Non-linear and chaotic dynamical systems

Spectral methods for non-linear dynamics

Predicting the behaviour of an individual orbit of a non-linear dynamical system can be notoriously difficult, especially in the presence of chaotic effects. However, the statistics of many orbits (with initial conditions drawn from a given distribution) can often be predicted. In recent work [2,3], it was shown that the statistics of a chaotic recursion relation can be predicted by constructing an associated sequence of periodic elliptic operators. For such operators, the density of states is well understood, can be computed straightforwardly and explicit formulas can often be derived. The example studied was a non-linear recursion relation which can be related to a sequence of periodic operators generated by a Fibonacci tiling rule. The link was used to derive an explicit formula for the limiting distribution of orbits. This distribution contains characteristic features of the associated operators’ densities of states, such as Van Hove singularities near to critical values.

Models for cochlear amplification

The human cochlea uses an amplification mechanism in the form of a non-linear feedback loop. This mechanism amplifies quieter sounds much more strongly than louder sounds, allowing humans to encode sounds over a wide range of volumes with relative efficiency. Our work [1] showed how this non-linear amplification could be integrated into a PDE model for a cochlea-inspired graded metamaterial. The resulting active metamaterial took the form of a coupled array of Hopf resonators (critically-poised Hopf resonators being the preferred canonical models for cochlear amplification).