Constructing models from time series with nontrivial dynamics involves the problem of how to choose the best model from within a class of models, or to choose between competing classes. This paper discusses a method of building nonlinear models of possibly chaotic systems from data, while maintaining good robustness against noise. The models that are built are close to the simplest possible according to a description length criterion. The method will deliver a linear model if that has shorter description length than a nonlinear model. We show how our models can be used for prediction, smoothing and interpolation in the usual way. We also show how to apply the results to identification of chaos by detecting the presence of homoclinic orbits directly from time series. 1 The Model Selection Problem As our understanding of chaotic and other nonlinear phenomena has grown, it has become apparent that linear models are inadequate to model most dynamical processes. Nevertheless, linear models...

. We present three distinct concepts of what constitutes a scienti. ..c understanding of a dynamical system . The development of each of these paradigms has resulted in a signi...cant expansion in the kind of system that can be investigated. In particular, the recently-developed "algorithm ic modelling paradigm" has allowed us to enlarge the domain of discourse to include complex real-world processes that cannot be necessarily be described by conventional dierential equations. 1. Introduction What do we mean when we say that we understand a dynamical system? In this essay, we identify three distinct paradigms for scienti...c understanding of dynamical systems. These paradigms are the models of knowing of the title. The introduction of new models of knowing has resulted in a signi...cant expansion in the kinds of systems that can be investigated scienti...cally. The ...rst paradigm, which we shall refer to as the Newtonian 1 , was established in the seventeenth century. Accor...