No Arabic abstract
I describe a method, particularly suitable to implementation by computer algebra, for the derivation of low-dimensional models of dynamical systems. The method is systematic and is based upon centre manifold theory. Computer code for the algorithm is relatively simple, robust and flexible. The method is applied to two examples: one a straightforward pitchfork bifurcation, and one being the dynamics of thin fluid films.
We develop further ideas on how to construct low-dimensional models of stochastic dynamical systems. The aim is to derive a consistent and accurate model from the originally high-dimensional system. This is done with the support of centre manifold theory and techniques. Aspects of several previous approaches are combined and extended: adiabatic elimination has previously been used, but centre manifold techniques simplify the noise by removing memory effects, and with less algebraic effort than normal forms; analysis of associated Fokker-Plank equations replace nonlinearly generated noise processes by their long-term equivalent white noise. The ideas are developed by examining a simple dynamical system which serves as a prototype of more interesting physical situations.
Recently, the place of the main programming language for scientific and engineering computations has been little by little taken by Julia. Some users want to work completely within the Julia framework as they work within the Python framework. There are libraries for Julia that cover the majority of scientific and engineering computations demands. The aim of this paper is to combine the usage of the Julia framework for numerical computations and for symbolic computations in mathematical modeling problems. The main functional domains determining various variants of the application of computer algebra systems are described. In each of these domains, generic representatives of computer algebra systems in Julia are distinguished. The conclusion is that it is possible (and even convenient) to use computer algebra systems within the Julia framework.
Burgers equation is one of the simplest nonlinear partial differential equations-it combines the basic processes of diffusion and nonlinear steepening. In some applications it is appropriate for the diffusion coefficient to be a time-dependent function. Using a Waynes transformation and centre manifold theory, we derive l-mode and 2-mode centre manifold models of the generalised Burgers equations for bounded smooth time dependent coefficients. These modellings give some interesting extensions to existing results such as the similarity solutions using the similarity method.
Consider the macroscale modelling of microscale spatiotemporal dynamics. Here we develop a new approach to ensure coarse scale discrete models preserve important self-adjoint properties of the fine scale dynamics. The first part explores the discretisation of microscale continuum dynamics. The second addresses how dynamics on a fine lattice are mapped to lattice a factor of two coarser (as in multigrids). Such mapping of discrete lattice dynamics may be iterated to empower us in future research to explore scale dependent emergent phenomena. The support of dynamical systems, centre manifold, theory ensures that the coarse scale modelling applies with a finite spectral gap, in a finite domain, and for all time. The accuracy of the models is limited by the asymptotic resolution of subgrid coarse scale processes, and is controlled by the level of truncation. As given examples demonstrate, the novel feature of the approach developed here is that it ensures the preservation of important conservation properties of the microscale dynamics.
We detail how incorporating physics into neural network design can significantly improve the learning and forecasting of dynamical systems, even nonlinear systems of many dimensions. A map building perspective elucidates the superiority of Hamiltonian neural networks over conventional neural networks. The results clarify the critical relation between data, dimension, and neural network learning performance.