Abstract
Dynamical systems on monoids have been recently proposed as minimal mathematical models for the intuitive notion of deterministic dynamics. This paper shows that any dynamical system DS L on a monoid L can be exhaustively decomposed into a family of mutually disconnected subsystems—the constituent systems of D S L . In addition, constituent systems are themselves indecomposable, even though they may very well be complex. Finally, this work also makes clear how any dynamical system DS L turns out to be identical to the sum of all its constituent systems. Constituent systems can thus be thought as the indecomposable, but possibly complex, building blocks to which the dynamics of an arbitrary complex system fully reduces. However, no further reduction of the constituents is possible, even if they are themselves complex.
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Notes
- 1.
The state space of a cellular automaton is discrete (i.e. finite or countably infinite) if the cellular automaton state space only includes finite configurations, that is to say, configurations where all but a finite number of cells are in the quiescent state. If this condition is not satisfied, the state space has the power of the continuum.
- 2.
My thanks to Tomasz Kowalski for pointing out to me the relation between vector spaces over fields and dynamical systems on monoids.
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Giunti, M. (2016). Decomposing Dynamical Systems. In: Minati, G., Abram, M., Pessa, E. (eds) Towards a Post-Bertalanffy Systemics. Contemporary Systems Thinking. Springer, Cham. https://doi.org/10.1007/978-3-319-24391-7_6
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DOI: https://doi.org/10.1007/978-3-319-24391-7_6
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