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Denis Gaidashev

Publications and source records attributed to Denis Gaidashev.

At least 19 recordsLinked to original sources
arXiv2025

Renormalization and blow ups for the nonlinear Schr\"odinger equation

Existence of finite-time blow ups in the classical one-dimensional nonlinear Schr\"odinger equation (NLS) (1) i \partial_t u + u_{x x} + |u|^{2r} u = 0, u(x,0) = u_0(x) has been one of the central problems in the studies of the singularity formation in the PDEs. We revisit this problem using an approach based on the ideas borrowed from Dynamical Systems. To that end, we reformulate the initial value problem for (1), with r \in \mathbb{N}, r \ge 1, as a fixed point problem for a certain renormalization operator, and use the ideas of apriori bounds to prove existence of a renormalization fixed point. Existence of such fixed points leads to existence of self-similar solutions of the form u(x,t) = (T-t)^{-{1 \over 2 r}} U((T-t)^{-{1 \over 2}} x), whose L^{2 r +2}-norms are bounded up-to a finite time T and whose energy blows up at T.

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arXiv2024

Wild attractors for Fibonacci maps

Existence of wild attractors -- attractors whose basin has a positive Lebesgue measure but is not a residual set -- has been one of central themes in one-dimensional dynamics. It has been demonstrated by H. Bruin et al. that Fibonacci maps with a sufficiently flat critical point admit a wild attractor. We propose a constructive trichotomy that describes possible scenarios for the Lebesgue measure of the Fibonacci attractor based on a computable criterion. We use this criterion, together with a computer-assisted proof of existence of a Fibonacci renormalization $2$-cycle for non-integer critical degrees, to demonstrate that Fibonacci maps do not have a wild attractor when the degree of the critical point is $d=3.8$ (and, conjecturally, for $2< d \le 3.8$), and do admit it when $d=5.1$ (and, conjecturally, for $d \ge 5.1$).

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arXiv2022

Renormalization and a-priori bounds for Leray self-similar solutions to the generalized mild Navier-Stokes equations

We demonstrate that the problem of existence of Leray self-similar blow up solutions in a generalized mild Navier-Stokes system with the fractional Laplacian $(-Δ)^{γ/2}$ can be stated as a fixed point problem for a "renormalization" operator. We proceed to construct {\it a-priori} bounds, that is a renormalization invariant precompact set in an appropriate weighted $L^p$-space. As a consequence of a-priori bounds, we prove existence of renormalization fixed points for $d \ge 2$ and $d<γ<2 d+2$, and existence of non-trivial Leray self-similar mild solutions in $C^\infty([0,T),(H^k)^d \cap (L^p)^d)$, $k>0, p \ge 2$, whose $(L^p)^d$-norm becomes unbounded in finite time $T$.

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arXiv2022

Renormalization and existence of the finite-time blow up solutions for a one-dimensional analogue of the Navier-Stokes equations

The one-dimensional quasi-geostrophic equation is the one-dimensional Fourier-space analogue of the famous Navier-Stokes equations. In their work Li and Sinai have proposed a renormalization approach to the problem of existence of finite-time blow up solutions of this equation. In this setting, existence of finite time blow ups is a consequence of existence of a fixed point for a certain renormalization operator on an appropriate functional space. They have provided a proof of existence of complex-valued finite time blow up solutions of the quasi-geostrophic equation. In this paper we revisit the renormalization problem for the quasi-geostrophic blow ups, prove existence of a family of renormalization fixed points, and deduce existence of real $C^\infty([0,T),C^\infty(\mathbb{R}) \cap L^2(\mathbb{R}))$ solutions to the quasi-geostrophic equation whose energy and enstrophy become unbounded in finite time, different from those found in the previous work of Li and Sinai.

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arXiv2022

On some spectral properties of stochastic similarity matrices for data clustering

Clustering in image analysis is a central technique that allows to classify elements of an image. We describe a simple clustering technique that uses the method of similarity matrices. We expand upon recent results in spectral analysis for Gaussian mixture distributions, and in particular, provide conditions for the existence of a spectral gap between the leading and remaining eigenvalues for matrices with entries from a Gaussian mixture with two real univariate components. Furthermore, we describe an algorithm in which a collection of image elements is treated as a dynamical system in which the existence of the mentioned spectral gap results in an efficient clustering.

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arXiv2020

Complex a priori bounds for Lorenz maps

We construct complex a-priori bounds for certain infinitely renormalizable Lorenz maps. As a corollary, we show that renormalization is a real-analytic operator on the corresponding space of Lorenz maps.

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arXiv2019

Renormalization and Siegel disks for complex Hénon maps

We use hyperbolicity of golden-mean renormalization of dissipative Hénon-like maps to prove that the boundaries of Siegel disks of sufficiently dissipative quadratic complex Hénon maps with golden-mean rotation number are topological circles. Conditionally on an appropriate renormalization hyperbolicity property, we derive the same result for Siegel disks of Hénon maps with all eventually periodic rotation numbers.

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arXiv2019

Renormalization of almost commuting pairs

In this paper we give a new prove of hyperbolicity of renormalization of critical circle maps using the formalism of almost-commuting pairs. We extend renormalization to two-dimensional dissipative maps of the annulus which are small perturbations of one-dimensional critical circle maps. Finally, we demontsrate that a two-dimensional map which lies in the stable set of the renormalization operator possesses an attractor which is topologically a circle. Such a circle is critical: the dynamics on it is topologically, but not smoothly, conjugate to a rigid rotation.

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arXiv2018

On the geometry of period doubling invariant sets for area-preserving maps

The geometry of the period doubling Cantor sets of strongly dissipative infinitely renormalizable H\'enon-like maps has been shown to be unbounded by M. Lyubich, M. Martens and A. de Carvalho, although the measure of unbounded "spots" in the Cantor set has been demonstrated to be zero. We show that an even more extreme situation takes places for infinitely renormalizable area-preserving H\'enon-like maps: both bounded and unbounded geometries exist on subsets of positive measure.

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arXiv2017

Exponentially small splitting of separatrices near a period-doubling bifurcation in area-preserving maps

We consider the conservative Hénon family at the period-doubling bifurcation of its fixed point and demonstrate that the separatrices of the fixed saddle point nearing the bifurcation split exponentially: given that $λ_+$ is the smaller of the eigenvalues of the saddle point, the angle between the separatrices along the homoclinic orbit satisfies $$\sin α= O(e^{-{π^2 \over \log |λ_+|}})+ O\left( e^{-2 (1-κ) {π^2 \over \log |λ_+|}} \right),$$ for any positive $κ<1$.

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arXiv2016

Renormalization for Lorenz maps of monotone combinatorial types

Lorenz maps are maps of the unit interval with one critical point of order rho>1, and a discontinuity at that point. They appear as return maps of leafs of sections of the geometric Lorenz flow. We construct real a priori bounds for renormalizable Lorenz maps with certain monotone combinatorics, and use these bounds to show existence of periodic points of renormalization, as well as existence of Cantor attractors for dynamics of infinitely renormalizable Lorenz maps.

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arXiv2016

Golden mean Siegel disk universality and renormalization

We provide a computer-assisted proof of one of the central open questions in one-dimensional renormalization theory -- universality of the golden-mean Siegel disks. We further show that for every function in the stable manifold of the golden-mean renormalization fixed point the boundary of the Siegel disk is a quasicircle which coincides with the closure of the critical orbit, and that the dynamics on the boundary of the Siegel disk is rigid. Furthermore, we extend the renormalization from one-dimensional analytic maps with a golden-mean Siegel disk to two-dimensional dissipative Hénon-like maps and show that the renormalization hyperbolicity result still holds in this setting.

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arXiv2014

Rigidity for infinitely renormalizable area-preserving maps

The period doubling Cantor sets of strongly dissipative Henon-like maps with different average Jacobian are not smoothly conjugated. The Jacobian Rigidity Conjecture says that the period doubling Cantor sets of two-dimensional Henon-like maps with the same average Jacobian are smoothly conjugated. This conjecture is true for average Jacobian zero, e.g. the one-dimensional case. The other extreme case is when the maps preserve area, e.g. the average Jacobian is one. Indeed, the period doubling Cantor set of area-preserving maps in the universality class of the Eckmann-Koch-Wittwer renormalization fixed point are smoothly conjugated.

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arXiv2014

Spectral properties of renormalization for area-preserving maps

Area-preserving maps have been observed to undergo a universal period-doubling cascade, analogous to the famous Feigenbaum-Coullet-Tresser period doubling cascade in one-dimensional dynamics. A renormalization approach has been used by Eckmann, Koch and Wittwer in a computer-assisted proof of existence of a conservative renormalization fixed point. Furthermore, it has been shown by Gaidashev, Johnson and Martens that infinitely renormalizable maps in a neighborhood of this fixed point admit invariant Cantor sets with vanishing Lyapunov exponents on which dynamics for any two maps is smoothly conjugate. This rigidity is a consequence of an interplay between the decay of geometry and the convergence rate of renormalization towards the fixed point. In this paper we prove a result which is crucial for a demonstration of rigidity: that an upper bound on this convergence rate of renormalizations of infinitely renormalizable maps is sufficiently small.

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arXiv2014

On the scaling ratios for Siegel disks

The boundary of the Siegel disk of a quadratic polynomial with an irrationally indifferent fixed point and the rotation number whose continued fraction expansion is preperiodic has been observed to be self-similar with a certain scaling ratio. The restriction of the dynamics of the quadratic polynomial to the boundary of the Siegel disk is known to be quasisymmetrically conjugate to the rigid rotation with the same rotation number. The geometry of this self-similarity is universal for a large class of holomorphic maps. A renormalization explanation of this universality has been proposed in the literature. In this paper we provide an estimate on the quasisymmetric constant of the conjugacy, and use it to prove bounds on the scaling ratio $λ$ of the form $$α^γ\le |λ| \le C δ^s,$$ where $s$ is the period of the continued fraction, and $α\in (0,1)$ depends on the rotation number in an explicit way, while $C>1$, $δ\in (0,1)$ and $γ\in (0,1)$ depend only on the maximum of the integers in the continued fraction expansion of the rotation number.

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arXiv2011

A numerical study of infinitely renormalizable area-preserving maps

It has been shown in (Gaidashev et al, 2010) and (Gaidashev et al, 2011) that infinitely renormalizable area-preserving maps admit invariant Cantor sets with a maximal Lyapunov exponent equal to zero. Furthermore, the dynamics on these Cantor sets for any two infinitely renormalizable maps is conjugated by a transformation that extends to a differentiable function whose derivative is Holder continuous of exponent alpha>0. In this paper we investigate numerically the specific value of alpha. We also present numerical evidence that the normalized derivative cocycle with the base dynamics in the Cantor set is ergodic. Finally, we compute renormalization eigenvalues to a high accuracy to support a conjecture that the renormalization spectrum is real.

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arXiv2010

On Analytic Perturbations of a Family of Feigenbaum-like Equations

We prove existence of solutions $(ϕ,λ)$ of a family of of Feigenbaum-like equations \label{family} ϕ(x)={1+\eps \over λ} ϕ(ϕ(λx)) -\eps x +τ(x), where $\eps$ is a small real number and $τ$ is analytic and small on some complex neighborhood of $(-1,1)$ and real-valued on $\fR$. The family $(\ref{family})$ appears in the context of period-doubling renormalization for area-preserving maps (cf. \cite{GK}). Our proof is a development of ideas of H. Epstein (cf \cite{Eps1}, \cite{Eps2}, \cite{Eps3}) adopted to deal with some significant complications that arise from the presence of terms $\eps x +τ(x)$ in the equation $(\ref{family})$. The method relies on a construction of novel {\it a-priori} bounds for unimodal functions which turn out to be very tight. We also obtain good bounds on the scaling parameter $λ$. A byproduct of the method is a new proof of the existence of a Feigenbaum-Coullet-Tresser function.

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