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Wieslaw Krawcewicz

Publications and source records attributed to Wieslaw Krawcewicz.

14 recordsLinked to original sources
arXiv2025

Global Bifurcation In Four-Component Bose-Einstein Condensates In Space

We analyze a system of coupled Bose-Einstein condensates in the domain of a unitary ball in $\mathbb{R}^3$. The coupling is due to atom-to-atom interactions that occur between different gas components. The multi-component Bose-Einstein condensate is described by a system of Gross-Pitaevskii equations, which has an explicit trivial branch of constant solutions bifurcating from the zero-solution. Our main theorem establishes that this trivial branch undergoes multiple global bifurcations at any critical values with kernels of dimensions at least $3(2k+1)$, for $k \in \mathbb{N}^+$. Handling these high dimension kernels poses a challenge from the perspective of bifurcation theory. Our methodology, which relies on the $G$-equivariant gradient degree, effectively manages these complexities and establishes the existence of at least two global branch in the particular case of $k = 0$ and at least six branches in the case of $k = 1$.

math.AP↗
arXiv2025

Global Bifurcation of Spiral Wave Solutions to the Complex Ginzburg-Landau Equation

We use the $\mathbb T^2$-equivariant degree to establish the existence of unbounded branches of rotating spiral wave solutions with any number of arms for the complex Ginzburg Landau equation GLe on the planar unit disc, leveraging the spatial symmetries inherent to the problem and avoiding limiting constraints encountered in previous studies (Dai 2021) that utilized the classical Leray-Schauder degree. Our results provide rigorous mathematical justification for the formation and persistence of these fundamental patterns, which are ubiquitous in physical, chemical, and biological systems but have, until now, eluded formal proof under general conditions.

math.AP↗
arXiv2024

Global Bifurcation in Symmetric Systems of Nonlinear Wave Equations

In this paper, we use the equivariant degree theory to establish a global bifurcation result for the existence of non-stationary branches of solutions to a nonlinear, two-parameter family of hyperbolic wave equations with local delay and non-trivial damping. As a motivating example, we consider an application of our result to a system of $N$ identical vibrating strings with dihedral coupling relations.

math.AP↗
arXiv2020

Subharmonic Solutions In Reversible Non-Autonomous Differential Equations

We study the existence of subharmonic solutions in the system $\ddot {u}(t)=f(t,u(t))$, where $u(t)\in\mathbb{R}^{k}$ and $f$ is an even and $p$-periodic function in time. Under some additional symmetry conditions on the function $f$, the problem of finding $mp$-periodic solutions can be reformulated in a functional space as a $Γ\times\mathbb{Z}_{2}\times D_{m}% $-equivariant equation, where the group $Γ\times\mathbb{Z}_{2}$ acts on the space $\mathbb{R}^{k}$ and $D_{m}$ acts on $u(t)$ by time-shifts and reflection. We apply Brouwer equivariant degree to prove the existence of an infinite number of subharmonic solutions for the function $f$ that satisfies additional hypothesis on linear behavior near zero and the Nagumo condition at infinity. We also discuss the bifurcation of subharmonic solutions when the system depends on an extra parameter.

math.DS↗
arXiv2020

Periodic Solutions to Reversible Second Order Autonomous DDEs in Prescribed Symmetric Nonconvex Domains

The existence and spatio-temporal patterns of $2π$-periodic solutions to second order reversible equivariant autonomous systems with commensurate delays are studied using the Brouwer $O(2) \times Γ\times \mathbb Z_2$-equivariant degree theory. The solutions are supposed to take their values in a prescribed symmetric domain $D$, while $O(2)$ is related to the reversal symmetry combined with the autonomous form of the system. The group $Γ$ reflects symmetries of $D$ and/or possible coupling in the corresponding network of identical oscillators, and $\mathbb Z_2$ is related to the oddness of the right-hand side. Abstract results, based on the use of Gauss curvature of $\partial D$, Hartman-Nagumo type {\it a priori bounds} and Brouwer equivariant degree techniques, are supported by a concrete example with $Γ= D_8$ -- the dihedral group of order $16$.

math.DS↗
arXiv2020

Existence and Spatio-Temporal Patterns of Periodic Solutions to Second Order Non-Autonomous Equivariant Delayed Systems

Existence and spatio-temporal symmetric patterns of periodic solutions to second order reversible equivariant non-autonomous periodic systems with multiple delays are studied under the Hartman-Nagumo growth conditions. The method is based on using the Brouwer $D_1 \times \mathbb Z_2\times Γ$-equivariant degree theory, where $D_1$ is related to the reversing symmetry, $\mathbb Z_2$ is related to the oddness of the right-hand-side and $Γ$ reflects the symmetric character of the coupling in the corresponding network. Abstract results are supported by a concrete example with $Γ= D_n$ -- the dihedral group of order $2n$.

math.DS↗
arXiv2018

Nonlinear Vibrations in the Fullerene Molecule $C_{60}$

In this paper we analyze nonlinear dynamics of the fullerene molecule. We prove the existence of global branches of periodic solutions emerging from an icosahedral equilibrium (nonlinear normal modes). We also determine the symmetric properties of the branches of nonlinear normal modes for maximal orbit types. We find several solutions which are standing and rotating waves that propagate along the molecule with icosahedral, tetrahedral, pentagonal and triangular symmetries. We complement our theoretical results with numerical computations using Newton's method.

math.DS↗
arXiv2018

Solutions of Fixed Period in the Nonlinear Wave Equation on Networks

The wave equation on network is defined by $\partial_{tt}u=Δ_{G}u+g(u)$, where $u\in\mathbb{R}^{n}$ and the graph Laplacian $Δ_{G}$ is an operator on functions on $n$ vertices. We suppose that $g:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ is an odd continuous function that satisfies $g(0)=g^{\prime }(0)=0$ and the Nagumo condition. Assuming that the graph is invariant by a subgroup of permutations $Γ$, using a $Γ$-equivariant topological invariant we prove the existence of multiple non-constant $p$-periodic solutions characterized by their symmetries.

math.DS↗
arXiv2018

Symmetries of Nonlinear Vibrations in Tetrahedral Molecular Configurations

We study nonlinear vibrational modes of oscillations for tetrahedral configurations of particles. In the case of tetraphosphorus, the interaction of atoms is given by bond stretching and van der Waals forces. Using equivariant gradient degree, we present a full topological classification of the spatio-temporal symmetries of the periodic solutions. This procedure gives all the symmetries of the nonlinear vibrations for general force fields.

math.DS↗
arXiv2017

Hopf Bifurcation of Relative Periodic Solutions: Case Study of a Ring of Passively Mode-Locked Lasers

In this paper, we consider an equivariant Hopf bifurcation of relative periodic solutions from relative equilibria in systems of functional differential equations respecting $Γ\times S^1$-spatial symmetries. The existence of branches of relative periodic solutions together with their symmetric classification is established using the equivariant twisted $Γ\times S^1$-degree with one free parameter. As a case study, we consider a delay differential model of coupled identical passively mode-locked semiconductor lasers with the dihedral symmetry group $Γ=D_8$.

math.DS↗
arXiv2017

Dihedral Molecular Configurations Interacting by Lennard-Jones and Coulomb Forces

In this paper, we investigate periodic vibrations of a group of particles with a dihedral configuration in the plane governed by the Lennard-Jones and Coulomb forces. Using the gradient equivariant degree, we provide a full topological classification of the periodic solutions with both temporal and spatial symmetries. In the process, we provide with general formulae for the spectrum of the linearized system which allows us to obtain the critical frequencies of the particle motions which indicate the set of all critical periods of small amplitude periodic solutions emerging from a given stationary symmetric orbit of solutions.

math.DS↗
arXiv2016

Multiple Periodic Solutions for $Γ$-symmetric Newtonian Systems

The existence of periodic solutions in $Γ$-symmetric Newtonian systems $\ddot{x}=-\nabla f(x)$ can be effectively studied by means of the $(Γ\times O(2))$-equivariant gradient degree with values in the Euler ring $U(Γ\times O(2))$. In this paper, we show that in the case of $Γ$ being a finite group, the Euler ring $U(Γ\times O(2))$ and the related basic degrees are effectively computable using Euler ring homomorphisms, the Burnside ring $A(Γ\times O(2))$ and the reduced $(Γ\times O(2))$-degree with no free parameters. We present several examples of Newtonian systems with various symmetries, for which we show existence of multiple periodic solutions. We also provide exact value of the equivariant topological invariant for those problems.

math.DS↗
arXiv2016

Non-invasive stabilization of periodic orbits in $O_4$-symmetrically coupled Van der Pol oscillators

Pyragas time delayed feedback control has proven itself as an effective tool to non-invasively stabilize periodic solutions. In a number of publications, this method was adapted to equivariant settings and applied to stabilize branches of small periodic solutions in systems of symmetrically coupled Landau oscillators near a Hopf bifurcation point. The form of the control ensures the non-invasiveness property, hence reducing the problem to finding a set of the gain matrices, which would guarantee the stabilization. In this paper, we apply this method to a system of Van der Pol oscillators coupled in a cube-like configuration leading to $O_4$-equivariance. We discuss group theoretic restrictions which help to shape our choice of control. Furthermore, we explicitly describe the domains in the parameter space for which the periodic solutions are stable.

math.OC↗