Mapping Nonlinear Lattice Equations onto Cellular Neural Networks

In the last years completely integrable Hamiltonian systems were of great interest because of their physical nature, e.g., the existence of soliton solutions, and their relation to eigenvalue and sorting problems. But until recently, they found little interest among electrical engineers because lossless circuits are difficult to realize as physical systems. However, if we are only interested in the “signals” associated with Hamiltonian systems, and not in conserving the energy in the 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 (CNN’s) come very close to some Hamiltonian systems; therefore, they are potentially useful for simulating or realizing such systems. In this paper, we will show how to map two one-dimensional nonlinear lattices, the Fermi-Pasta-Ulam lattice and the Toda lattice, onto a CNN. We demonstrate 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 interesting phenomenon is due to a special symmetry in the CNN system of ODE’s.