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Piotr Zgliczyński

Publications and source records attributed to Piotr Zgliczyński.

36 records · Page 2Linked to original sources
arXiv2019

On non-autonomously forced Burgers equation with periodic and Dirichlet boundary conditions

We study the non-autonomously forced Burgers equation $$ u_t(x,t) + u(x,t)u_x(x,t) - u_{xx}(x,t) = f(x,t) $$ on the space interval $(0,1)$ with two sets of the boundary conditions: the Dirichlet and periodic ones. For both situations we prove that there exists the unique $H^1$ bounded trajectory of this equation defined for all $t\in \mathbb{R}$. Moreover we demonstrate that this trajectory attracts all trajectories both in pullback and forward sense. We also prove that for the Dirichlet case this attraction is exponential.

math.AP↗
arXiv2017

Geometric proof of the Grobman-Hartman Theorem

We give geometric proofs for Grobman-Hartman theorem for diffeomorphisms and ODEs. Proofs use covering relations and cone conditions for maps and isolating segments and cone condition for ODEs. We prove also the Hölder condition for the conjugating homeomorphims.

math.DS↗
arXiv2017

Shadowing of non-transversal heteroclinic chains

We present a new result about the shadowing of nontransversal chain of heteroclinic connections based on the idea of dropping dimensions. We illustrate this new mechanism with several examples. As an application we discuss this mechanism in a simplification of a toy model system derived by Colliander et al. in the context of cubic defocusing nonlinear Schrödinger equation.

math.DS↗
arXiv2016

Existence of periodic solutions of the FitzHugh-Nagumo equations for an explicit range of the small parameter

The FitzHugh-Nagumo model describing propagation of nerve impulses in axon is given by fast-slow reaction-diffusion equations, with dependence on a parameter $ε$ representing the ratio of time scales. It is well known that for all sufficiently small $ε>0$ the system possesses a periodic traveling wave. With aid of computer-assisted rigorous computations, we prove the existence of this periodic orbit in the traveling wave equation for an explicit range $ε\in (0, 0.0015]$. Our approach is based on a novel method of combination of topological techniques of covering relations and isolating segments, for which we provide a self-contained theory. We show that the range of existence is wide enough, so the upper bound can be reached by standard validated continuation procedures. In particular, for the range $ε\in [1.5 \times 10^{-4}, 0.0015]$ we perform a rigorous continuation based on covering relations and not specifically tailored to the fast-slow setting. Moreover, we confirm that for $ε=0.0015$ the classical interval Newton-Moore method applied to a sequence of Poincaré maps already succeeds. Techniques described in this paper can be adapted to other fast-slow systems of similar structure.

math.DS↗
arXiv2016

Hyperbolicity and Averaging for the Srzednicki-Wójcik equation

For the Srzednicki-Wójcik equation, the planar nonautonomous ODE parameterized by $κ\in \mathbb{R}$, $$ z'=\overline{z}(1+ |z|^2 \exp(i κt)), \qquad z(t) \in \mathbb{C} $$ using averaging we show how the region of hyperbolicty grows with $|κ|$. Based on this we give bounds on the sizes of bounded orbits.

math.CA↗
arXiv2016

On the Petras algorithm for verified integration of piecewise analytic functions

We consider the algorithm for verified integration of piecewise analytic functions given by Petras. The analysis of the algorithm contained in Patras' paper is limited to a narrow class of functions and gives upper bounds only. We present an estimation of the complexity (measured by a number of evaluations of an integrand) of the algorithm, both upper and lower bounds, for a wider class of functions. We show examples with complexity $Θ(|\ln\eps|/\eps^{p-1})$, for any $p >1$, where $\eps$ is the desired accuracy of the computed integral.

math.NA↗
arXiv2016

Stabilizing the Long-time Behavior of the Navier-Stokes Equations and Damped Euler Systems by Fast Oscillating Forces

The paper studies the issue of stability of solutions to the Navier-Stokes and damped Euler systems in periodic boxes. We show that under action of fast oscillating-in- time external forces all two dimensional regular solutions converge to a time periodic flow. Unexpectedly, effects of stabilization can be also obtained for systems with stationary forces with large total momentum (average of the velocity). Thanks to the Galilean transformation and space boundary conditions, the stationary force changes into one with time oscillations. In the three dimensional case we show an analogical result for weak solutions to the Navier- Stokes equations.

math.AP↗
arXiv2015

Stabilizing effect of large average initial velocity in forced dissipative PDEs invariant with respect to Galilean transformations

We describe a topological method to study the dynamics of dissipative PDEs on a torus with rapidly oscillating forcing terms. We show that a dissipative PDE, which is invariant with respect to Galilean transformations, with a large average initial velocity can be reduced to a problem with rapidly oscillating forcing terms. We apply the technique to the Burgers equation, and the incompressible 2D Navier-Stokes equations with a time-dependent forcing. We prove that for a large initial average speed the equation admits a bounded eternal solution, which attracts all other solutions forward in time. For the incompressible 3D Navier-Stokes equations we establish existence of a locally attracting solution.

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arXiv2015

Connecting orbits for a singular nonautonomous real Ginzburg-Landau type equation

We propose a method for computation of stable and unstable sets associated to hyperbolic equilibria of nonautonomous ODEs and for computation of specific type of connecting orbits in nonautonomous singular ODEs. We apply the method to a certain a singular nonautonomous real Ginzburg-Landau type equation, which that arises from the problem of formation of spots in the Swift-Hohenberg equation.

math.DS↗
arXiv2015

Rigorous numerics for PDEs with indefinite tail: existence of a periodic solution of the Boussinesq equation with time-dependent forcing

We consider the Boussinesq PDE perturbed by a time-dependent forcing. Even though there is no smoothing effect for arbitrary smooth initial data, we are able to apply the method of self-consistent bounds to deduce the existence of smooth classical periodic solutions in the vicinity of 0. The proof is non-perturbative and relies on construction of periodic isolating segments in the Galerkin projections.

math.AP↗
arXiv2015

On rigorous estimates of eigenspaces and eigenvalues of a matrix

We present a method of cones for rigorous estimations of eigenvectors, eigenspaces and eigenvalues of a matrix. The key notion is the cone-domination and is inspired by ideas from hyperbolic dynamical systems. We present theorems which allow to rigorously locate the spectrum of the matrix and the eigenspaces, also multidimensional ones in case of eigenvalues of multiplicity greater than one or clusters of close eigenvalues. In case of isolated eigenvalue we show that the our method give the same or better estimates than ones known in literature.

math.DS↗
arXiv2015

Geometric proof for normally hyperbolic invariant manifolds

We present a new proof of the existence of normally hyperbolic manifolds and their whiskers for maps. Our result is not perturbative. Based on the bounds on the map and its derivative, we establish the existence of the manifold within a given neighbourhood. Our proof follows from a graph transform type method and is performed in the state space of the system. We do not require the map to be invertible. From our method follows also the smoothness of the established manifolds, which depends on the smoothness of the map, as well as rate conditions, which follow from bounds on the derivative of the map. Our method is tailor made for rigorous, interval arithmetic based, computer assisted validation of the needed assumptions.

math.DS↗
arXiv2014

Existence of globally attracting solutions for one-dimensional viscous Burgers equation with nonautonomous forcing - a computer assisted proof

We prove the existence of globally attracting solutions of the viscous Burgers equation with periodic boundary conditions on the line for some particular choices of viscosity and non-autonomous forcing. The attract- ing solution is periodic if the forcing is periodic. The method is general and can be applied to other similar partial differential equations. The proof is computer assisted.

math.DS↗