Chaos is a complex behavioural pattern associated with nonlinear systems. Its association with the analytical complexity of the generating system is thought to be minimal. Our experiments show that this is not necessarily true and that, with the increase of analytical complexity, newer classes of dynamical behaviour can be seen. We call these new classes emergent because the system we study remains the same so far as the physics it represents is concerned, while we apply methods to increase intrinsically its degrees of freedom. We propose four techniques for the latter: (complex)ification, splitting, signal-flow-graph products, and interpolation/extrapolation. Some interesting new behavioural patterns which emerge experimentally and which we report here are: attractor symmetries, duality between time and parameters, and a purportedly new route to chaos, named after Gibbs.