This work analyzes bifurcation delay and front propagation in the one-dimensional real Ginzburg-Landau equation (RGLE) with periodic boundary conditions on monotonically growing or shrinking domains. First, we obtain closed-form expressions for the delay of primary bifurcations on a growing domain and show that the additional domain growth before the appearance of a pattern is independent of the growth time scale. We also quantify primary bifurcation delay on a shrinking domain; the time scale of domain compression is reflected in the additional compression before the pattern decays. For secondary bifurcations such as the Eckhaus instability, we obtain a lower bound on the delay of phase slips due to a time-dependent domain. We also construct a heuristic model to classify regimes with arrested phase slips, i.e. phase slips that fail to develop. Then, we study how fronts are influenced by a time-dependent domain. We derive expressions for the velocity and profile of homogeneous fronts on a time-dependent domain. We also derive the natural ``asymptotic'' velocity and front profile and show that these deviate from predictions based on the marginal stability criterion familiar from fixed domain theory. This difference arises because the time-dependence of the domain lifts the degeneracy of the spatial eigenvalues associated with speed selection and represents a fundamental distinction from the fixed domain theory that we verify using direct numerical simulations. The effect of a growing domain on pattern-spreading and Eckhaus front velocities is inspected qualitatively and found to be similar to that of homogeneous fronts. These more complex fronts can also experience delayed onset. Lastly, we show that dilution -- an effect present when the order parameter is conserved -- increases bifurcation delay and amplifies changes in the homogeneous front velocity on time-dependent domains.

Discrete integrable systems are closely related to orthogonal polynomials and isospectral matrix transformations. In this paper, we use these relationships to propose a nonautonomous time-discretization of the Camassa-Holm (CH) peakon equation, which describes the motion of peakon waves, which are soliton waves with sharp peaks. We then validate our time-discretization, and clarify its asymptotic behavior as the discrete-time goes to infinity. We present numerical examples to demonstrate that the proposed discrete equation captures peakon wave motions.

We consider the Riemann--Hilbert (RH) approach to the construction of periodic finite-band solutions to the focusing nonlinear Schr\"odinger (NLS) equation, addressing the question of how the RH problem parameters can be retrieved from the solution. Within the RH approach, a finite-band solution to the NLS equation is given in terms of the solution of an associated RH problem, the jump conditions for which are characterized by specifying the endpoints of the arcs defining the contour of the RH problem and the constants (so-called phases) involved in the jump matrices. In our work, we solve the problem of retrieving the phases given the solution of the NLS equation evaluated at a fixed time. Our findings are corroborated by numerical examples of phases computation, demonstrating the viability of the method proposed.

We elaborate a systematic way to obtain higher order contributions in the nonlinear steepest descent method for Riemann-Hilbert problem associated with homogeneous Painleve II equation. The problem is reformulated as a matrix factorization problem on two circles and can be solved perturbatively reducing it to finite systems of algebraic linear equations. The method is applied to find explicitly long-time asymptotic behaviour for tau function of Painleve II equation.

Cellular Automata (CA) have long been foundational in simulating dynamical systems computationally. With recent innovations, this model class has been brought into the realm of deep learning by parameterizing the CA's update rule using an artificial neural network, termed Neural Cellular Automata (NCA). This allows NCAs to be trained via gradient descent, enabling them to evolve into specific shapes, generate textures, and mimic behaviors such as swarming. However, a limitation of traditional NCAs is their inability to exhibit sufficiently complex behaviors, restricting their potential in creative and modeling tasks. Our research explores enhancing the NCA framework by incorporating multiple neighborhoods and introducing structured noise for seed states. This approach is inspired by techniques that have historically amplified the expressiveness of classical continuous CA. All code and example videos are publicly available on https://github.com/MagnusPetersen/MNNCA.

N-body systems characterized by inverse square attractive forces may display a self similar collapse known as the gravo-thermal catastrophe. In star clusters, collapse is halted by binary stars, and a large fraction of Milky Way clusters may have already reached this phase. It has been speculated -- with guidance from simulations -- that macroscopic variables such as central density and velocity dispersion are governed post-collapse by an effective, low-dimensional system of ODEs. It is still hard to distinguish chaotic, low dimensional motion, from high dimensional stochastic noise. Here we apply three machine learning tools to state-of-the-art dynamical simulations to constrain the post collapse dynamics: topological data analysis (TDA) on a lag embedding of the relevant time series, Sparse Identification of Nonlinear Dynamics (SINDY), and Tests of Accuracy with Random Points (TARP).

The geometric approach to nonequilibrium thermodynamics is promising for understanding the role of dissipation and thermodynamic trade-off relations. This paper proposes a geometric framework for studying the nonequilibrium thermodynamics of reaction-diffusion systems. Based on this framework, we obtain several decompositions of the entropy production rate with respect to conservativeness, spatial structure, and wavenumber. We also generalize optimal transport theory to reaction-diffusion systems and derive several trade-off relations, including thermodynamic speed limits and thermodynamic uncertainty relations. The thermodynamic trade-off relations obtained in this paper shed light on thermodynamic dissipation in pattern formation. We numerically demonstrate our results using the Fisher$\unicode{x2013}$Kolmogorov$\unicode{x2013}$Petrovsky$\unicode{x2013}$Piskunov equation and the Brusselator model and discuss how the spatial pattern affects unavoidable dissipation.

The clarion call for causal reduction in the study of capital markets is intensifying. However, in self-referencing and open systems such as capital markets, the idea of unidirectional causation (if applicable) may be limiting at best, and unstable or fallacious at worst. In this research, we critically assess the use of scientific deduction and causal inference within the study of empirical finance and econometrics. We then demonstrate the idea of competing causal chains using a toy model adapted from ecological predator/prey relationships. From this, we develop the alternative view that the study of empirical finance, and the risks contained therein, may be better appreciated once we admit that our current arsenal of quantitative finance tools may be limited to ex post causal inference under popular assumptions. Where these assumptions are challenged, for example in a recognizable reflexive context, the prescription of unidirectional causation proves deeply problematic.

Geometry-inspired measures (such as discrete Ricci curvatures) and topological data analysis (TDA) based methods (such as persistent homology) have become attractive tools for characterizing the higher-order structure of networks representing the financial systems. In this study, our goal is to perform a comparative analysis of both these approaches, especially by assessing the fragility and systemic risk in the Indian stock markets, which is known for its high volatility and risk. To achieve this goal, we analyze the time series of daily log-returns of stocks comprising the National Stock Exchange (NSE) and the Bombay Stock Exchange (BSE). Specifically, our aim is to monitor the changes in standard network measures, edge-centric discrete Ricci curvatures, and persistent homology based topological measures computed from cross-correlation matrices of stocks. In this study, the edge-centric discrete Ricci curvatures have been employed for the first time in the analysis of the Indian stock markets. The Indian stock markets are known to be less diverse in comparison to the US market, and hence provides us an interesting example. Our results point that, among the persistent homology based topological measures, persistent entropy is simple and more robust than $L^1$-norm and $L^2$-norm of persistence landscape. In a broader comparison between network analysis and TDA, we highlight that the network analysis is sensitive to the way of constructing the networks (threshold or minimum spanning tree), as well as the threshold values used to construct the correlation-based threshold networks. On the other hand, the persistent homology is a more robust approach and is able to capture the higher-order interactions and eliminate noisy data in financial systems, since it does not take into account a single value of threshold but rather a range of values.