Many real systems have been identified as potentially chaotic (e.g., trains of heart beats, records of multiple populations in an ecological system, weather and climate data). In such cases, when “healthy” is defined by the measured trajectory being confined to a “strange attractor” in a “phase space,” how can we define and measure degradation? The simplest course is to define degradation as when the attractor changes. The earliest warning of such a change would be when a trajectory leaves an attractor, without an external forcing, but there seems little written on the problem of identifying whether a short trajectory belongs to a given attractor (at least to within a neighborhood defined by stochastic disturbances and uncertainty in the parameters of the defining equations).
This chapter explores the problem of testing short trajectories in the neighborhood of the Lorenz attractor. Although there are some conditions necessary for developing a working test, it turns out that the Lorenz attractor is perhaps one of the kindest examples to explore, being easily visualized as well as having significant work done on its statistical properties. Further work with more complex attractors is necessary before this approach can be considered general. However, the proposed test shows the potential of not only detecting degradation, but of being a general partial goodness of fit test for a short trajectory to a neighborhood of an attractor, even for high-dimensional attractors. In addition, the construction of the test suggests a dimension reduction scheme useful for exploring deficiencies in the attractor in describing the trajectory.