We explore pattern formation in an active fluid system involving two chemical species that regulate active stress: a fast-diffusing species ($A$) and a slow-diffusing species ($I$). The growth of species $A$ is modelled using a nonlinear logistic term. Through linear stability analysis, we derive phase diagrams illustrating the various dynamical regimes in parameter space. Our findings indicate that an increase in the P\'eclet number results in the destabilisation of the uniform steady state. In contrast, counter-intuitively, an increase in the nonlinear growth parameter of $A$ actually stabilises the homogeneous steady-state regime. Additionally, we observe that greater asymmetry between the species leads to three distinct dynamical phases, while low asymmetry fails to produce oscillatory instability. Numerical simulations conducted in instability regimes show patterns that range from irregular, arrhythmic configurations at high P\'eclet numbers to both transient and robust symmetry-breaking chimera states. Notably, these chimera patterns are more prevalent in the oscillatory instability regime, and our stability analysis indicates that this regime is the most extensive for high nonlinear growth parameters and moderately high P\'eclet numbers. Further, we also find soliton-like structures where aggregations of species $A$ merge, and new aggregations spontaneously emerge, and these patterns are prevalent in the phase of stationary instability. Overall, our study illustrates that a diverse array of patterns can emerge in active matter influenced by nonlinear growth in a chemical species, with chimeras being particularly dominant when the nonlinear growth parameter is elevated.

We consider an interacting particle system on star graphs. As in the case of the Kdv equation, we have infinitely many invariants ( martingale invariants). It enables us to obtain the limiting distribution of the Markov chain. Each of the martingale invariants is a homogeneous polynomial with coefficients of Narayana numbers.The identity for the enumeration of plane unlabeled trees, which gives Narayana numbers, becomes the key identity to obtain the probability of death states by a change of variables.

Analytical results are presented for the structure of networks that evolve via a preferential-attachment-random-deletion (PARD) model in the regime of overall network growth and in the regime of overall contraction. The phase transition between the two regimes is studied. At each time step a node addition and preferential attachment step takes place with probability $P_{\rm add}$, and a random node deletion step takes place with probability $P_{\rm del} = 1 - P_{\rm add}$. The balance between growth and contraction is captured by the parameter $\eta = P_{\rm add} - P_{\rm del}$, which in the regime of overall network growth satisfies $0 < \eta \le 1$ and in the regime of overall network contraction $-1 \le \eta < 0$. Using the master equation and computer simulations we show that for $-1 < \eta < 0$ the time-dependent degree distribution $P_t(k)$ converges towards a stationary form $P_{\rm st}(k)$ which exhibits an exponential tail. This is in contrast with the power-law tail of the stationary degree distribution obtained for $0 < \eta \le 1$. Thus, the PARD model has a phase transition at $\eta=0$, which separates between two structurally distinct phases. At the transition, for $\eta=0$, the degree distribution exhibits a stretched exponential tail. While the stationary degree distribution in the phase of overall growth represents an asymptotic state, in the phase of overall contraction $P_{\rm st}(k)$ represents an intermediate asymptotic state of a finite life span, which disappears when the network vanishes.

Recently, there has been an increasing interest in employing dynamical systems as solvers of NP-complete problems. In this paper, we present accurate implementations of two continuous-time dynamical solvers, known in the literature as analog SAT and digital memcomputing, using advanced numerical integration algorithms of SPICE circuit simulators. For this purpose, we have developed Python scripts that convert Boolean satisfiability (SAT) problems into electronic circuits representing the analog SAT and digital memcomputing dynamical systems. Our Python scripts process conjunctive normal form (CNF) files and create netlists that can be directly imported into LTspice. We explore the SPICE implementations of analog SAT and digital memcomputing solvers by applying these to a selected set of problems and present some interesting and potentially useful findings related to digital memcomputing and analog SAT.

We analyzed mathematically a model describing flux jumps in superconductivity in a 1D configuration. Three effects occur from fastest to slowest: Joule heating, magnetic relaxation and temperature diffusion. Adimensionalising the equations showed that magnetic field fronts penetrate the material as inhomogeneous Burgers fronts. An additional global term pushes the magnetic field and is responsible for flux jumps. We considered a medium temperature for which the heat capacity of a sample can be taken as a constant and a low temperature where heat capacity depends on temperature causing a nonlinear temperature evolution. As expected, we found that flux jumps occur mostly at low temperature. To understand flux trapping, we examined external magnetic field pulses of different amplitudes and duration. We found that flux trapping is maximal for medium amplitudes and low temperatures.

Lagrangian multiform theory is a variational framework for integrable systems. In this article we introduce a new formulation which is based on symplectic geometry and which treats position, momentum and time coordinates of a finite-dimensional integrable hierarchy on an equal footing. This formulation allows a streamlined one-step derivation of both the multi-time Euler-Lagrange equations and the closure relation (encoding integrability). We argue that any Lagrangian one-form for a finite-dimensional system can be recast in our new framework. We also show that our new framework easily extends to non-commuting flows corresponding to nonabelian Lie groups. Thus Hamiltonian Lie group actions can be derived from a variational principle.

Networks model the architecture backbone of complex systems. The backbone itself can change over time leading to what is called `temporal networks'. Interpreting temporal networks as trajectories in graph space of a latent graph dynamics has recently enabled the extension of concepts and tools from dynamical systems and time series to networks. Here we address temporal networks with unlabelled nodes, a case that has received relatively little attention so far. Situations in which node labelling cannot be tracked over time often emerge in practice due to technical challenges, or privacy constraints. In unlabelled temporal networks there is no one-to-one matching between a network snapshot and its adjacency matrix. Characterizing the dynamical properties of such unlabelled network trajectories is nontrivial. We here exploit graph invariants to extend to the unlabelled setting network-dynamical quantifiers of linear correlations and dynamical instability. In particular, we focus on autocorrelation functions and the sensitive dependence on initial conditions. We show with synthetic graph dynamics that the measures are capable of recovering and estimating these dynamical fingerprints even when node labels are unavailable. We also validate the methods for some empirical temporal networks with removed node labels.

We explore the relation between a classical periodic Hamiltonian system and an associated discrete quantum system on a torus in phase space. The model is a sinusoidally perturbed Harper model and is similar to the sinusoidally perturbed pendulum. Separatrices connecting hyperbolic fixed points in the unperturbed classical system become chaotic under sinusoidal perturbation. We numerically compute eigenstates of the Floquet propagator for the associated quantum system. For each Floquet eigenstate we compute a Husimi distribution in phase space and an energy and energy dispersion from the expectation value of the unperturbed Hamiltonian operator. The Husimi distribution of each Floquet eigenstate resembles a classical orbit with a similar energy and similar energy dispersion. Chaotic orbits in the classical system are related to Floquet eigenstates that appear ergodic. For a hybrid regular and chaotic system, we use the energy dispersion to separate the Floquet eigenstates into ergodic and integrable subspaces. The distribution of quasi-energies in the ergodic subspace resembles that of a random matrix model. The width of a chaotic region in the classical system is estimated by integrating the perturbation along a separatrix orbit. We derive a related expression for the associated quantum system from the averaged perturbation in the interaction representation evaluated at states with energy close to the separatrix.