Non-Linear Dynamics

author: Colm Connaughton, Mathematics Institute, University of Warwick
published: April 3, 2017,   recorded: March 2017,   views: 8


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For many non-specialists, notions of nonlinearity, chaos and complexity are often blurred together in scientific discourse. In reality, chaos is only one of a rich set of important phenomena observed in nonlinear dynamical systems. This lecture will provide a non-mathematical introduction to some of the key concepts of nonlinear dynamics and try to clarify how they relate to complexity science. We will begin by discussing why nonlinearity is important and generic and why nonlinear dynamical systems with a small number of variables are helpful in modelling complex systems with large numbers of variables. We will then explain the key ideas of phase space, trajectories and fixed points with some illustrative examples from ecological and epidemiological modelling. We will then introduce the notion of stability of fixed points. This is important because stable fixed points are the simplest examples of attractors: structures in phase space that determine the typical long-time behaviour of a dynamical system. More interesting examples of attractors include periodic cycles and the strange attractors associated with chaotic dynamics. The final part of the lecture will discuss bifurcations, one of the most important concepts in nonlinear science. A bifurcation is a change in the structure or stability of the attractor(s) of a dynamical system as an external parameter is varied. Bifurcations are important because they are associated with qualitative changes in the typical long-time behaviour of a dynamical system. We will try to illustrate how different types of bifurcations are connected to essentially nonlinear phenomena like tipping points, multi-stability, hysteresis and emergence of nonlinear oscillations.

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