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Explicit construction of chaotic attractors in Glass networks

Roderick Edwards 1 Etienne Farcot 2, 3 Eric Foxall 1 
2 VIRTUAL PLANTS - Modeling plant morphogenesis at different scales, from genes to phenotype
CRISAM - Inria Sophia Antipolis - Méditerranée , INRA - Institut National de la Recherche Agronomique, UMR AGAP - Amélioration génétique et adaptation des plantes méditerranéennes et tropicales
Abstract : Chaotic dynamics have been observed in example piecewise-affine models of gene regulatory networks. Here we show how the underlying Poincaré maps can be explicitly constructed. To do this, we proceed in two steps. First, we consider a limit case, where some parameters tend to ∞, and then consider the case with finite parameters as a perturbation of the previous one. We provide a detailed example of this construction, in 3-d, with several thresholds per variable. This construction is essentially a topological horseshoe map. We show that the limit situation is conjugate to the golden mean shift, and is thus chaotic. Then, we show that chaos is preserved for large parameters, relying on the structural stability of the return map in the limit case. We also describe a method to embed systems with several thresholds into binary systems, of higher dimensions. This shows that all results found for systems having several thresholds remain valid in the binary case.
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Roderick Edwards, Etienne Farcot, Eric Foxall. Explicit construction of chaotic attractors in Glass networks. Chaos, Solitons and Fractals, Elsevier, 2012, 45 (5), pp.666-680. ⟨10.1016/j.chaos.2012.02.018⟩. ⟨hal-00828842⟩



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