  Two novel phenomena for unidirectionally coupled 3-cell Hopfield neural
networks (HNNs) are investigated. The first one is the persistence of chaos,
which means the permanency of sensitivity and infinitely many unstable periodic
oscillations in the response HNN even if the networks are not synchronized in
the generalized sense. Doubling of chaotic attractors is the second phenomenon
realized in this study. It can be achieved when the response network possesses
two stable point attractors in the absence of the driving. This feature leads
to the formation of two coexisting chaotic attractors with disjoint basins.
Lyapunov functions are utilized to deduce the presence of an invariant region,
and the sensitivity is rigorously proved. The absence of synchronization is
approved via the auxiliary system approach and analysis of conditional Lyapunov
exponents. Additionally, quadruple and octuple coexisting chaotic attractors
are demonstrated, and the formation of hyperchaos is discussed.
