Influence of feedback in wave based chaotic networks

The effective connectivity between input and output channels in complex coupled networks based on the transfer of electro-magnetic waves (RF, microwave or optical) plays an important role in many fields. The transfer of the waves between connected nodes in complex coupled networks is strongly influenced by the presence of non-reciprocal feedback channels with uni-directional amplification. The introduction of such non-reciprocal channels breaks the time reversal invariance of the connectivity matrix between the input and the output channels of the network. The feedback channels are intrinsically changing the spectral properties of the connectivity matrix and hence the associated static and dynamic transfer properties of the ingoing and outgoing ports of the network.

Here we use a random matrix model to represent the complex and fully chaotic network without feedback. Extra feedback connections are introduced that change the reciprocity and time reversal invariance of the reference network and hence the connectivity.

Exploiting a modern electrical engineering description of multi-port complex circuits in terms of the S-matrix of internal multi-port sections, we show by computer simulations the influence of feedback for increasing network complexity. The results are interpreted using a random matrix approach for the multi-port connectivity matrix. The effect of amplifying bi-directional or uni-directional connecting loop on the properties of the resulting S-matrix are computed for increasing network size. The signal transfer capacity of a multi channel the network based on the Smatrix shows for smaller networks a strong influence of the extra connected loop. In particular for the uni-directional amplifying feedback loop.