### Spot Patterns in the 2-D Schnakenberg Model with Localized Heterogeneities

A hybrid asymptotic-numerical theory is developed to analyze the effect of different types of localized heterogeneities on the existence, linear stability, and slow dynamics of localized spot patterns for the two-component Schnakenberg reaction-diffusion model in a 2-D domain. Two distinct types of localized heterogeneities are considered: a strong localized perturbation of a spatially uniform feed rate and the effect of removing a small hole in the domain, through which the chemical species can leak out. Our hybrid theory reveals a wide range of novel phenomena such as, saddle-node bifurcations for quasi-equilibrium spot patterns that otherwise would not occur for a homogeneous medium, a new type of spot solution pinned at the concentration point of the feed rate, spot self-replication behavior leading to the creation of more than two new spots, and the existence of a creation-annihilation attractor with at most three spots. Depending on the type of localized heterogeneity introduced, localized spots are either repelled or attracted towards the localized defect on asymptotically long time scales. Results for slow spot dynamics and detailed predictions of various instabilities of quasi-equilibrium spot patterns, all based on our hybrid asymptotic-numerical theory, are illustrated and confirmed through extensive full PDE numerical simulations.

### Instability in large bounded domains -- branched versus unbranched resonances

We study instabilities in large bounded domains for prototypical model problems in the presence of transport and negative nonlinear feedback. In the most common scenario, bifurcation diagrams are steep, with a jump to finite amplitude at the transition from convective to absolute instability. We identify and analyze a generic alternate scenario, where this transition is smooth, with a gradual increase. Transitions in both cases are mediated by propagating fronts whose speed is determined by the linear dispersion relation. In the most common scenario, front speeds are determined by branched resonances, while in an alternate scenario studied here, front speeds are determined by unbranched resonances and typically involve long-range interaction with boundaries.

### Vortex filaments on arrays of coupled oscillators in the regime of nonlinear resonance

Numerical simulations support the possibility of long-time existence of vortex structures in the form of quantized filaments on arrays of coupled, weakly dissipative nonlinear oscillators within a three-dimensional domain, under a resonance external driving applied at the domain boundary. Qualitatively established are the ranges of parameters characterizing the system and the external signal, which are favorable for creating a modulation-stable quasi-uniform energy background, a crucial factor for realization of this phenomenon.

### Uncertainty Quantification of Multi-Scale Resilience in Nonlinear Complex Networks using Arbitrary Polynomial Chaos

In an increasing connected world, resilience is an important ability for a system to retain its original function when perturbations happen. Even though we understand small-scale resilience well, our understanding of large-scale networked resilience is limited. Recent research in network-level resilience and node-level resilience pattern has advanced our understanding of the relationship between topology and dynamics across network scales. However, the effect of uncertainty in a large-scale networked system is not clear, especially when uncertainties cascade between connected nodes. In order to quantify resilience uncertainty across the network resolutions (macro to micro), we develop an arbitrary polynomial chaos (aPC) expansion method to estimate the resilience subject to parameter uncertainties with arbitrary distributions. For the first time and of particular importance, is our ability to identify the probability of a node in losing its resilience and how the different model parameters contribute to this risk. We test this using a generic networked bi-stable system and this will aid practitioners to both understand macro-scale behaviour and make micro-scale interventions.

### Instantaneous phase synchronization of two decoupled quantum limit-cycle oscillators induced by conditional photon detection

We show that conditional photon detection induces instantaneous phase synchronization between two decoupled quantum limit-cycle oscillators. We consider two quantum van der Pol oscillators without mutual coupling, each with an additional linearly coupled bath, and perform continuous measurement of photon counting on the output fields of the two baths interacting through a beam splitter. It is observed that in-phase or anti-phase coherence of the two decoupled oscillators instantaneously increases after the photon detection and then decreases gradually in the weak quantum regime or quickly in the strong quantum regime until the next photon detection occurs. In the strong quantum regime, quantum entanglement also increases after the photon detection and quickly disappears. We derive the analytical upper bounds for the increases in the quantum entanglement and phase coherence by the conditional photon detection in the quantum limit.

### Genuine Nonlinearity and its Connection to the Modified Korteweg - de Vries Equation in Phase Dynamics

The study of hyperbolic waves involves various notions which help characterise how these structures evolve. One important facet is the notion of \emph{genuine nonlinearity}, namely the ability for shocks and rarefactions to form instead of contact discontinuities. In the context of the Whitham Modulation equations, this paper demonstrate that a loss of genuine nonlinearity leads to the appearance of a dispersive set of dynamics in the form of the modified Korteweg de-Vries equation governing the evolution of the waves instead. Its form is universal in the sense that its coefficients can be written entirely using linear properties of the underlying waves such as the conservation laws and linear dispersion relation. This insight is applied to two systems of physical interest, one an optical model and the other a stratified hydrodynamics experiment, to demonstrate how it can be used to provide insight into how waves in these systems evolve when genuine nonlinearity is lost.

### Improving Delay Based Reservoir Computing via Eigenvalue Analysis

We analyze the reservoir computation capability of the Lang-Kobayashi system by comparing the numerically computed recall capabilities and the eigenvalue spectrum. We show that these two quantities are deeply connected, and thus the reservoir computing performance is predictable by analyzing the eigenvalue spectrum. Our results suggest that any dynamical system used as a reservoir can be analyzed in this way as long as the reservoir perturbations are sufficiently small. Optimal performance is found for a system with the eigenvalues having real parts close to zero and off-resonant imaginary parts.

### Bump attractors and waves in networks of leaky integrate-and-fire neurons

Bump attractors are wandering localised patterns observed in in vivo experiments of spatially-extended neurobiological networks. They are important for the brain's navigational system and specific memory tasks. A bump attractor is characterised by a core in which neurons fire frequently, while those away from the core do not fire. We uncover a relationship between bump attractors and travelling waves in a classical network of excitable, leaky integrate-and-fire neurons. This relationship bears strong similarities to the one between complex spatiotemporal patterns and waves at the onset of pipe turbulence. Waves in the spiking network are determined by a firing set, that is, the collection of times at which neurons reach a threshold and fire as the wave propagates. We define and study analytical properties of the voltage mapping, an operator transforming a solution's firing set into its spatiotemporal profile. This operator allows us to construct localised travelling waves with an arbitrary number of spikes at the core, and to study their linear stability. A homogeneous "laminar" state exists in the network, and it is linearly stable for all values of the principal control parameter. Sufficiently wide disturbances to the homogeneous state elicit the bump attractor. We show that one can construct waves with a seemingly arbitrary number of spikes at the core; the higher the number of spikes, the slower the wave, and the more its profile resembles a stationary bump. As in the fluid-dynamical analogy, such waves coexist with the homogeneous state, are unstable, and the solution branches to which they belong are disconnected from the laminar state; we provide evidence that the dynamics of the bump attractor displays echoes of the unstable waves, which form its building blocks.

### Classical dynamical density functional theory: from fundamentals to applications

Classical dynamical density functional theory (DDFT) is one of the cornerstones of modern statistical mechanics. It is an extension of the highly successful method of classical density functional theory (DFT) to nonequilibrium systems. Originally developed for the treatment of simple and complex fluids, DDFT is now applied in fields as diverse as hydrodynamics, materials science, chemistry, biology, and plasma physics. In this review, we give a broad overview over classical DDFT. We explain its theoretical foundations and the ways in which it can be derived. The relations between the different forms of deterministic and stochastic DDFT as well as between DDFT and related theories, such as quantum-mechanical time-dependent DFT, mode coupling theory, and phase field crystal models, are clarified. Moreover, we discuss the wide spectrum of extensions of DDFT, which covers methods with additional order parameters (like extended DDFT), exact approaches (like power functional theory), and systems with more complex dynamics (like active matter). Finally, the large variety of applications, ranging from fluid mechanics and polymer physics to solidification, pattern formation, biophysics, and electrochemistry, is presented.

### A time-dependent energy-momentum method

We devise a generalisation of the energy momentum-method for studying the stability of non-autonomous Hamiltonian systems with a Lie group of Hamiltonian symmetries. A generalisation of the relative equilibrium point notion to a non-autonomous realm is provided and studied. Relative equilibrium points of non-autonomous Hamiltonian systems are described via foliated Lie systems, which opens a new field of application of such differential equations. We reduce non-autonomous Hamiltonian systems via the Marsden-Weinstein theorem and we provide conditions ensuring the stability of the projection of relative equilibrium points to the reduced space. As an application, we study the stability of relative equilibrium points for a class of mechanical systems, which covers rigid bodies as a particular instance.

### Anomalous heat transport in classical many-body systems: overview and perspectives

In this review paper we aim at illustrating recent achievements in anomalous heat diffusion, while highlighting open problems and research perspectives. We briefly recall the main features of the phenomenon for low-dimensional classical anharmonic chains and outline some recent developments on perturbed integrable systems, and on the effect of long-range forces and magnetic fields. Some selected applications to heat transfer in material science at the nanoscale are described. In the second part, we discuss of the role of anomalous conduction on coupled transport and describe how systems with anomalous (thermal) diffusion allow a much better power-efficiency trade-off for the conversion of thermal to particle current.

### Temperature Dependent Non-linear Damping in Palladium Nano-mechanical Resonators

Advances in nano-fabrication techniques has made it feasible to observe damping phenomena beyond the linear regime in nano-mechanical systems. In this work, we report cubic non-linear damping in palladium nano-mechanical resonators. Nano-scale palladium beams exposed to a $H_2$ atmosphere become softer and display enhanced Duffing non-linearity as well as non-linear damping at ultra low temperatures. The damping is highest at the lowest temperatures of $\sim 110\: mK$ and decreases when warmed up-to $\sim 1\textrm{ }K$. We experimentally demonstrate for the first time a temperature dependent non-linear damping in a nano-mechanical system below 1 K. It is consistent with a predicted two phonon mediated non-linear Akhiezer scenario for ballistic phonons with mean free path comparable to the beam thickness. This opens up new possibilities to engineer non-linear phenomena at low temperatures.

### Not sure? Handling hesitancy of COVID-19 vaccines

From the moment the first COVID-19 vaccines are rolled out, there will need to be a large fraction of the global population ready in line. It is therefore crucial to start managing the growing global hesitancy to any such COVID-19 vaccine. The current approach of trying to convince the "no"s cannot work quickly enough, nor can the current policy of trying to find, remove and/or rebut all the individual pieces of COVID and vaccine misinformation. Instead, we show how this can be done in a simpler way by moving away from chasing misinformation content and focusing instead on managing the "yes--no--not-sure" hesitancy ecosystem.