Regions of multistability in some low-dimensional logistic models with excitation type coupling

A naive model of many networked logistic maps with an excitation type coupling [Neural Networks, vol. 20, 102–108 (2007)], which is an extension of other low dimensional models, has been recently proposed to mimic the waking-sleeping bistability found in brain systems. Although the dynamics of large and complex aggregates of elementary components can not be understood nor extrapolated from the properties of a few components, some patterns of behavior could be conserved independently of the topology and of the number of coupled units. Following this insight, we have collected several of those systems where a few logistic maps are coupled under a similar mutual excitation scheme. The regions of bi- and multistability of these systems are sketched and reported.

💡 Research Summary

The paper investigates the emergence of bi‑ and multistability in low‑dimensional networks of logistic maps that are coupled through a purely excitatory scheme. Building on the earlier proposal of an “excitation‑type coupling” (Neural Networks, vol. 20, 102‑108, 2007), the authors ask whether the essential dynamical features responsible for the waking‑sleeping bistability observed in large brain networks can already be seen in systems composed of only a few interacting units.

Model definition. Each node i evolves according to the classic logistic map
 x_i^{t+1}=r x_i^{t}(1−x_i^{t})
with growth parameter r∈