scholarly journals 2D piecewise affine models approximate real continuous dynamics up to invariant sets**This work was supported in part by the projects GeMCo (ANR 2010 BLANC020101), RESET (Bioinformatique, ANR-11-BINF-0005), and by the LABEX SIGNALIFE (ANR-ll-LABX-0028-01).

2016 ◽  
Vol 49 (18) ◽  
pp. 1060-1065 ◽  
Author(s):  
Madalena Chaves ◽  
Jean-Luc Gouzé
Keyword(s):  
Invariant Sets ◽  
Piecewise Affine ◽  
Affine Models

A Greedy Approach to Identification of Piecewise Affine Models

2003 ◽  
pp. 97-112 ◽  
Author(s):  
Alberto Bemporad ◽  
Andrea Garulli ◽  
Simone Paoletti ◽  
Antonio Vicino
Keyword(s):  
Greedy Approach ◽  
Piecewise Affine ◽  
Affine Models

On the geometry and regularity of invariant sets of piecewise-affine automorphisms on the Euclidean space

2019 ◽  
Vol 40 (8) ◽  
pp. 2183-2218
Author(s):  
C. SİNAN GÜNTÜRK ◽  
NGUYEN T. THAO
Keyword(s):  
Euclidean Space ◽  
Rate Of Convergence ◽  
Linear Part ◽  
Invariant Sets ◽  
Measurable Partition ◽  
Ergodic Averages ◽  
Piecewise Affine ◽  
Affine Maps ◽  
Orbit Closures ◽  
Borel Measurable

In this paper, we derive geometric and analytic properties of invariant sets, including orbit closures, of a large class of piecewise-affine maps $T$ on $\mathbb{R}^{d}$. We assume that (i) $T$ consists of finitely many affine maps defined on a Borel measurable partition of $\mathbb{R}^{d}$, (ii) there is a lattice $\mathscr{L}\subset \mathbb{R}^{d}$ that contains all of the mutual differences of the translation vectors of these affine maps, and (iii) all of the affine maps have the same linear part that is an automorphism of $\mathscr{L}$. We prove that finite-volume invariant sets of such piecewise-affine maps always consist of translational tiles relative to this lattice, up to some multiplicity. When the partition is Jordan measurable, we show that closures of bounded orbits of $T$ are invariant and yield Jordan measurable tiles, again up to some multiplicity. In the latter case, we show that compact $T$-invariant sets also consist of Jordan measurable tiles. We then utilize these results to quantify the rate of convergence of ergodic averages for $T$ in the case of bounded single tiles.

scholarly journals Discrete time piecewise affine models of genetic regulatory networks

2006 ◽  
Vol 52 (4) ◽  
pp. 524-570 ◽  
Author(s):  
R. Coutinho ◽  
B. Fernandez ◽  
R. Lima ◽  
A. Meyroneinc
Keyword(s):  
Discrete Time ◽  
Regulatory Networks ◽  
Genetic Regulatory Networks ◽  
Piecewise Affine ◽  
Affine Models
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