View from the Pennines: Strange Nonchaotic Attractors

Living just under the moor, I have always known that drainage can cause problems. There are a number of obvious culverts near the house, and a well and sump in the front garden. But it was not until we excavated to create more space around the front door that we discovered the true extent of the drainage system, which must be Victorian if not older. There is a culvert running parallel to the house fed by several smaller drains which run directly down the slope of the hill. They are independent of the well, and lead to a larger culvert. The culverts are lined with dry stone walling, and have large ag stones on top, presumably to prevent earth falling into them. The extent of the system is a surprise, and shows how important drainage has been to the house and barn. Life is full of such surprises, which often provide the opportunity to re-evaluate the assumptions we make about things we take for granted. After a while, the surprise wears o and the lesson learned is assimilated into a new picture of the world. Over the past few years I have experienced this in my understanding of objects in dynamical systems called strange nonchaotic attractors (SNA), and which have properties often associated with chaotic dynamics despite their name. Examples of SNA are found in many di erence equations of the form