992 research outputs found
Identification of excitable media using a scalar coupled map lattice model
The identification problem for excitable media is investigated in this paper. A new scalar coupled map lattice (SCML) model is introduced and the orthogonal least squares algorithm is employed to determinate the structure of the SCML model and to estimate the associated parameters. A simulated pattern and a pattern observed directly from a real Belousov-Zhabotinsky reaction are identified. The identified SCML models are shown to possess almost the same local dynamics as the original systems and are able to provide good long term predictions
Boolean Delay Equations: A simple way of looking at complex systems
Boolean Delay Equations (BDEs) are semi-discrete dynamical models with
Boolean-valued variables that evolve in continuous time. Systems of BDEs can be
classified into conservative or dissipative, in a manner that parallels the
classification of ordinary or partial differential equations. Solutions to
certain conservative BDEs exhibit growth of complexity in time. They represent
therewith metaphors for biological evolution or human history. Dissipative BDEs
are structurally stable and exhibit multiple equilibria and limit cycles, as
well as more complex, fractal solution sets, such as Devil's staircases and
``fractal sunbursts``. All known solutions of dissipative BDEs have stationary
variance. BDE systems of this type, both free and forced, have been used as
highly idealized models of climate change on interannual, interdecadal and
paleoclimatic time scales. BDEs are also being used as flexible, highly
efficient models of colliding cascades in earthquake modeling and prediction,
as well as in genetics. In this paper we review the theory of systems of BDEs
and illustrate their applications to climatic and solid earth problems. The
former have used small systems of BDEs, while the latter have used large
networks of BDEs. We moreover introduce BDEs with an infinite number of
variables distributed in space (``partial BDEs``) and discuss connections with
other types of dynamical systems, including cellular automata and Boolean
networks. This research-and-review paper concludes with a set of open
questions.Comment: Latex, 67 pages with 15 eps figures. Revised version, in particular
the discussion on partial BDEs is updated and enlarge
Learning spatio-temporal patterns with Neural Cellular Automata
Neural Cellular Automata (NCA) are a powerful combination of machine learning
and mechanistic modelling. We train NCA to learn complex dynamics from time
series of images and PDE trajectories. Our method is designed to identify
underlying local rules that govern large scale dynamic emergent behaviours.
Previous work on NCA focuses on learning rules that give stationary emergent
structures. We extend NCA to capture both transient and stable structures
within the same system, as well as learning rules that capture the dynamics of
Turing pattern formation in nonlinear Partial Differential Equations (PDEs). We
demonstrate that NCA can generalise very well beyond their PDE training data,
we show how to constrain NCA to respect given symmetries, and we explore the
effects of associated hyperparameters on model performance and stability. Being
able to learn arbitrary dynamics gives NCA great potential as a data driven
modelling framework, especially for modelling biological pattern formation.Comment: For videos referenced in appendix, see:
https://github.com/AlexDR1998/NCA/tree/main/Video
Model of the Belousov-Zhabotinsky reaction
The article describes results of the modified model of the
Belousov-Zhabotinsky reaction, which resembles rather well the limit set
observed upon experimental performance of the reaction in the Petri dish. We
discuss the concept of the ignition of circular waves and show that only the
asymmetrical ignition leads to the formation of spiral structures. From the
qualitative assumptions on the behavior of dynamic systems, we conclude that
the Belousov-Zhabotinsky reaction likely forms a regular grid.Comment: 17 pages, 12 figure
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