Complexity in linearly coupled dynamical networks : Some unusual phenomena in energy accumulation 1

This paper addresses the energy accumulation problem, in terms of the H2 norm, of linearly coupled dynamical networks. An interesting outer-coupling relationship is constructed, under which the H2 norm of the newly constructed network with column-input and row-output shaped matrices increases exponentially fast with the node number N : it increases generally much faster than 2 when N is large while the H2 norm of each node is 1. However, the H2 norm of the network with a diffusive coupling is equal to γ2N , i.e., increasing linearly, when the network is stable, where γ2 is the H2 norm of a single node. And the H2 norm of the network with antisymmetrical coupling also increases, but rather slowly, with the node number N . Other networks with block-diagonal-input and block-diagonal-output matrices behave similarly. It demonstrates that the changes of H2 norms in different networks are very complicated, despite the fact that the networks are linear. Finally, the influence of the H2 norm of the locally linearized network on the output of a network with Lur’e nodes is discussed.