Related papers: Macroscale, slowly varying, models emerge from the…
Multiscale dynamics are ubiquitous in applications of modern science. Because of time scale separation between relatively small set of slowly evolving variables and (typically) much larger set of rapidly changing variables, direct numerical…
Many real-world systems studied are governed by complex, nonlinear dynamics. By modeling these dynamics, we can gain insight into how these systems work, make predictions about how they will behave, and develop strategies for controlling…
Some existing models of the atherosclerosis development are discussed and a new improved mathematical model, which takes into account new experimental results about diverse roles of macrophages in atherosclerosis, is proposed. Using technic…
The theory of slow manifolds is an important tool in the study of deterministic dynamical systems, giving a practical method by which to reduce the number of relevant degrees of freedom in a model, thereby often resulting in a considerable…
This article establishes the foundation for a new theory of invariant/integral manifolds for non-autonomous dynamical systems. Current rigorous support for dimensional reduction modelling of slow-fast systems is limited by the rare events…
Developments in dynamical systems theory provides new support for the discretisation of \pde{}s and other microscale systems. By systematically resolving subgrid microscale dynamics the new approach constructs asymptotically accurate,…
The long-term dynamics of many dynamical systems evolve on an attracting, invariant "slow manifold" that can be parameterized by a few observable variables. Yet a simulation using the full model of the problem requires initial values for…
We present a rigorous framework for the local analysis of canards and slow passages through bifurcations in a wide class of infinite-dimensional dynamical systems with time-scale separation. The framework is applicable to models where an…
This paper introduces a novel method for approximating the dynamics of a large autonomous system projected onto a fixed subspace. The core contribution is a novel recursive algorithm to construct an effective time-dependent generator that…
Empirical modelling often aims for the simplest model consistent with the data. A new technique is presented which quantifies the consistency of the model dynamics as a function of location in state space. As is well-known, traditional…
We develop a diffusion approximation for systems subject to fast random resetting by small amplitudes. Equivalently, this describes systems with frequent but small catastrophes. We demonstrate the validity of the approximation by computing…
While existing mathematical descriptions can accurately account for phenomena at microscopic scales (e.g. molecular dynamics), these are often high-dimensional, stochastic and their applicability over macroscopic time scales of physical…
I prove that a centre manifold approach to creating finite difference models will consistently model linear dynamics as the grid spacing becomes small. Using such tools of dynamical systems theory gives new assurances about the quality of…
To ease analysis and simulation we make low-dimensional models of complicated dynamical systems. Centre manifold theory provides a systematic basis for the reduction of dimensionality from some detailed dynamical prescription down to a…
Constructing numerical models of noisy partial differential equations is very delicate. Our long term aim is to use modern dynamical systems theory to derive discretisations of dissipative stochastic partial differential equations. As a…
Dynamical systems that evolve continuously over time are ubiquitous throughout science and engineering. Machine learning (ML) provides data-driven approaches to model and predict the dynamics of such systems. A core issue with this approach…
This paper introduces coordinate-independent methods for analysing multiscale dynamical systems using numerical techniques based on the transfer operator and its adjoint. In particular, we present a method for testing whether an arbitrary…
The paper introduces a novel topological method for prediction and modeling for a nonlinear time--series that exhibit recurring patterns. According to the model, global manifold of the reconstructed state--space can be approximated by a few…
Dynamic facilitation theory assumes short-ranged dynamic constraints to be the essential feature of supercooled liquids and draws much of its conclusions from the study of kinetically constrained models. While deceptively simple, these…
In systems removed from equilibrium, intrinsic microscopic fluctuations become correlated over distances comparable to the characteristic macroscopic length over which the external constraint is exerted. In order to investigate this…