Mathematicians at the University of Liverpool and Imperial College have solved a problem that had remained open for more than four decades. This problem is known as "Density of Axiom A in the Arnol’d Family".
Phase-locking is a phenomenon wherein two oscillating systems interact with each other (such as two pendulum clocks hanging on the same wall) and eventually synchronize. This was first observed by Huygens in the 17th century, and has since been seen to occur in a diverse range of systems, including the interaction between rates of breathing and heartbeats in humans.
In the 1960s, the famous Russian mathematician Vladimir Arnol’d introduced a simple model to study phase-locking phenomena, now known as the Arnol’d Family. The model is meant to describes a periodic motion on a circle affected by an external force, and depends on two parameters: a rotation parameter, which determines the underlying motion, and a forcing parameter, which determines the size of the external influence. The family has been used as a simplified model for a large number of physical and biological systems, such as a beating heart.
The above picture shows a 2-dimensional representation of the Arnol’d family. Each point on the picture corresponds to a different choice of parameters, with the rotation parameter drawn horizontally and the forcing parameter vertically.
White regions indicate parameters where phase-locking phenomena lead to a system that is stable under perturbations – this type of behaviour is referred to as Axiom A (following terminology establish by Fields Medalist Stephen Smale in the 1960s). Experiments suggest that this behaviour is dense - wherever we look in the picture, sufficiently high magnification will reveal some white areas.
When the forcing parameter is small (more precisely: for parameters below the grey line in the picture), this was already verified and explained by Arnol’d. The general case remained open – but has now been solved by Lasse Rempe-Gillen (University of Liverpool) and Sebastian van Strien (Imperial College).
The work has passed peer review and is now accepted for publication in the Duke Mathematical Journal, one of the leading scholarly journals in mathematics. The research was partly funded by EPSRC and the Leverhulme Trust, and builds on fundamental results due to Lyubich; Graczyk and Świątek; Kozlovski, Shen and van Strien; and Rempe-Gillen.
Research in this field relies heavily on the use of complex numbers – an extension of the usual number line that fills an entire two-dimensional plane. Allowing variables and parameters to take complex numbers reveals a rich structure that would otherwise stay hidden. (See the desktop backgroun above, which shows such a complex extension.)
An interactive exhibit explaining the result and its context has been developed at the University of Liverpool and has been shown at the Festival of Mathematics and its Applications in Manchester on July 3-4, 2014. Promotional images and further information are available upon request.