Detecting chaos in complex systems
Chaos theory describes the behaviours of complex systems, those with so many moving elements that it is impossible to begin to explain the dynamics without powerful computers. Despite their seeming unpredictability, the behaviours of these systems are not completely random and, in the everyday use of the word, chaotic. Rather, they are deterministic and non-linear. However, because they are complex, we can never know all the initial conditions of the system in sufficient detail to define the evolution of the states of the system. Because they also exhibit sensitive dependence, meaning that a very small change in initial conditions can cause drastic changes in output, lack of knowledge about the initial conditions can lead to, well, chaos. Within the scope of the project 'Topological aspects of dynamical independence and chaos' (TOPDSC), EU-funded scientists extended the mathematics used to detect whether a system is chaotic. They examined local and global properties and interactions among elements in certain mathematical (compact metric) spaces. The emphasis was on the global property of chaotic mixing, in particular in relation to fluid flow. Scientists developed characterisations of the local aspects of mixing expressed in terms of weakly mixing sub-spaces. Among the many results were a relation between weakly mixing sets and various types of chaos. The team also studied definitions of chaos in terms of local aspects such as pairs and n-tuples, ordered sequences of elements in which n is a non-negative integer. TOPDSC discovered many new phenomena and mathematical descriptions in chaos theory and proved the practicality of their results. With a focus on successful application of results, the project has extended understanding of topological aspects of chaos in systems with complicated dynamics.
Keywords
Chaos, complex systems, topological, dynamical independence, n-tuples
