity maps of an n-dimensional unit ball B^n (n >/- 2) form a residual set among all orientation-preserving C^r -diffeomorphisms B^n \to R^n. In other words, any generic n-dimensional dynamical phenomenon can be obtained by iterations of C^r -close to identity maps, with the same dimension of the phase space. As an application, we show that any C^r-generic two-dimensional map which belongs to the Newhouse domain (i.e., it has a wild hyperbolic set, so it is not uniformly-hyperbolic, nor uniformly partially-hyperbolic) and which neither contracts, nor expands areas, is C^r -universal in the sense that its iterations, after an appropriate coordinate transformation, C^r -approximate every orientation-preserving two-dimensional diffeomorphism arbitrarily well. In particular, every such universal map has an infinite set of coexisting hyperbolic attractors and repellers

math.DS↗