Explicit construction of chaotic attractors in Glass networks

Chaotic dynamics have been observed in example piecewise-affine models of gene regulatory networks. Here we show how the underlying Poincare maps can be explicitly constructed. To do this, we proceed in two steps. First, we consider a limit case, where some parameters tend to infinity, 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. (C) 2012 Elsevier Ltd. All rights reserved.

在基因调控网络的分段仿射模型示例中观察到了混沌动力学。在此我们展示如何明确构建基础的庞加莱映射。为此，我们分两步进行。首先，我们考虑一种极限情况，其中某些参数趋于无穷，然后将具有有限参数的情况视为前一种情况的扰动。我们提供了这种构建的一个详细的三维示例，每个变量有几个阈值。这种构建本质上是一种拓扑马蹄映射。我们表明极限情况与黄金分割移位共轭，因此是混沌的。然后，我们表明对于大参数混沌是保持的，这依赖于极限情况下返回映射的结构稳定性。我们还描述了一种将具有几个阈值的系统嵌入到更高维二进制系统的方法。这表明对于具有几个阈值的系统所发现的所有结果在二进制情况下仍然有效。(C) 2012爱思唯尔有限公司。保留所有权利。