Mathematics
of Simulations |
Sequential dynamical systems (SDS), which came out of the research done in
FOSS allows us to analyze the generic features of simulations. The main features
and constituents of simulations are:
Entities/Objects/Actors
Examples of entities in a simulation are, e.g.,
a car in a traffic simulation, a molecule in gas dynamics simulations, a station
in an IP network simulation and so on. An entity usually has a state reflecting
its properties.
Communication links
Entities communicate or exchange full or
partial information about their states. Typically, entities only communicate with
nearby entities.
Update rules
Each entity has a rule or a set of rules for they update
their state. An entity replacing its old state with the state computed by its
local function is referred to as an update of that entity.
Update mechanism
A computer simulation has a rule or mechanism
that schedules the individual updates of entities. Some common choices include
sequential updating, parallel updating, random updating and event-driven updating.
Typically, we will have perfect knowledge about each object in isolation. The
complex dynamics of the simulation results from composition of local rules. The
goal of SDS research is to obtain information about the global dynamics of a simulation
based on the local information we have, and without implementing and running the
actual computer simulation.
Sequential Dynamical Systems
Our research is centered around an abstraction of the above description of
a computer simulation. The objects in an SDS are the vertices of some graph Y,
and each of them has associated a binary state. The edges of the graph Y represents
the communication links among objects. For each object there is a (symmetric)
local function. The objects are update sequentially according to a fixed schedule
or permutation.
SDS capture essential features of simulations such as scheduling and dependencies.
They allow us to study e.g. equivalence of schedules in an abstract setting, and
thus addressing the question of, e.g. validity. The nature of the work on
SDS falls into the categories of algebra, combinatorics, dynamical systems and
probability theory.
Some topics and questions we have studied include:
- Defining "inequivalence" of dynamical systems and determine the number of
inequivalent dynamical systems that can be obtained through rescheduling.
- Investigate the effects of symmetries of the Y-graph on the corresponding
phase portraits/phase spaces of SDS.
- Designing a "relative" theory in the sense of analyzing how changes in the
Y-graph affect the underlying dynamical system. Effects of stochasticity in one
or more of the constituents of an SDS. Dynamical system issues such as fixed
points and structure of periodic orbits.
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