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Systems modeled by partial differential equations (PDEs) are at least as ubiquitous as systems that are by nature finite-dimensional and modeled by ordinary differential equations (ODEs). And yet, systematic and readily usable…

In this paper we introduce a formalism that allows to describe the response of a part of a biochemical system in terms of renewal equations. In particular, we examine under which conditions the interactions between the different parts of a…

The dynamics of systems biological processes are usually modeled by a system of ordinary differential equations (ODEs) with many unknown parameters that need to be inferred from noisy and sparse measurements. Here, we introduce…

Causal discovery is a data-driven paradigm for analyzing complex systems, while physics-based models, such as ordinary differential equations (ODEs), provide mechanistic structure for real-world dynamical processes. Integrating these…

This letter presents a high-dimensional analysis of the training dynamics for a single-layer nonlinear contrastive learning model. The empirical distribution of the model weights converges to a deterministic measure governed by a…

This work concerns the exponential stabilization of underactuated linear homogeneous systems of m parabolic partial differential equations (PDEs) in cascade (reaction-diffusion systems), where only the first state is controlled either…

Learning complex trajectories from demonstrations in robotic tasks has been effectively addressed through the utilization of Dynamical Systems (DS). State-of-the-art DS learning methods ensure stability of the generated trajectories;…

Fractional differential equations (FDEs) are an extension of the theory of fractional calculus. However, due to the difficulty in finding analytical solutions, there have not been extensive applications of FDEs until recent decades. With…

The interpretation of numerical methods, such as finite difference methods for differential equations, as point estimators suggests that formal uncertainty quantification can also be performed in this context. Competing statistical…

The multiple futile cycle is an important building block in networks of chemical reactions arising in molecular biology. A typical process which it describes is the addition of n phosphate groups to a protein. It can be modelled by a system…

In this work, we are motivated by a recent variant of the nonlinear Schrodinger (NLS) equation describing cold, dilute atomic condensates with quantum fluctuation effects. Our goal is to develop robust numerical methods capable of…

Equilibrium and nonequilibrium systems exhibit power-law singularities close to their critical and bifurcation points respectively. A recent study has shown that biochemical nonequilibrium models with positive feedback belong to the…

Computer algebra methods for analyzing reaction networks often rely on the assumption of mass-action kinetics, which transform the governing ODEs into polynomial systems amenable to techniques such as Gr\"obner basis computation and related…

Anticipating bifurcation-induced transitions in dynamical systems has gained relevance in various fields of the natural, social, and economic sciences. Before the annihilation of a system's equilibrium point by means of a bifurcation, the…

Ordinary differential equations (ODEs) are widely used to describe the time evolution of natural phenomena across various scientific fields. Estimating the parameters of these systems from data is a challenging task, particularly when…

Biochemical molecules interact through modification and binding reactions, giving raise to a combinatorial number of possible biochemical species. The time-dependent evolution of concentrations of the species is commonly described by a…

Noise-induced transitions between metastable fixed points in systems evolving on multiple time scales are analyzed in situations where the time scale separation gives rise to a slow manifold with bifurcation. This analysis is performed…

Fast-slow dynamical systems have subsystems that evolve on vastly different timescales, and bifurcations in such systems can arise due to changes in any or all subsystems. We classify bifurcations of the critical set (the equilibria of the…

Statistical early warning signs can be used to identify an approaching bifurcation in stochastic dynamical systems and are now regularly employed in applications concerned with the identification of potential rapid, non-linear change or…

There is growing interest in anticipating critical transitions in natural systems, often pursued through statistical detection of early warning signals associated with dynamical bifurcations. In stochastic dynamical systems, such signals…