Mapping nonlinear lattice equations onto cellular neural networks

If one is only interested in the signals associated with Hamiltonian systems, and not in conserving the energy in individual circuit elements (nonlinear inductors and capacitors), then such systems can be built as analog circuits, which implement some signal flow graphs. Under certain restrictions, cellular neural networks (CNNs) come very close to some Hamiltonian systems; therefore, they are potentially useful for simulating or realizing such systems. It is shown how to map two one-dimensional nonlinear lattices, the Fermi-Pasta-Ulam lattice and the Toda lattice, onto a CNN. It is demonstrated for the Toda lattice what happens if the signals are driven beyond the linear region of the output function. Though the system is no longer Hamiltonian, numerical experiments reveal the existence of solitons for special initial conditions. This phenomenon is due to a special symmetry in the CNN system of ordinary differential equations