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D. Turaev

Publications and source records attributed to D. Turaev.

12 recordsLinked to original sources
arXiv2024

Desynchronization of temporal solitons in Kerr cavities with pulsed injection

A numerical and analytical study was conducted to investigate the bifurcation mechanisms that cause desynchronization between the soliton repetition frequency and the frequency of external pulsed injection in a Kerr cavity described by the Lugiato-Lefever equation. The results suggest that desynchronization typically occurs through an Andronov-Hopf bifurcation. Additionally, a simple and intuitive criterion for this bifurcation to occur is proposed.

physics.optics↗
arXiv2023

Smooth Persistence of Attractors for Set-Valued Dynamical Systems: A Boundary Map Approach

We study the problem of persistence of attractors with smooth boundary for a class of set-valued dynamical systems that naturally arise in the context of random and control dynamical systems, as well as in systems modeling the dynamical propagation of uncertainty. In order to tackle the inherent difficulties associated to the multi-valued structure of such dynamical systems, we introduce a single-valued map, the so-called boundary map, which is a contactomorphism of the unit-tangent bundle of the state space, with the following characteristic property: boundaries of attractors of the set-valued dynamical system correspond in a unique way to invariant Legendrian manifolds of this map. We show how the underlying contact geometry guarantees the smooth persistence of such attractors under perturbations of the set-valued dynamical system, provided that the associated boundary map is normally hyperbolic at the unit normal bundle of the boundary.

math.DS↗
arXiv2022

Stable motions of high energy particles interacting via a repelling potential

The motion of N particles interacting by a smooth repelling potential and confined to a compact d-dimensional region is proved to be, under mild conditions, non-ergodic for all sufficiently large energies. Specifically, choreographic solutions, for which all particles follow approximately the same path close to an elliptic periodic orbit of the single-particle system, are proved to be KAM stable in the high energy limit. Finally, it is proved that the motion of N repelling particles in a rectangular box is non-ergodic at high energies for a generic choice of interacting potential: there exists a KAM-stable periodic motion by which the particles move fast only in one direction, each on its own path, yet in synchrony with all the other parallel moving particles.

math.DS↗
arXiv2014

Oscillating mushrooms: adiabatic theory for a non-ergodic system

Can elliptic islands contribute to sustained energy growth as parameters of a Hamiltonian system slowly vary with time? In this paper we show that a mushroom billiard with a periodically oscillating boundary accelerates the particle inside it exponentially fast. We provide an estimate for the rate of acceleration. Our numerical experiments confirms the theory. We suggest that a similar mechanism applies to general systems with mixed phase space.

math.DS↗
arXiv2010

Analytical Proof of Space-Time Chaos in Ginzburg-Landau Equations

We prove that the attractor of the 1D quintic complex Ginzburg-Landau equation with a broken phase symmetry has strictly positive space-time entropy for an open set of parameter values. The result is obtained by studying chaotic oscillations in grids of weakly interacting solitons in a class of Ginzburg-Landau type equations. We provide an analytic proof for the existence of two-soliton configurations with chaotic temporal behavior, and construct solutions which are closed to a grid of such chaotic soliton pairs, with every pair in the grid well spatially separated from the neighboring ones for all time. The temporal evolution of the well-separated multi-soliton structures is described by a weakly coupled lattice dynamical system (LDS) for the coordinates and phases of the solitons. We develop a version of normal hyperbolicity theory for the weakly coupled LDSs with continuous time and establish for them the existence of space-time chaotic patterns similar to the Sinai-Bunimovich chaos in discrete-time LDSs. While the LDS part of the theory may be of independent interest, the main difficulty addressed in the paper concerns with lifting the space-time chaotic solutions of the LDS back to the initial PDE. The equations we consider here are space-time autonomous, i.e. we impose no spatial or temporal modulation which could prevent the individual solitons in the grid from drifting towards each other and destroying the well-separated grid structure in a finite time. We however manage to show that the set of space-time chaotic solutions for which the random soliton drift is arrested is large enough, so the corresponding space-time entropy is strictly positive.

math.DS↗
arXiv2007

Foam-like structure of the Universe

On the quantum stage spacetime had the foam-like structure. When the Universe cools, the foam structure tempers and does not disappear. We show that effects caused by the foamed structure mimic very well the observed Dark Matter phenomena. Moreover, we show that in a foamed space photons undergo a chaotic scattering and together with every discrete source of radiation we should observe a diffuse halo. We show that the distribution of the diffuse halo of radiation around a point-like source repeats exactly the distribution of dark matter around the same source, i.e. the DM halos are sources of the diffuse radiation.

astro-ph↗
arXiv2006

The Universal Rotation Curve of Spiral Galaxies

The observed strong dark-to-luminous matter coupling is described by a bias relation between visible and dark matter sources. We discuss the bias which emerges in the case where the topological structure of the Universe at very large distances does not match properly that of the Friedman space. With the use of such "topological" bias, we construct the Universal Rotation Curve (URC) for spirals which occurs to be in a striking agreement with the empirically known URC. We also show that the topological bias explains the origin of the Tully-Fisher relation ($ L\sim V^{4}$) and predicts peculiar oscillations in the URC with a characteristic length $\sim \sqrt{L}$.

astro-ph↗
arXiv2005

Approximating multi-dimensional Hamiltonian flows by billiards

Consider a family of smooth potentials $V_ε$, which, in the limit $ε\to0$, become a singular hard-wall potential of a multi-dimensional billiard. We define auxiliary billiard domains that asymptote, as $ε\to0$ to the original billiard, and provide asymptotic expansion of the smooth Hamiltonian solution in terms of these billiard approximations. The asymptotic expansion includes error estimates in the $C^{r}$ norm and an iteration scheme for improving this approximation. Applying this theory to smooth potentials which limit to the multi-dimensional close to ellipsoidal billiards, we predict when the separatrix splitting persists for various types of potentials.

nlin.CD↗
arXiv2002

Soft billiards with corners

We develop a framework for dealing with smooth approximations to billiards with corners in the two-dimensional setting. Let a polygonal trajectory in a billiard start and end up at the same billiard's corner point. We prove that smooth Hamiltonian flows which limit to this billiard have a nearby periodic orbit if and only if the polygon angles at the corner are ''acceptable''. The criterion for a corner polygon to be acceptable depends on the smooth potential behavior at the corners, which is expressed in terms of a {scattering function}. We define such an asymptotic scattering function and prove the existence of it, explain how it can be calculated and predict some of its properties. In particular, we show that it is non-monotone for some potentials in some phase space regions. We prove that when the smooth system has a limiting periodic orbit it is hyperbolic provided the scattering function is not extremal there. We then prove that if the scattering function is extremal, the smooth system has elliptic periodic orbits limiting to the corner polygon, and, furthermore, that the return map near these periodic orbits is conjugate to a small perturbation of the Henon map and therefore has elliptic islands. We find from the scaling that the island size is typically algebraic in the smoothing parameter and exponentially small in the number of reflections of the polygon orbit.

nlin.CD↗
arXiv2002

On modification of the Newton's law of gravity at very large distances

We discuss a Modified Field Theory (MOFT) in which the number of fields can vary. It is shown that when the number of fields is conserved MOFT reduces to the standard field theory but interaction constants undergo an additional renormalization and acquire a dependence on spatial scales. In particular, the renormalization of the gravitational constant leads to the deviation of the law of gravity from the Newton's law in some range of scales $r_{\min} r_{\max}$ acquires a new constant value $G^{\prime}\sim Gr_{\max}/r_{\min}$. From the dynamical standpoint this looks as if every point source is surrounded with a halo of dark matter. It is also shown that if the maximal scale $r_{\max}$ is absent, the homogeneity of the dark matter in the Universe is consistent with a fractal distribution of baryons in space, in which the luminous matter is located on thin two-dimensional surfaces separated by empty regions of ever growing size.

astro-ph↗
arXiv2001

Corrections to the Newton and Coulomb potentials caused by effects of spacetime foam

We use an extended quantum field theory (EQFT) hep-th/9911168 to explore possible observational effects of the spacetime. It is shown that as it was expected the spacetime foam can provide quantum bose fields with a cutoff at very small scales, if the energy of zero - point fluctuations of fields is taken into account. It is also shown that EQFT changes the behaviour of massless fields at very large scales (in the classical region). We show that as $r\gg 1/μ$ the Coulomb and Newton forces acquire the behaviour $\sim 1/r$ (instead of $1/r^{2}$).

hep-th↗