Abstract
A new theory of constructing optimal truncated Low-Dimensional Dynamical Systems (LDDS), either based on known databases or directly from partial differential equations, is presented. Applying the new theory to four examples, i.e., the one--dimensional linear heat transfer equation, the nonlinear Burgers' equation and the two-dimensional Navier-Stokes equations with closed or open domains, it is shown that the optimal truncated LDDS in which the projecting errors have already been reduced to the minimum, can be constructed. Depending upon different optimal conditions, different LDDS can be found. Within the framework of optimal truncated LDDS, the initial bases chosen for iterations are the crucial factors in the optimization processes. The nonlinear Galerkin method can significantly improve the results, after the optimal bases have been gotten.
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