Chapter 7 - Nonlinear Dynamics

Abstract

The study of the evolution with time of nonlinear systems is called nonlinear dynamics. Examples in metamorphic geology are the buckling and boudinage of layers embedded in nonlinear materials, the behaviour of coupled mineral reactions and the formation of joint systems. Nonlinear systems evolve through a number of stages: (1) A uniform state becomes unstable and fluctuations away from the uniform state begin to grow in the form of some kind of pattern either in space or time or both. This initial growth stage can be characterised using a linear stability analysis. (2) The initial growth of spatiotemporal patterns is commonly sinusoidal and exponential. (3) The exponential sinusoidal growth is interrupted in many systems by one or both of the onset of nonlinear saturation that slows the growth rate or of a bifurcation where the system switches to a different kind of behaviour. Typical bifurcation behaviour comprises switches from periodic to localised to chaotic behaviour. We explore two archetype systems described by the Swift-Hohenberg equation and reaction-diffusion equations. Some nonlinear systems are better described by considering the states that minimise an energy function such as the Helmholtz energy; for nonlinear systems such a function is non-convex and may consist of many local minima. Minimisation of a non-convex energy function leads to the formation of fractal and multifractal geometries. We explore such concepts with the use of wavelets. A related analysis of some systems introduces the concept called snakes and ladders where the system undergoes successive bifurcations as it evolves sequentially from one energy minimum into a neighbouring energy minimum.