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Daniel Wilczak

Publications and source records attributed to Daniel Wilczak.

14 recordsLinked to original sources
arXiv2025

A mechanism for growth of topological entropy and global changes of the shape of chaotic attractors

The theoretical and numerical understanding of the key concept of topological entropy is an important problem in dynamical systems. Most studies have been carried out on maps (discrete-time systems). We analyse a scenario of global changes of the structure of an attractor in continuous-time systems leading to an unbounded growth of the topological entropy of the underlying dynamical system. As an example, we consider the classical Roessler system. We show that for an explicit range of parameters a chaotic attractor exists. We also prove the existence of a sequence of bifurcations leading to the growth of the topological entropy. The proofs are computer-aided.

math.DS↗
arXiv2025

Symbolic dynamics for the Kuramoto-Sivashinsky PDE on the line II

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.

math.DS↗
arXiv2025

Self-consistent bounds method for dissipative PDEs

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.

math.AP↗
arXiv2024

Make Interval Bound Propagation great again

In various scenarios motivated by real life, such as medical data analysis, autonomous driving, and adversarial training, we are interested in robust deep networks. A network is robust when a relatively small perturbation of the input cannot lead to drastic changes in output (like change of class, etc.). This falls under the broader scope field of Neural Network Certification (NNC). Two crucial problems in NNC are of profound interest to the scientific community: how to calculate the robustness of a given pre-trained network and how to construct robust networks. The common approach to constructing robust networks is Interval Bound Propagation (IBP). This paper demonstrates that IBP is sub-optimal in the first case due to its susceptibility to the wrapping effect. Even for linear activation, IBP gives strongly sub-optimal bounds. Consequently, one should use strategies immune to the wrapping effect to obtain bounds close to optimal ones. We adapt two classical approaches dedicated to strict computations -- Dubleton Arithmetic and Affine Arithmetic -- to mitigate the wrapping effect in neural networks. These techniques yield precise results for networks with linear activation functions, thus resisting the wrapping effect. As a result, we achieve bounds significantly closer to the optimal level than IBPs.

cs.LG↗
arXiv2024

Continuation and bifurcations of periodic orbits and symbolic dynamics in the Swift-Hohenberg equation

Steady states of the Swift--Hohenberg equation are studied. For the associated four--dimensional ODE we prove that on the energy level $E=0$ two smooth branches of even periodic solutions are created through the saddle-node bifurcation. We also show that these orbits satisfy certain geometric properties, which implies that the system has positive topological entropy for an explicit and wide range of parameter values of the system. The proof is computer-assisted and it uses rigorous computation of bounds on certain Poincaré map and its higher order derivatives.

math.DS↗
arXiv2021

Recent advances in rigorous computation of Poincaré maps

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.

math.NA↗
arXiv2019

Validated numerics for period-tupling and touch-and-go bifurcations of symmetric periodic orbits in reversible systems

We propose a general framework for computer-assisted verification of the presence of symmetry breaking, period-tupling and touch-and-go bifurcations of symmetric periodic orbits for reversible maps. The framework is then adopted to Poincaré maps in reversible autonomous Hamiltonian systems. In order to justify the applicability of the method, we study bifurcations of halo orbits in the Circular Restricted Three Body Problem. We give a computer-assisted proof of the existence of wide branches of halo orbits bifurcating from $L_{1,2,3}$-Lyapunov families and for wide range of mass parameter. For two physically relevant mass parameters we prove, that halo orbits undergo multiple period doubling, quadrupling and third-order touch-and-go bifurcations.

math.DS↗
arXiv2015

An implicit algorithm for validated enclosures of the solutions to variational equations for ODEs

We propose a new algorithm for computing validated bounds for the solutions to the first order variational equations associated to ODEs. These validated solutions are the kernel of numerics computer-assisted proofs in dynamical systems literature. The method uses a high-order Taylor method as a predictor step and an implicit method based on the Hermite-Obreshkov interpolation as a corrector step. The proposed algorithm is an improvement of the $C^1$-Lohner algorithm proposed by Zgliczy\'nski and it provides sharper bounds. As an application of the algorithm, we give a computer-assisted proof of the existence of an attractor set in the R\"ossler system, and we show that the attractor contains an invariant and uniformly hyperbolic subset on which the dynamics is chaotic, that is, conjugated to subshift of finite type with positive topological entropy.

math.NA↗
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↗
arXiv2007

Period doubling in the Rossler system - a computer assisted proof

The goal of this paper is to show how to produce a piece of rigorous bifurcation diagram of periodic orbits for an ODE. We study the Rossler system, one of the textbook examples of ODEs generating nontrivial dynamics, for the parameter range containing two period doubling bifurcations.

math.DS↗