Publications

Dynamics of a time-delayed relay system

Published in Physical Review E, 2024

We study the dynamics of a piecewise-linear second-order delay differential equation that is representative of feedback systems with relays (switches) that actuate after a fixed delay. The system under study exhibits strong multirhythmicity, the coexistence of many stable periodic solutions for the same values of the parameters. We present a detailed study of these periodic solutions and their bifurcations. Starting from an integro-differential model, we show how to reduce the system to a set of finite-dimensional maps. We then demonstrate that the parameter regions of existence of periodic solutions can be understood in terms of discontinuity induced bifurcations and their stability is determined by smooth bifurcations. Using this technique we are able to show that slowly oscillating solutions are always stable if they exist. We also demonstrate the coexistence of stable periodic solutions with quasiperiodic solutions.

Recommended citation: Lucas Illing, Pierce Ryan, and Andreas Amann. Dynamics of a time-delayed relay system. Phys. Rev. E , 109:014223, 2024. https://journals.aps.org/pre/abstract/10.1103/PhysRevE.109.014223

Nonlinear dynamics & stochastic processes in cybersecurity applications

Published in University College Cork, 2023

The Internet is an extremely complex system which has a significant impact on the world we live in. In this thesis, we formalise Internet-based problems as mathematical models to better understand their dynamics. Modelling these problems requires dynamical features such as time delay, periodic forcing, switching and stochasticity. We study several dynamical systems which employ a combination of these features from Internet applications, including targeted ransomware, data networks, and signal processing. We also study a climate science system which shares features with the signal processing system and exhibits similar dynamics. Stochasticity is found to be critical in the modelling of the negotiations involved in targeted ransomware, while time delay is a crucial feature in the modelling of data networks. The signal processing and climate science systems give rise to extremely rich dynamics, which we are able to study analytically due to the presence of switching. This yields further insights into related smooth systems.

Recommended citation: Pierce Ryan, "Nonlinear dynamics & stochastic processes in cybersecurity applications", University College Cork, 2023. https://cora.ucc.ie/items/d836b8ab-70b3-479b-a951-359480721c8a

Dynamics of Targeted Ransomware Negotiation

Published in IEEE Access, 2022

In this paper, we consider how the development of targeted ransomware has affected the dynamics of ransomware negotiations to better understand how to respond to ransomware attacks. We construct a model of ransomware negotiations as an asymmetric non-cooperative two-player game. In particular, our model considers the investments that a malicious actor must make in order to conduct a successful targeted ransomware attack. We demonstrate how imperfect information is a crucial feature for replicating observed real-world behaviour. Furthermore, we present optimal strategies for both the malicious actor and the target, and demonstrate how imperfect information results in a non-trivial optimal strategy for the malicious actor.

Recommended citation: P. Ryan, J. Fokker, S. Healy and A. Amann, Dynamics of targeted ransomware negotiation, IEEE Access, 10:3283632844, 2022 https://ieeexplore.ieee.org/document/9738625

Border-collision bifurcations in a driven time-delay system

Published in Chaos: An Interdisciplinary Journal of Nonlinear Science, 2020

We show that a simple piecewise-linear system with time delay and periodic forcing gives rise to a rich bifurcation structure of torus bifurcations and Arnold tongues, as well as multistability across a significant portion of the parameter space. The simplicity of our model enables us to study the dynamical features analytically. Specifically, these features are explained in terms of border collision bifurcations of an associated Poincaré map. Given that time delay and periodic forcing are common ingredients in mathematical models, this analysis provides widely applicable insight.

Recommended citation: Border-collision bifurcations in a driven time-delay system, Chaos: An Interdisciplinary Journal of Nonlinear Science, 30(2):023121, 2020. https://doi.org/10.1063/1.5119982