CyberSec Research

The understanding of non-linear effects in circular storage rings and colliders based on superconducting magnets is a key issue for the luminosity the beam lifetime optimisation. A detailed analysis of the multidimensional phase space requires a large computing effort when many variants of the magnetic lattice, representing the realisation of magnetic errors or configurations for performance optimisation, have to be considered. Dynamic indicators for chaos detection have proven to be very effective in finding and distinguishing the weakly-chaotic regions of phase space where diffusion takes place and regions that remain stable over time scales in the order of multiple hours of continuous operation. This paper explores the use of advanced chaos indicators, including the Fast Lyapunov Indicator with Birkhoff weights and the Reverse Error Method, in realistic lattice models for the CERN Large Hadron Collider (LHC). Their convergence, predictive power, and potential to define a magnetic lattice quality factor linked to long-term dynamic aperture are assessed. The results demonstrate the efficiency of these indicators in identifying chaotic dynamics, offering valuable insights of these chaos indicators for optimising accelerator lattices with reduced computational cost compared to the classical approach based on CPU-demanding long-term tracking campaigns.

Reservoir computing can embed attractors into random neural networks (RNNs), generating a ``mirror'' of a target attractor because of its inherent symmetrical constraints. In these RNNs, we report that an attractor-merging crisis accompanied by intermittency emerges simply by adjusting the global parameter. We further reveal its underlying mechanism through a detailed analysis of the phase-space structure and demonstrate that this bifurcation scenario is intrinsic to a general class of RNNs, independent of training data.

Multifunctionality is ubiquitous in biological neurons. Several studies have translated the concept to artificial neural networks as well. Recently, multifunctionality in reservoir computing (RC) has gained the widespread attention of researchers. Multistable dynamics of the reservoir can be configured to capture multiple tasks, each by one of the co-existing attractors. However, there are several limitations in the applicability of this approach. So far, multifunctional RC has been shown to be able to reconstruct different attractor climates only when the attractors are well separated in the phase space. We propose a more flexible reservoir computing scheme capable of multifunctioning beyond the earlier limitations. The proposed architecture holds striking similarity with the multifunctional biological neural networks and showcases superior performance. It is capable of learning multiple chaotic attractors with overlapping phase space. We successfully train the RC to achieve multifunctionality with wide range of tasks.

Tipping behaviour can occur when an equilibrium loses stability in response to a slowly varying parameter crossing a bifurcation threshold, or where noise drives a system from one attractor to another, or some combination of these effects. Similar behaviour can be expected when a multistable system is forced by a chaotic deterministic system rather than by noise. In this context, the chaotic tipping window was recently introduced and investigated for discrete-time dynamics. In this paper, we find tipping windows for continuous-time nonlinear systems forced by chaos. We characterise the tipping window in terms of forcing by unstable periodic orbits of the chaos, and we show how the location and structure of this window depend on the relative timescales between the forcing and the responding system. We illustrate this by finding tipping windows for two examples of coupled bistable ODEs forced with chaos. Additionally, we describe the dynamic tipping window in the setting of a changing system parameter.

The epidemiological dynamics of Mycoplasma pneumoniae are characterized by complex and poorly understood multiannual cycles, posing challenges for forecasting. Using Bayesian methods to fit a seasonally forced transmission model to long-term surveillance data from Denmark (1958-1995, 2010-2025), we investigate the mechanisms driving recurrent outbreaks of M. pneumoniae. The period of the multiannual cycles (predominantly approx. 5 years in Denmark) are explained as a consequence of the interaction of two time-scales in the system, one intrinsic and one extrinsic (seasonal). While it provides an excellent fit to shorter time series (a few decades), we find that the deterministic model eventually settles into an annual cycle, failing to reproduce the observed 4-5-year periodicity long-term. Upon further analysis, the system is found to exhibit transient chaos and thus high sensitivity to stochasticity. We show that environmental (but not purely demographic) stochasticity can sustain the multi-year cycles via stochastic resonance. The disruptive effects of COVID-19 non-pharmaceutical interventions (NPIs) on M. pneumoniae circulation constitute a natural experiment on the effects of large perturbations. Consequently, the effects of NPIs are included in the model and medium-term predictions are explored. Our findings highlight the intrinsic sensitivity of M. pneumoniae dynamics to perturbations and interventions, underscoring the limitations of deterministic epidemic models for long-term prediction. More generally, our results emphasize the potential role of stochasticity as a driver of complex cycles across endemic and recurring pathogens.

The bacteria metabolic process of open nonlinear dissipative system far from equilibrium point is modeled using classical methods of synergetics. The invariant measure and its convergence in the phase space of the system was obtained in strange attractor mode. The distribution of point density of trajectory intersection of phase space cells with maximum invariant measure and convergence in time of its average value was obtained. The result concluded is that the value of an invariant measure can be a characteristic of the transitional process of adaptation of cell metabolic process to change outside environment.

We present novel Hamiltonian descriptions of some three-dimensional systems including two well-known systems describing the three-wave-interaction problem and some well-known chaotic systems, namely, the Chen, L\"u, and Qi systems. We show that all of these systems can be described in a Hamiltonian framework in which the Poisson matrix $\mathcal{J}$ is supplemented by a resistance matrix $\mathcal{R}$. While such resistive-Hamiltonian systems are manifestly non-conservative, we construct higher-degree Poisson matrices via the Jordan product as $\mathcal{N} = \mathcal{J} \mathcal{R} + \mathcal{R} \mathcal{J}$, thereby leading to new bi-Hamiltonian systems. Finally, we discuss conformal Hamiltonian dynamics on Poisson manifolds and demonstrate that by appropriately choosing the underlying parameters, the reduced three-wave-interaction model as well as the Chen and L\"u systems can be described in this manner where the concomitant non-conservative part of the dynamics is described with the aid of the Euler vector field.

The eigenstate thermalization hypothesis (ETH) underpins much of our modern understanding of the thermalization of closed quantum many-body systems. Here, we investigate the statistical properties of observables in the eigenbasis of the Lindbladian operator of a Markovian open quantum system. We demonstrate the validity of a Lindbladian ETH ansatz through extensive numerical simulations of several physical models. To highlight the robustness of Lindbladian ETH, we consider what we dub the dilute-click regime of the model, in which one postselects only quantum trajectories with a finite fraction of quantum jumps. The average dynamics are generated by a non-trace-preserving Liouvillian, and we show that the Lindbladian ETH ansatz still holds in this case. On the other hand, the no-click limit is a singular point at which the Lindbladian reduces to a doubled non-Hermitian Hamiltonian and Lindbladian ETH breaks down.

Ising machines show promise as ultrafast hardware for optimizations encoded in Ising Hamiltonians but fall short in terms of success rate and performance scaling. Here, we propose a novel Ising machine that exploits the three-dimensional nature of nonlinear polarization oscillators to counteract these limitations. Based on the evolution of the optical polarization in third-order nonlinear media, the high-dimensional machine reaches the Ising ground state by the mechanism of dimensional collapse: the dynamics on the Poincar\'e sphere undergoes a self-induced collapse into polarization fixed points mapping Ising spins. The photonic setup consists of polarization-modulated pulses in a $\chi^{(3)}$ crystal subject to iterative feedback. We numerically demonstrate that its high-dimensional operation leads to an enhanced success probability on benchmark graphs and an exponential improvement in performance scaling with respect to coherent Ising machines. The proposed polarization Ising machine paves the way for a new class of Ising solvers with enhanced computing capabilities.

There is general agreement that fostering trust and cooperation within the AI development ecosystem is essential to promote the adoption of trustworthy AI systems. By embedding Large Language Model (LLM) agents within an evolutionary game-theoretic framework, this paper investigates the complex interplay between AI developers, regulators and users, modelling their strategic choices under different regulatory scenarios. Evolutionary game theory (EGT) is used to quantitatively model the dilemmas faced by each actor, and LLMs provide additional degrees of complexity and nuances and enable repeated games and incorporation of personality traits. Our research identifies emerging behaviours of strategic AI agents, which tend to adopt more "pessimistic" (not trusting and defective) stances than pure game-theoretic agents. We observe that, in case of full trust by users, incentives are effective to promote effective regulation; however, conditional trust may deteriorate the "social pact". Establishing a virtuous feedback between users' trust and regulators' reputation thus appears to be key to nudge developers towards creating safe AI. However, the level at which this trust emerges may depend on the specific LLM used for testing. Our results thus provide guidance for AI regulation systems, and help predict the outcome of strategic LLM agents, should they be used to aid regulation itself.

The chaotic dynamics of small-scale vorticity plays a key role in understanding and controlling turbulence, with direct implications for energy transfer, mixing, and coherent structure evolution. However, measuring or controlling its dynamics remains a major conceptual and experimental challenge due to its transient and chaotic nature. Here we use a combination of experiments, theory and simulations to show that small magnetic particles of different densities, exploring flow regions of distinct vorticity statistics, can act as effective probes for measuring and forcing turbulence at its smallest scale. The interplay between the magnetic torque, from an externally controllable magnetic field, and hydrodynamic stresses, from small-scale turbulent vorticity, reveals an extremely rich phenomenology. Notably, we present the first observation of stochastic resonance for particles in turbulence: turbulent fluctuations, effectively acting as noise, counterintuitively enhance the particle rotational response to external forcing. We identify a pronounced resonant peak in particle rotational phase-lag when the applied magnetic field matches the characteristic intensity of small-scale vortices. Furthermore, we uncover a novel symmetry-breaking mechanism: an oscillating magnetic field with zero-mean angular velocity remarkably induces net particle rotation in turbulence with zero-mean vorticity, as turbulent fluctuations aid the particle in "surfing" the magnetic field. Our findings offer insights into flexible particle manipulation in complex flows and open up a novel magnetic resonance-based approach for measuring vorticity: magnetic particles as probes emit detectable magnetic fields, enabling turbulence quantification even under optically-inaccessible conditions.

A network of coupled time-varying systems, where individual nodes are interconnected through links, is a modeling framework widely used by many disciplines. For identical nodes displaying a complex behavior known as chaos, clusters of nodes or the entire network can synchronize for a range of coupling strengths. Here, we demonstrate that small differences in the nodes give rise to desynchronization events, known as bubbling, in regimes where synchronization is expected. Thus, small unit heterogeneity in all real systems has an unexpected and outsized effect on the network dynamics. We present a theoretical analysis of bubbling in chaotic oscillator networks and predict when bubble-free behavior is expected. Our work demonstrates that the domain of network synchronization is much smaller than expected and is replaced by epochs of synchronization interspersed with extreme events. Our findings have important implications for real-world systems where synchronized behavior is crucial for system functionality.

Predicting future observations plays a central role in machine learning, biology, economics, and many other fields. It lies at the heart of organizational principles such as the variational free energy principle and has even been shown -- based on the second law of thermodynamics -- to be necessary for reaching the fundamental energetic limits of sequential information processing. While the usefulness of the predictive paradigm is undisputed, complex adaptive systems that interact with their environment are more than just predictive machines: they have the power to act upon their environment and cause change. In this work, we develop a framework to analyze the thermodynamics of information processing in percept-action loops -- a model of agent-environment interaction -- allowing us to investigate the thermodynamic implications of actions and percepts on equal footing. To this end, we introduce the concept of work capacity -- the maximum rate at which an agent can expect to extract work from its environment. Our results reveal that neither of two previously established design principles for work-efficient agents -- maximizing predictive power and forgetting past actions -- remains optimal in environments where actions have observable consequences. Instead, a trade-off emerges: work-efficient agents must balance prediction and forgetting, as remembering past actions can reduce the available free energy. This highlights a fundamental departure from the thermodynamics of passive observation, suggesting that prediction and energy efficiency may be at odds in active learning systems.

We explore the concept of scaling invariance in a type of dynamical systems that undergo a transition from order (regularity) to disorder (chaos). The systems are described by a two-dimensional, nonlinear mapping that preserves the area in the phase space. The key variables are the action and the angle, as usual from Hamiltonian systems. The transition is influenced by a control parameter giving the form of the order parameter. We observe a scaling invariance in the average squared action within the chaotic region, providing evidence that this change from regularity (integrability) to chaos (non-integrability) is akin to a second-order or continuous phase transition. As the order parameter approaches zero, its response against the variation of the control parameter (susceptibility) becomes increasingly pronounced (indeed diverging), resembling a phase transition. These findings could not be obtained without a seminal paper on Phys. Rev. Lett. {\bf 2004}, {\em 93}, 014101.

A type of chaos called laminar chaos was found in singularly perturbed dynamical systems with periodically [Phys. Rev. Lett. 120, 084102 (2018)] and quasiperiodically [Phys. Rev. E 107, 014205 (2023)] time-varying delay. Compared to high-dimensional turbulent chaos that is typically found in such systems with large constant delay, laminar chaos is a very low-dimensional phenomenon. It is characterized by a time series with nearly constant laminar phases that are interrupted by irregular bursts, where the intensity level of the laminar phases varies chaotically from phase to phase. In this paper, we demonstrate that laminar chaos, and its generalizations, can also be observed in systems with random and chaotically time-varying delay. Moreover, while for periodic and quasiperiodic delays the appearance of (generalized) laminar chaos and turbulent chaos depends in a fractal manner on the delay parameters, it turns out that short-time correlated random and chaotic delays lead to (generalized) laminar chaos in almost the whole delay parameter space, where the properties of circle maps with quenched disorder play a crucial role. It follows that introducing such a delay variation typically leads to a drastic reduction of the dimension of the chaotic attractor of the considered systems. We investigate the dynamical properties and generalize the known methods for detecting laminar chaos in experimental time series to random and chaotically time-varying delay.

Vertical thermal convection system exhibits weak turbulence and spatio-temporally chaotic behavior. In this system, we report seven equilibria and 26 periodic orbits, all new and linearly unstable. These orbits, together with four previously studied in Zheng et al. (2024) bring the number of periodic orbit branches computed so far to 30, all solutions to the fully non-linear three-dimensional Navier-Stokes equations. These new invariant solutions capture intricate flow patterns including straight, oblique, wavy, skewed and distorted convection rolls, as well as bursts and defects in rolls. Most of the solution branches show rich spatial and/or spatio-temporal symmetries. The bifurcation-theoretic organisation of these solutions are discussed; the bifurcation scenarios include Hopf, pitchfork, saddle-node, period-doubling, period-halving, global homoclinic and heteroclinic bifurcations, as well as isolas. Given this large number of unstable orbits, our results may pave the way to quantitatively describing transitional fluid turbulence using periodic orbit theory.

Reservoir computers can be used to predict time series generated by spatio-temporal chaotic systems. Using multiple reservoirs in parallel has shown improved performances for these predictions, by effectively reducing the input dimensionality of each reservoir. Similarly, one may further reduce the dimensionality of the input data by transforming to a lower-dimensional latent space. Combining both approaches, we show that using dimensionality-reduced latent space predictions for parallel reservoir computing not only reduces computational costs, but also leads to better prediction results for small to medium reservoir sizes. This synergetic approach is illustrated and evaluated on the basis of the prediction of the one-dimensional Kuramoto-Sivashinsky equation.

The interstellar medium (ISM) of our Galaxy is magnetized, compressible and turbulent, influencing many key ISM properties, like star formation, cosmic ray transport, and metal and phase mixing. Yet, basic statistics describing compressible, magnetized turbulence remain uncertain. Utilizing grid resolutions up to $10,080^3$ cells, we simulate highly-compressible, magnetized ISM-style turbulence with a magnetic field maintained by a small-scale dynamo. We measure two coexisting kinetic energy cascades, $\mathcal{E}_{{\rm kin}}(k) \propto k^{-n}$, in the turbulence, separating the plasma into scales that are non-locally interacting, supersonic and weakly magnetized $(n=2.01\pm 0.03\approx 2)$ and locally interacting, subsonic and highly magnetized $(n=1.465\pm 0.002\approx 3/2)$, where $k$ is the wavenumber. We show that the $3/2$ spectrum can be explained with scale-dependent kinetic energy fluxes and velocity-magnetic field alignment. On the highly magnetized modes, the magnetic energy spectrum forms a local cascade $(n=1.798\pm 0.001\approx 9/5)$, deviating from any known \textit{ab initio} theory. With a new generation of radio telescopes coming online, these results provide a means to directly test if the ISM in our Galaxy is maintained by the compressible turbulent motions from within it.