Many natural systems are organized as networks, in which the nodes interact in a time-dependent fashion. The object of our study is to relate connectivity to the temporal behavior of a network in which the nodes are (real or complex) logistic maps, coupled according to a connectivity scheme that obeys certain constrains, but also incorporates random aspects. We investigate in particular the relationship between the system architecture and possible dynamics. In the current paper we focus on establishing the framework, terminology and pertinent questions for low-dimensional networks. A subsequent paper will further address the relationship between hardwiring and dynamics in high-dimensional networks.
For networks of both complex and real node-maps, we define extensions of the Julia and Mandelbrot sets traditionally defined in the context of single map iterations. For three different model networks, we use a combination of analytical and numerical tools to illustrate how the system behavior (measured via topological properties of the Julia sets) changes when perturbing the underlying adjacency graph. We differentiate between the effects on dynamics of different perturbations that directly modulate network connectivity: increasing/decreasing edge weights, and altering edge configuration by adding, deleting or moving edges. We discuss the implications of considering a rigorous extension of Fatou-Julia theory known to apply for iterations of single maps, to iterations of ensembles of maps coupled as nodes in a network.
Beyond Linear Additive and Hill Functions: A General Logistic Reformulation of Delay-Coupled Gene Regulatory Networks with Equilibrium Analysis, Hopf Bifurcation, and Lipschitz Stability
Study of Low-dimensional Nonlinear Fractional Difference Equations of Complex Order
本文研究了1维和2维的复阶分数映射，即$0<α_0<1$的$α_0e^{i r π/2}$和$0\le r<1$.在二维中，我们研究了H{é}non和Lozi映射，在$1d$中，我们研究了Logistic映射、帐篷映射、高斯映射、圆映射和Bernoulli映射。$2d$中的推广可以用两种不同的方法来完成，这两种方法对于分数阶是不等价的，并且会导致不同的分叉图。我们观察到，光滑的映射，如Logistic，高斯，和H{é}non映射不显示混沌，而不连续的映射，如Lozi，Bernoulli，和圆映射显示混沌。帐篷映射是连续的，但不可微，它也显示出混沌。在$2d$中，我们发现表现混沌的复分数阶映射也表现多稳定性。因此，可以推断，复分数阶的光滑映射往往比不连续或不可微的映射表现出更规则的行为。
