Geometric Representations of Nonlinear Dynamics

Coupled lattices of various types can be created. In the cellular automata approach, a variable that can take on discrete values constitutes the elements which are coupled together. The coupling occurs via rules that simulate physical processes such as biological interactions, diffusion, and so forth. Coupled map lattices take this one step further and assign to each lattice element a difference equation that, when iterated, produces a discrete dynamical system. Coupled ODE lattices represent the next step in complexity, and accuracy, for a coupled lattice here, an ODE or system of ODEs are coupled together, again by a choice of simple rules chosen to simulate the desired physical interaaions. 

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Computational techniques are centrally important at every stage of investigation of nonlinear dynamical systems. We have reviewed the main theoretical and computational tools used in studying these problems among these are bifurcation and stability analysis, numerical techniques for the solution of ordinary differential equations and partial differential equations, continuation methods, coupled lattice and cellular automata methods for the simulation of spatiotemporal phenomena, geometric representations of phase space attractors, and the numerical analysis of experimental data through the reconstruction of phase portraits, including the calculation of correlation dimensions and Lyapunov exponents from the data. 

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