  This work aims to propose and design a class of networks of coupled linear
and nonlinear oscillators, in which short bursts of exogenous excitation result
in sustained endogenous network activity that returns to a quiescent state only
after a characteristic time and along a different path than when originally
excited. The desired hysteretic behavior is obtained through the coupling of
self-excited oscillations with purposely designed rate laws for slowly-varying
nodal parameters, governed only by local interactions in the network. The
proposed architecture and the sought dynamics take inspiration from complex
biological systems that combine endogenous energy sources with a paradigm for
distributed sensing and information processing. In this paper, the network
design problem considers arbitrary topologies and investigates the dependence
of the desired response on model parameters, as well as on the placement of a
single nonlinear node in an otherwise linear network. Perturbation analysis in
various asymptotic parameter limits is used to define the proposed internal
dynamics. Parameter continuation techniques validate the asymptotic results
numerically and demonstrate their robustness over finite ranges of parameter
values. Both approaches suggest a nontrivial dependence of the optimal
distribution of nonlinearity on the network topology.
