# [Focus on] Complex systems Research activities of the CAS3C3 Team of
Laboratoire Jean Kuntzmann

Complex systems research activities of the CAS^{3}C^{3}
team are focused on
- Discrete dynamical systems.
- Parallel graph rewriting and Graph transformations.
- Graph models for biology and astrophysics.

### DEM-Systems

DEM-systems are a class of discrete dynamical systems with properties of both cellular automata and L-systems. They are defined as sequences on a one-dimensional loop with rules governing dynamics in which new sites can be created, depending on the states of a neighbourhood of sites, and complex behaviour can be generated. Unlike for CA, finite initial sequences can produce positive spatial entropy over time.

However, even in cases where the entropy is zero, considerable complexity is possible, especially when the sequence length grows to infinity, and we demonstrate and study behaviours of DEM-systems including fragmentation of sequences, self-reproducing patterns, self-similar but irregular patterns, patterns that not only produce new sites but produce producers of new sites, and sequences whose growth rate is sublinear, linear, quadratic, cubic, or exponential.

Références:
hal-00961656
hal-00953772
hal-01326782

### Parallel graph rewriting with overlapping rules

Considering an initial graph g and a system of rewriting rules R={l_{i} -> r_{i}, i=1... n}, we rewrite the graph g into a graph g'
by using, simultaneously, the rules of R whose left-hand sides, l_{i}, match subgraphs of g.
All the occurrences of the l_{i} in g are replacing by an instance of r_{i}.
In order to deal with match overlapping, we introduce the notion of pregraphs and follow the rewriting modulo approach.

GPaR is a parallel graph rewriting software implemented in C++ with a graphical user interface. It can be used in a large variety of rewriting problems including cellular automata, L-systems and
fractal systems.

Références:
hal-01408834
hal-02084261
hal-01985043
hal-01898363

### Split thickness of graphs

We examine the planar split thickness of a graph, that is, the smallest k
such that the graph is k-splittable into a planar graph.
A k-split operation substitutes
a vertex v by at most k new vertices such that each neighbor of v
is connected to at least one of the new vertices.

hal-01819362
hal-01326779

### Simulating aggregates of bivalents in 2n=40
mouse meiotic spermatocytes through
inhomogeneous percolation processes

We show that an inhomogeneous Bernoulli site percolation process running upon a fullerene's dual C'_{1200} can be used for representing bivalents attached to the nuclear envelope in mouse Mus M. Domesticus 2n=40 meiotic spermatocytes during pachytene. It is shown that the induced clustering generated by overlapping percolation domains correctly reproduces the probability distribution observed in the experiments (data) after fine tuning the parameters.

Références:
hal-01823737
hal-01814944
hal-01982363

### Simulation of hierarchical n-body systems based on
dynamical trees

We present an algorithm to simulate the gravitational interaction
of a large number of point masses. This problem is known as the n-body
problem in physics and astronomy. The algorithm detects and uses hierarchical
structure present in the current state of the point masses to speed up
the computation. The structure is represented as a dynamical tree.

Références:
hal-00769677

Last modified: Thu Dec 5 11:38:40 CET 2019