arXiv subjects
Publications and source records attributed to Piotr Zgliczyński.
We provide a computer-assisted proof of the exact count of classes of central configurations for five bodies for several sets of mass values that are exceptional from the point of view of the finiteness results of Albouy and Kaloshin in the planar case and of Hampton and Jensen in the spatial case.
We consider the problem governed by the gradient ODE $x'=\nabla F(x)$ in $\mathbb{R}^d$ on which we assume that it has a finite number of hyperbolic equilibria whose stable and unstable manifolds intersect transversally. This problem is perturbed by the memory term $$x'(t)=\nabla F(x(t))+\varepsilon\int_{-\infty}^t M(t-s)x(s)\, ds$$ where $\varepsilon>0$ is a small constant. The key result is that the structure of connections between the equilibria of the unperturbed problem is exactly preserved for a small $\varepsilon>0$.
In the $n$-body problem, when a~cluster of bodies tends to a collision, then its normalized shape curve converges to the set of normalized central configurations, which has $SO(2)$ symmetry in the planar case. This leaves a possibility that the normalized shape curve tends to the circle obtained by rotation of some central configuration instead of a particular point on it. This is the \emph{infinite spin problem} which concerns the rotational behavior of total collision orbits in the $n$-body problem. The question also makes sense for partial collision. We show that the infinite spin is not possible if the limiting circle is isolated from other connected components of the set of normalized central configurations. Our approach extends the method from recent work for total collision by Moeckel and Montgomery, which was based on a combination of the center manifold theorem with Łojasiewicz inequality. To that we add a shadowing result for pseudo-orbits near normally hyperbolic manifold and careful estimates on the influence of other bodies on the cluster of colliding bodies.
We prove the existence of infinite number of homoclinic and heteroclinic orbits to two periodic orbits for the Kuramoto-Sivashinsky PDE on the line with odd and periodic boundary conditions and for some fixed parameter value of the system. The proof is computer assisted and it is based on a new algorithm for rigorous integration of the variational equation for a class of dissipative PDEs on the torus.
We discuss the method of self-consistent bounds for dissipative PDEs with periodic boundary conditions. We prove convergence theorems for a class of dissipative PDEs, which constitute a theoretical basis of a general framework for construction of an algorithm that computes bounds for the solutions of the underlying PDE and its dependence on initial conditions. We also show, that the classical examples of parabolic PDEs including Kuramoto-Sivashinsky equation and the Navier-Stokes on the torus fit into this framework.
We present a characterisation of a blender based on the topological alignment of certain sets in phase space in combination with cone conditions. Importantly, the required conditions can be verified by checking properties of a single iterate of the diffeomorphism, which is achieved by finding finite series of sets that form suitable sequences of alignments. This characterisation is applicable in arbitrary dimension. Moreover, the approach naturally extends to establishing C1-persistent heterodimensional cycles. Our setup is flexible and allows for a rigorous, computer-assisted validation based on interval arithmetic.
We present a method for the complete analysis of the dynamics of dissipative Partial Differential Equations (PDEs) undergoing a pitchfork bifurcation. We apply our technique to the Kuramoto--Sivashinsky PDE on the line to obtain a computer-assisted proof of the creation of two symmetric branches of non-symmetric fixed points and heteroclinic connections between the symmetric fixed point and the new ones. The range of parameters is given explicitly and is large enough to allow for the rigorous continuation of the fixed points and heteroclinic connections created during the bifurcation.
We investigate the Brusselator system with diffusion and Dirichlet boundary conditions on one dimensional space interval. Our proof demonstrates that, for certain parameter values, a periodic orbit exists. This proof is computer-assisted and rooted in the rigorous integration of partial differential equations. Additionally, we present the evidence of the occurrence of period-doubling bifurcation.
We present a Lohner-type algorithm for rigorous integration of systems of Delay Differential Equations (DDEs) with multiple delays and its application in computation of Poincaré maps to study the dynamics of some bounded, eternal solutions. The algorithm is based on a piecewise Taylor representation of the solutions in the phase-space and it exploits the smoothing of solutions occurring in DDEs to produces enclosures of solutions of a high order. We apply the topological techniques to prove various kinds of dynamical behavior, for example, existence of (apparently) unstable periodic orbits in Mackey-Glass Equation (in the regime of parameters where chaos is numerically observed) and persistence of symbolic dynamics in a delay-perturbed chaotic ODE (the Rössler system).
We study the problem of planar central configurations with $N$ heavy bodies and $k$ bodies with arbitrary small masses. We derive the equation which describe the limit of light masses going to zero, which can be seen as the equation for central configurations in the anisotropic plane. Using computer rigorous computations we compute all central configurations for $N=2$ and $k=3,4$ and for the derived limit problem. We show that the results are consistent.
We extend Sharkovskii's theorem to the cases of $N$-dimensional maps which are close to 1D maps, with an attracting $n$-periodic orbit. We prove that, with relatively weak topological assumptions, there exist also $m$-periodic orbits for all $m\triangleright n$ in Sharkovskii's order, in the nearby. We also show, as an example of application, how to obtain such a result for the Rössler system with an attracting periodic orbit, for four sets of parameter values. The proofs are computer-assisted.
Consider the Restricted Planar Circular 3 Body Problem with both realistic mass ratio and Jacobi constant for the Sun-Jupiter pair. We prove the existence of all possible combinations of past and future final motions. In particular, we obtain the existence of oscillatory motions. All the constructed trajectories cross the orbit of Jupiter but avoid close encounters with it. The proof relies on the method of correctly aligned windows and is computer assisted.
In this article we present recent advances on interval methods for rigorous computation of Poincaré maps. We also discuss the impact of choice of Poincaré section and coordinate system on obtained bounds for computing Poincaré map nearby fixed points.
We prove the existence of $n$-periodic orbits for almost all $n\in\mathbb{N}$ in the Rössler system with attracting periodic orbit, for two sets of parameters. The proofs are computer-assisted.
We propose a method to integrate dissipative PDEs rigorously forward in time with the use of Finite Element Method (FEM). The technique is based on the Galerkin projection on the FEM space and estimates on the residual terms. The proposed approach is illustrated on a periodically forced one-dimensional Burgers equation with Dirichlet conditions. For two particular choices of the forcing we prove the existence of the periodic globally attracting trajectory and give precise bounds on its shape.
We present the CAPD::DynSys library for rigorous numerical analysis of dynamical systems. The basic interface is described together with several interesting case studies illustrating how it can be used for computer-assisted proofs in dynamics of ODEs.
We give a computer assisted proof of the full listing of central configuration for $n$-body problem for Newtonian potential on the plane for $n=5,6,7$ with equal masses. We show all these central configurations have a reflective symmetry with respect to some line. For $n=8,9,10$ we establish the existence of central configurations without any reflectional symmetry.
The rotation of Hyperion is often modelled by equations of motion of an ellipsoidal satellite. The model is expected to be chaotic for large range of parameters. The paper contains a rigorous computer-assisted proof of the existence of symbolic dynamics in its dynamics by the use of CAPD C++ library.