Dimensional implications of dynamical data on manifolds to empirical KL analysis

We explore the approximation of attracting manifolds of complex systems using dimension reducing methods. Complex systems having high-dimensional dynamics typically are initially analyzed by exploring techniques to reduce the dimension. Linear techniques, such as Galerkin projection methods, and nonlinear techniques, such as center manifold reduction are just some of the examples used to approximate the manifolds on which the attractors lie. In general, if the manifold is not highly curved, then both linear and nonlinear methods approximate the surface well. However, if the manifold curvature changes significantly with respect to parametric variations, then linear techniques may fail to give an accurate model of the manifold. This may not be a surprise in itself, but it is a fact so often overlooked or misunderstood when utilizing the popular KL method, that we offer this explicit study of the effects and consequences. Here we show that certain dimensions defined by linear methods are highly sensitive when modeled in situations where the attracting manifolds have large parametric curvature. Specifically, we show how manifold curvature mediates the dimension when using a linear basis set as a model. We punctuate our results with the definition of what we call, a “curvature induced parameter,” dCI. Both finite- and infinite-dimensional models are used to illustrate the theory.