2,827 research outputs found
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
Integrable 1D Toda cellular automata
First, we recall the algebro-geometric method of construction of finite field
valued solutions of the discrete KP equation and next we perform a reduction of
the dKP equation to the discrete 1D Toda equation. This gives a method of
construction of solutions of the discrete 1D Toda equation taking values in a
finite field.Comment: 9 pages, 2 figures; Corrected typo
Quantumness of discrete Hamiltonian cellular automata
We summarize a recent study of discrete (integer-valued) Hamiltonian cellular
automata (CA) showing that their dynamics can only be consistently defined, if
it is linear in the same sense as unitary evolution described by the
Schr\"odinger equation. This allows to construct an invertible map between such
CA and continuous quantum mechanical models, which incorporate a fundamental
scale. Presently, we emphasize general aspects of these findings, the
construction of admissible CA observables, and the existence of solutions of
the modified dispersion relation for stationary states.Comment: 4 pages; invited talk at the symposium "Wigner 111 - Colourful and
Deep" (Budapest, November 2013), to appear in EPJ Web of Conference
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