  Linearized catalytic reaction equations modeling e.g. the dynamics of genetic
regulatory networks under the constraint that expression levels, i.e. molecular
concentrations of nucleic material are positive, exhibit nontrivial dynamical
properties, which depend on the average connectivity of the reaction network.
In these systems the inflation of the edge of chaos and multi-stability have
been demonstrated to exist. The positivity constraint introduces a nonlinearity
which makes chaotic dynamics possible. Despite the simplicity of such minimally
nonlinear systems, their basic properties allow to understand fundamental
dynamical properties of complex biological reaction networks. We analyze the
Lyapunov spectrum, determine the probability to find stationary oscillating
solutions, demonstrate the effect of the nonlinearity on the effective in- and
out-degree of the active interaction network and study how the frequency
distributions of oscillatory modes of such system depend on the average
connectivity.
